s2sphere¶
Python implementation of a part of the C++ S2 geometry library. Install with:
pip install s2sphere
The S2 geometry library is explained in more detail in this presentation by Octavian Procopiuc. It maps a sphere to a 1D index. This enables scalable proximity searches using distributed indices such as with MongoDB and App Engine Datastore. The test cases of this package are extensive and reflect those in the original S2 geometry library.
Links: Documentation, GitHub, Issue Tracker
Contents¶
Examples¶
One of the standard applications is to get a set of S2 cells at various levels covering a rectangle in \((lat, lng)\) coordinates:
import s2sphere
r = s2sphere.RegionCoverer()
p1 = s2sphere.LatLng.from_degrees(33, 122)
p2 = s2sphere.LatLng.from_degrees(33.1, 122.1)
cell_ids = r.get_covering(s2sphere.LatLngRect.from_point_pair(p1, p2))
print(cell_ids)
which prints this list:
[9291041754864156672, 9291043953887412224, 9291044503643226112, 9291045878032760832, 9291047252422295552, 9291047802178109440, 9291051650468806656, 9291052200224620544]
API¶

class
s2sphere.
Angle
(radians=0)[source]¶ A onedimensional angle (as opposed to a twodimensional solid angle).
It has methods for converting angles to or from radians and degrees.
Parameters: radians (float) – angle in radians see
S1Angle

radians
¶

degrees
¶


class
s2sphere.
AreaMetric
(deriv)[source]¶ Area metric. A 2D specialization of
s2sphere.Metric
.Preconfigured instances of this class are
s2sphere.AVG_AREA
,s2sphere.MIN_AREA
,s2sphere.MAX_AREA
.see
S2::AreaMetric

class
s2sphere.
Cap
(axis=Point: (1, 0, 0), height=1)[source]¶ A spherical cap, which is a portion of a sphere cut off by a plane.
see
S2Cap

ROUND_UP
= 1.0000000000000002¶


class
s2sphere.
Cell
(cell_id=None)[source]¶ Cell
see
S2Cell

get_edge
(k)[source]¶ the kth edge
Return the inwardfacing normal of the great circle passing through the edge from vertex k to vertex k+1 (mod 4). The normals returned by GetEdgeRaw are not necessarily unit length.

get_vertex
(k)[source]¶ Return the kth vertex of the cell (k = 0,1,2,3).
Vertices are returned in CCW order. The points returned by GetVertexRaw are not necessarily unit length.

exact_area
()[source]¶ cell area in steradians accurate to 6 digits but slow to compute
Return the area of this cell as accurately as possible. This method is more expensive but it is accurate to 6 digits of precision even for leaf cells (whose area is approximately 1e18).


class
s2sphere.
CellId
(id_=0)[source]¶ S2 cell id
The 64bit ID has:
 3 bits to encode the face
 060 bits to encode the position
 a 1
The final 1 is the least significant bit (lsb) in the underlying integer representation and is returned with
s2sphere.CellId.lsb()
.see
S2CellId

LINEAR_PROJECTION
= 0¶

TAN_PROJECTION
= 1¶

QUADRATIC_PROJECTION
= 2¶

PROJECTION
= 2¶

FACE_BITS
= 3¶

NUM_FACES
= 6¶

MAX_LEVEL
= 30¶

POS_BITS
= 61¶

MAX_SIZE
= 1073741824¶

WRAP_OFFSET
= 13835058055282163712L¶

classmethod
walk
(level)[source]¶ Walk along a Hilbert curve at the given level.
This function does not exist in the SWIG bindings of the original C++ library. It provides a more Pythonic way to iterate over cells.
Returns: Iterator over instances of CellId
s.

classmethod
walk_fast
(level)[source]¶ Walk along a Hilbert curve at the given level.
This function does not exist in the SWIG bindings of the original C++ library. It provides a more Pythonic way to iterate over cells.
Use with caution: this repeatedly mutates a single instance with a changing
id
. If you save the object, it will change out from underneath you.Returns: Iterator over ids in the same instance of CellId
.

get_vertex_neighbors
(level)[source]¶ Return the neighbors of closest vertex to this cell.
Normally there are four neighbors, but the closest vertex may only have three neighbors if it is one of the 8 cube vertices.

to_token
()[source]¶ A unique string token for this cell id.
This is a hex encoded version of the cell id with the right zeros stripped of.

class
s2sphere.
CellUnion
(cell_ids=None, raw=True)[source]¶ Cell Union
see
S2CellUnion

class
s2sphere.
LatLng
(lat, lng)[source]¶ A point on a sphere in latitutelongitude coordinates.
see
S2LatLng

class
s2sphere.
LatLngRect
(*args)[source]¶ A rectangle in latitudelongitude space.
see
S2LatLngRect

class
s2sphere.
LengthMetric
(deriv)[source]¶ Length metric. A 1D specialization of
s2sphere.Metric
.Preconfigured instances of this class are
s2sphere.AVG_ANGLE_SPAN
,s2sphere.MIN_ANGLE_SPAN
,s2sphere.MAX_ANGLE_SPAN
,s2sphere.AVG_EDGE
,s2sphere.MIN_EDGE
,s2sphere.MAX_EDGE
,s2sphere.AVG_DIAG
,s2sphere.MIN_DIAG
,s2sphere.MAX_DIAG
,s2sphere.AVG_WIDTH
,s2sphere.MIN_WIDTH
ands2sphere.MAX_WIDTH
.see
S2::LengthMetric

class
s2sphere.
LineInterval
(lo=1, hi=0)[source]¶ Line Interval in R1
see
R1Interval

class
s2sphere.
Metric
(deriv, dim)[source]¶ Metric
The classes
s2sphere.LengthMetric
ands2sphere.AreaMetric
are specializations of this class.see
S2::Metric

get_value
(level)[source]¶ The value of this metric at a given level.
Returns: Depending on whether this is used in one or two dimensions, this is an angle in radians or a solid angle in steradians.

get_closest_level
(value)[source]¶ Closest cell level according to the given value.
Return the level at which the metric has approximately the given value. For example,
s2sphere.AVG_EDGE.get_closest_level(0.1)
returns the level at which the average cell edge length is approximately 0.1. The return value is always a valid level.Parameters: value – Depending on whether this is used in one or two dimensions, this is an angle in radians or a solid angle in steradians.

get_min_level
(value)[source]¶ Minimum cell level for given value.
Return the minimum level such that the metric is at most the given value, or
s2sphere.CellId.MAX_LEVEL
if there is no such level. For example,s2sphere.MAX_DIAG.get_min_level(0.1)
returns the minimum level such that all cell diagonal lengths are 0.1 or smaller. The return value is always a valid level.Parameters: value – Depending on whether this is used in one or two dimensions, this is an angle in radians or a solid angle in steradians.

get_max_level
(value)[source]¶ Maximum cell level for given value.
Return the maximum level such that the metric is at least the given value, or zero if there is no such level. For example,
s2sphere.MIN_WIDTH.get_max_level(0.1)
returns the maximum level such that all cells have a minimum width of 0.1 or larger. The return value is always a valid level.Parameters: value – Depending on whether this is used in one or two dimensions, this is an angle in radians or a solid angle in steradians.


class
s2sphere.
Point
(x, y, z)[source]¶ A point in 3d Euclidean space.
“Normalized” points are points on the unit sphere.
see
S2Point

class
s2sphere.
RegionCoverer
[source]¶ Region Coverer
see
S2RegionCoverer

RegionCoverer.
min_level
¶

RegionCoverer.
max_level
¶

RegionCoverer.
level_mod
¶

RegionCoverer.
max_cells
¶


class
s2sphere.
SphereInterval
(lo=3.141592653589793, hi=3.141592653589793, args_checked=False)[source]¶ Interval in S1
see
S1Interval

complement
()[source]¶ Return the complement of the interior of the interval.
An interval and its complement have the same boundary but do not share any interior values. The complement operator is not a bijection, since the complement of a singleton interval (containing a single value) is the same as the complement of an empty interval.

Utilities¶

s2sphere.
area
(a, b, c)[source]¶ Area of the triangle (a, b, c).
see
S2::Area()

s2sphere.
get_u_norm
(face, u)[source]¶ Vector normal to the positive vaxis and the plane through the origin.
The vector is normal to the positive vaxis and a plane that contains the origin and the vaxis.
The righthanded normal (not necessarily unit length) for an edge in the direction of the positive vaxis at the given uvalue on the given face. (This vector is perpendicular to the plane through the sphere origin that contains the given edge.)
Return type: Point see
S2::GetUNorm()

s2sphere.
get_v_norm
(face, v)[source]¶ Vector normal to the positive uaxis and the plane through the origin.
The vector is normal to the positive uaxis and a plane that contains the origin and the uaxis.
Return the righthanded normal (not necessarily unit length) for an edge in the direction of the positive uaxis at the given vvalue on the given face.
see
S2::GetVNorm()

s2sphere.
girard_area
(a, b, c)[source]¶ see
S2::GirardArea()

s2sphere.
origin
()[source]¶ A unique and empirically chosen reference point.
see
S2::Origin()

s2sphere.
ortho
(a)[source]¶ see
S2::Ortho()

s2sphere.
robust_cross_prod
(a, b)[source]¶ A numerically more robust cross product.
The direction of \(a \times b\) becomes unstable as \((a + b)\) or \((a  b)\) approaches zero. This leads to situations where \(a \times b\) is not very orthogonal to \(a\) and/or \(b\). We could fix this using GramSchmidt, but we also want \(b \times a =  a \times b\).
The easiest fix is to just compute the cross product of \((b+a)\) and \((ba)\). Mathematically, this cross product is exactly twice the cross product of \(a\) and \(b\), but it has the numerical advantage that \((b+a)\) and \((ba)\) are always perpendicular (since \(a\) and \(b\) are unit length). This yields a result that is nearly orthogonal to both \(a\) and \(b\) even if these two values differ only in the lowest bit of one component.

s2sphere.
simple_ccw
(a, b, c)[source]¶ Simple Counterclockwise test.
Return true if the points A, B, C are strictly counterclockwise. Return false if the points are clockwise or collinear (i.e. if they are all contained on some great circle).
Due to numerical errors, situations may arise that are mathematically impossible, e.g. ABC may be considered strictly CCW while BCA is not. However, the implementation guarantees the following:
If simple_ccw(a,b,c), then !simple_ccw(c,b,a) for all a,b,c.see
S2::SimpleCCW()
Developing¶
To develop, clone the repository and include git submodules recursively:
git clone recursive https://github.com/sidewalklabs/s2sphere
Tests require pip install numpy
and a build of the original C++ library:
# build and install the C++ library
cd tests/s2geometry/geometry
cmake .
make j4
make install
# build the C++ library's Python bindings
cd python
cmake . # see comment below for OSX
make j4
make install
# verify Python bindings
python v c 'import s2'
OSX requires extra setup:
# point to Brew's OpenSSL installation
export OPENSSL_ROOT_DIR=$(brew prefix openssl)
# tell the Python cmake which libraries to use
cmake DPYTHON_LIBRARY=$(pythonconfig prefix)/lib/libpython2.7.dylib DPYTHON_INCLUDE_DIR=$(pythonconfig prefix)/include/python2.7 .
Then install this module with the dependencies needed for running tests and generating docs:
# install
pip install e .[tests,docs]
Documentation¶
cd docs/sphinx
make html
Tests¶
# run tests that don't require the C library
flake8
nosetests vv
# tests that compare C and Python implementations
python tests/compare_implementations.py vv
C++ API¶
This is the API documentation of the original S2 geometry library extracted from its source code and included here for reference.

class
R1Interval
¶ An R1Interval represents a closed, bounded interval on the real line. It is capable of representing the empty interval (containing no points) and zerolength intervals (containing a single point).
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator.

static inline R1Interval
Empty
()¶ Returns an empty interval.

static R1Interval
FromPoint
(double p)¶ Convenience method to construct an interval containing a single point.

static R1Interval
FromPointPair
(double p1, double p2)¶ Convenience method to construct the minimal interval containing the two given points. This is equivalent to starting with an empty interval and calling AddPoint() twice, but it is more efficient.

double
lo
() const¶

double
hi
() const¶

double
bound
(int i) const¶

Vector2_d const &
bounds
() const¶

void
set_lo
(double p)¶ Methods to modify one endpoint of an existing R1Interval. Do not use these methods if you want to replace both endpoints of the interval; use a constructor instead. For example:
lat_bounds = R1Interval(lat_lo, lat_hi);

void
set_hi
(double p)¶

bool
is_empty
() const¶ Return true if the interval is empty, i.e. it contains no points.

double
GetCenter
() const¶ Return the center of the interval. For empty intervals, the result is arbitrary.

double
GetLength
() const¶ Return the length of the interval. The length of an empty interval is negative.

bool
Contains
(double p) const¶

bool
InteriorContains
(double p) const¶

bool
Contains
(R1Interval const &y) const¶ Return true if this interval contains the interval ‘y’.

bool
InteriorContains
(R1Interval const &y) const¶ Return true if the interior of this interval contains the entire interval ‘y’ (including its boundary).

bool
Intersects
(R1Interval const &y) const¶ Return true if this interval intersects the given interval, i.e. if they have any points in common.

bool
InteriorIntersects
(R1Interval const &y) const¶ Return true if the interior of this interval intersects any point of the given interval (including its boundary).

double
GetDirectedHausdorffDistance
(R1Interval const &y) const¶ Return the Hausdorff distance to the given interval ‘y’. For two R1Intervals x and y, this distance is defined as
h(x, y) = max_{p in x} min_{q in y} d(p, q).

void
AddPoint
(double p)¶ Expand the interval so that it contains the given point “p”.

R1Interval
Expanded
(double radius) const¶ Return an interval that contains all points with a distance “radius” of a point in this interval. Note that the expansion of an empty interval is always empty.

R1Interval
Union
(R1Interval const &y) const¶ Return the smallest interval that contains this interval and the given interval “y”.

R1Interval
Intersection
(R1Interval const &y) const¶ Return the intersection of this interval with the given interval. Empty intervals do not need to be specialcased.

bool
ApproxEquals
(R1Interval const &y, double max_error = 1e15) const¶ Return true if length of the symmetric difference between the two intervals is at most the given tolerance.

static inline R1Interval

class
S1Angle
¶ This class represents a onedimensional angle (as opposed to a twodimensional solid angle). It has methods for converting angles to or from radians, degrees, and the E5/E6/E7 representations (i.e. degrees multiplied by 1e5/1e6/1e7 and rounded to the nearest integer).
This class has builtin support for the E5, E6, and E7 representations. An E5 is the measure of an angle in degrees, multiplied by 10**5.
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator.

static inline S1Angle
Radians
(double radians)¶ These methods construct S1Angle objects from their measure in radians or degrees.

static inline S1Angle
UnsignedE6
(uint32 e6)¶ Convenience functions – to use when args have been fixed32s in protos.
The arguments are static_cast into int32, so very large unsigned values are treated as negative numbers.

double
radians
() const¶

double
degrees
() const¶

void
Normalize
()¶ Normalize this angle to the range (180, 180] degrees.

static inline S1Angle

class
S1Interval
¶ An S1Interval represents a closed interval on a unit circle (also known as a 1dimensional sphere). It is capable of representing the empty interval (containing no points), the full interval (containing all points), and zerolength intervals (containing a single point).
Points are represented by the angle they make with the positive xaxis in the range [Pi, Pi]. An interval is represented by its lower and upper bounds (both inclusive, since the interval is closed). The lower bound may be greater than the upper bound, in which case the interval is “inverted” (i.e. it passes through the point (1, 0)).
Note that the point (1, 0) has two valid representations, Pi and Pi. The normalized representation of this point internally is Pi, so that endpoints of normal intervals are in the range (Pi, Pi]. However, we take advantage of the point Pi to construct two special intervals: the Full() interval is [Pi, Pi], and the Empty() interval is [Pi, Pi].
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator.

static inline S1Interval
Empty
()¶ Returns the empty interval.

static inline S1Interval
Full
()¶ Returns the full interval.

static S1Interval
FromPoint
(double p)¶ Convenience method to construct an interval containing a single point.

static S1Interval
FromPointPair
(double p1, double p2)¶ Convenience method to construct the minimal interval containing the two given points. This is equivalent to starting with an empty interval and calling AddPoint() twice, but it is more efficient.

double
lo
() const¶

double
hi
() const¶

double
bound
(int i) const¶

Vector2_d const &
bounds
() const¶

void
set_lo
(double p)¶ Methods to modify one endpoint of an existing S1Interval. Requires that the resulting S1Interval is valid. This implies you cannot call this method on an Empty() or Full() interval, since these intervals do not have any endpoints.
Do not use these methods if you want to replace both endpoints of the interval; use a constructor instead. For example:
lng_bounds = S1Interval(lng_lo, lng_hi);

void
set_hi
(double p)¶

inline bool
is_valid
() const¶ An interval is valid if neither bound exceeds Pi in absolute value, and the value Pi appears only in the Empty() and Full() intervals.

bool
is_full
() const¶ Return true if the interval contains all points on the unit circle.

bool
is_empty
() const¶ Return true if the interval is empty, i.e. it contains no points.

bool
is_inverted
() const¶ Return true if lo() > hi(). (This is true for empty intervals.)

double
GetCenter
() const¶ Return the midpoint of the interval. For full and empty intervals, the result is arbitrary.

double
GetLength
() const¶ Return the length of the interval. The length of an empty interval is negative.

S1Interval
Complement
() const¶ Return the complement of the interior of the interval. An interval and its complement have the same boundary but do not share any interior values. The complement operator is not a bijection, since the complement of a singleton interval (containing a single value) is the same as the complement of an empty interval.

double
GetComplementCenter
() const¶ Return the midpoint of the complement of the interval. For full and empty intervals, the result is arbitrary. For a singleton interval (containing a single point), the result is its antipodal point on S1.

bool
Contains
(double p) const¶ Return true if the interval (which is closed) contains the point ‘p’.

bool
InteriorContains
(double p) const¶ Return true if the interior of the interval contains the point ‘p’.

bool
Contains
(S1Interval const &y) const¶ Return true if the interval contains the given interval ‘y’. Works for empty, full, and singleton intervals.

bool
InteriorContains
(S1Interval const &y) const¶ Returns true if the interior of this interval contains the entire interval ‘y’. Note that x.InteriorContains(x) is true only when x is the empty or full interval, and x.InteriorContains(S1Interval(p,p)) is equivalent to x.InteriorContains(p).

bool
Intersects
(S1Interval const &y) const¶ Return true if the two intervals contain any points in common. Note that the point +/Pi has two representations, so the intervals [Pi,3] and [2,Pi] intersect, for example.

bool
InteriorIntersects
(S1Interval const &y) const¶ Return true if the interior of this interval contains any point of the interval ‘y’ (including its boundary). Works for empty, full, and singleton intervals.

double
GetDirectedHausdorffDistance
(S1Interval const &y) const¶ Return the Hausdorff distance to the given interval ‘y’. For two S1Intervals x and y, this distance is defined by
h(x, y) = max_{p in x} min_{q in y} d(p, q),
where d(.,.) is measured along S1.

void
AddPoint
(double p)¶ Expand the interval by the minimum amount necessary so that it contains the given point “p” (an angle in the range [Pi, Pi]).

S1Interval
Expanded
(double radius) const¶ Return an interval that contains all points with a distance “radius” of a point in this interval. Note that the expansion of an empty interval is always empty. The radius must be nonnegative.

S1Interval
Union
(S1Interval const &y) const¶ Return the smallest interval that contains this interval and the given interval “y”.

S1Interval
Intersection
(S1Interval const &y) const¶ Return the smallest interval that contains the intersection of this interval with “y”. Note that the region of intersection may consist of two disjoint intervals.

bool
ApproxEquals
(S1Interval const &y, double max_error = 1e15) const¶ Return true if the length of the symmetric difference between the two intervals is at most the given tolerance.

static inline S1Interval

typedef Vector3_d
S2Point
¶ An S2Point represents a point on the unit sphere as a 3D vector. Usually points are normalized to be unit length, but some methods do not require this. See util/math/vector3inl.h for the methods available. Among other things, there are overloaded operators that make it convenient to write arithmetic expressions (e.g. (1x)*p1 + x*p2).

class
S2
¶ The S2 class is simply a namespace for constants and static utility functions related to spherical geometry, such as area calculations and edge intersection tests. The name “S2” is derived from the mathematical symbol for the twodimensional unit sphere (note that the “2” refers to the dimension of the surface, not the space it is embedded in).
This class also defines a framework for decomposing the unit sphere into a hierarchy of “cells”. Each cell is a quadrilateral bounded by four geodesics. The top level of the hierarchy is obtained by projecting the six faces of a cube onto the unit sphere, and lower levels are obtained by subdividing each cell into four children recursively.
This class specifies the details of how the cube faces are projected onto the unit sphere. This includes getting the face ordering and orientation correct so that sequentially increasing cell ids follow a continuous spacefilling curve over the entire sphere, and defining the transformation from cellspace to cubespace in order to make the cells more uniform in size.
This file also contains documentation of the various coordinate systems and conventions used.
This class is not threadsafe for loops and objects that use loops.

static inline S2Point
Origin
()¶ Return a unique “origin” on the sphere for operations that need a fixed reference point. In particular, this is the “point at infinity” used for pointinpolygon testing (by counting the number of edge crossings).
It shouldnot* be a point that is commonly used in edge tests in order to avoid triggering code to handle degenerate cases. (This rules out the north and south poles.) It should also not be on the boundary of any lowlevel S2Cell for the same reason.

static bool
IsUnitLength
(S2Point const &p)¶ Return true if the given point is approximately unit length (this is mainly useful for assertions).

static S2Point
Ortho
(S2Point const &a)¶ Return a unitlength vector that is orthogonal to “a”. Satisfies Ortho(a) = Ortho(a) for all a.

static void
GetFrame
(S2Point const &z, Matrix3x3_d *m)¶ Given a point “z” on the unit sphere, extend this into a righthanded coordinate frame of unitlength column vectors m = (x,y,z). Note that the vectors (x,y) are an orthonormal frame for the tangent space at “z”, while “z” itself is an orthonormal frame for the normal space at “z”.

static S2Point
ToFrame
(Matrix3x3_d const &m, S2Point const &p)¶ Given an orthonormal basis “m” of column vectors and a point “p”, return the coordinates of “p” with respect to the basis “m”. The resulting point “q” satisfies the identity (mq == p).

static S2Point
FromFrame
(Matrix3x3_d const &m, S2Point const &q)¶ Given an orthonormal basis “m” of column vectors and a point “q” with respect to that basis, return the equivalent point “p” with respect to the standard axisaligned basis. The result satisfies (p == mq).

static bool
ApproxEquals
(S2Point const &a, S2Point const &b, double max_error = 1e15)¶ the coordinates of “p” with respect to the basis “m”. The resulting point “r” satisfies the identity (mr == p). Return true if two points are within the given distance of each other (this is mainly useful for testing).

static S2Point
RobustCrossProd
(S2Point const &a, S2Point const &b)¶ Return a vector “c” that is orthogonal to the given unitlength vectors “a” and “b”. This function is similar to a.CrossProd(b) except that it does a better job of ensuring orthogonality when “a” is nearly parallel to “b”, and it returns a nonzero result even when a == b or a == b.
It satisfies the following properties (RCP == RobustCrossProd):
(1) RCP(a,b) != 0 for all a, b (2) RCP(b,a) == RCP(a,b) unless a == b or a == b (3) RCP(a,b) == RCP(a,b) unless a == b or a == b (4) RCP(a,b) == RCP(a,b) unless a == b or a == b

static bool
SimpleCCW
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Return true if the points A, B, C are strictly counterclockwise. Return false if the points are clockwise or collinear (i.e. if they are all contained on some great circle).
Due to numerical errors, situations may arise that are mathematically impossible, e.g. ABC may be considered strictly CCW while BCA is not. However, the implementation guarantees the following:
If SimpleCCW(a,b,c), then !SimpleCCW(c,b,a) for all a,b,c.

static int
RobustCCW
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Returns +1 if the points A, B, C are counterclockwise, 1 if the points are clockwise, and 0 if any two points are the same. This function is essentially like taking the sign of the determinant of ABC, except that it has additional logic to make sure that the above properties hold even when the three points are coplanar, and to deal with the limitations of floatingpoint arithmetic.
RobustCCW satisfies the following conditions:
 RobustCCW(a,b,c) == 0 if and only if a == b, b == c, or c == a
 RobustCCW(b,c,a) == RobustCCW(a,b,c) for all a,b,c
 RobustCCW(c,b,a) == RobustCCW(a,b,c) for all a,b,c
In other words:
 The result is zero if and only if two points are the same.
 Rotating the order of the arguments does not affect the result.
 Exchanging any two arguments inverts the result.
On the other hand, note that it is not true in general that RobustCCW(a,b,c) == RobustCCW(a,b,c), or any similar identities involving antipodal points.

static inline int
RobustCCW
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &a_cross_b)¶ A more efficient version of RobustCCW that allows the precomputed crossproduct of A and B to be specified. (Unlike the 3 argument version this method is also inlined.)

static inline int
TriageCCW
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &a_cross_b)¶ This version of RobustCCW returns +1 if the points are definitely CCW,
1 if they are definitely CW, and 0 if two points are identical or the
result is uncertain. Uncertain certain cases can be resolved, if desired, by calling ExpensiveCCW.
The purpose of this method is to allow additional cheap tests to be done, where possible, in order to avoid calling ExpensiveCCW unnecessarily.

static int
ExpensiveCCW
(S2Point const &a, S2Point const &b, S2Point const &c)¶ This function is invoked by RobustCCW() if the sign of the determinant is uncertain. It always returns a nonzero result unless two of the input points are the same. It uses a combination of multipleprecision arithmetic and symbolic perturbations to ensure that its results are always selfconsistent (cf. Simulation of Simplicity, Edelsbrunner and Muecke). The basic idea is to assign an infinitesmal symbolic perturbation to every possible S2Point such that no three S2Points are collinear and no four S2Points are coplanar. These perturbations are so small that they do not affect the sign of any determinant that was nonzero before the perturbations.
Unlike RobustCCW(), this method does not require the input points to be normalized.

static bool
OrderedCCW
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &o)¶ Given 4 points on the unit sphere, return true if the edges OA, OB, and OC are encountered in that order while sweeping CCW around the point O. You can think of this as testing whether A <= B <= C with respect to the CCW ordering around O that starts at A, or equivalently, whether B is contained in the range of angles (inclusive) that starts at A and extends CCW to C. Properties:
 If OrderedCCW(a,b,c,o) && OrderedCCW(b,a,c,o), then a == b
 If OrderedCCW(a,b,c,o) && OrderedCCW(a,c,b,o), then b == c
 If OrderedCCW(a,b,c,o) && OrderedCCW(c,b,a,o), then a == b == c
 If a == b or b == c, then OrderedCCW(a,b,c,o) is true
 Otherwise if a == c, then OrderedCCW(a,b,c,o) is false

static double
Angle
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Return the interior angle at the vertex B in the triangle ABC. The return value is always in the range [0, Pi]. The points do not need to be normalized. Ensures that Angle(a,b,c) == Angle(c,b,a) for all a,b,c.
The angle is undefined if A or C is diametrically opposite from B, and becomes numerically unstable as the length of edge AB or BC approaches 180 degrees.

static double
TurnAngle
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Return the exterior angle at the vertex B in the triangle ABC. The return value is positive if ABC is counterclockwise and negative otherwise. If you imagine an ant walking from A to B to C, this is the angle that the ant turns at vertex B (positive = left, negative = right). Ensures that TurnAngle(a,b,c) == TurnAngle(c,b,a) for all a,b,c.

static double
Area
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Return the area of triangle ABC. The method used is about twice as expensive as Girard’s formula, but it is numerically stable for both large and very small triangles. All points should be unit length. The area is always positive.
The triangle area is undefined if it contains two antipodal points, and becomes numerically unstable as the length of any edge approaches 180 degrees.

static double
GirardArea
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Return the area of the triangle computed using Girard’s formula. All points should be unit length. This is slightly faster than the Area() method above but is not accurate for very small triangles.

static double
SignedArea
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Like Area(), but returns a positive value for counterclockwise triangles and a negative value otherwise.

static S2Point
PlanarCentroid
(S2Point const &a, S2Point const &b, S2Point const &c)¶ About centroids:
There are several notions of the “centroid” of a triangle. First, there is the planar centroid, which is simply the centroid of the ordinary (nonspherical) triangle defined by the three vertices. Second, there is the surface centroid, which is defined as the intersection of the three medians of the spherical triangle. It is possible to show that this point is simply the planar centroid projected to the surface of the sphere. Finally, there is the true centroid (mass centroid), which is defined as the area integral over the spherical triangle of (x,y,z) divided by the triangle area. This is the point that the triangle would rotate around if it was spinning in empty space.
The best centroid for most purposes is the true centroid. Unlike the planar and surface centroids, the true centroid behaves linearly as regions are added or subtracted. That is, if you split a triangle into pieces and compute the average of their centroids (weighted by triangle area), the result equals the centroid of the original triangle. This is not true of the other centroids.
Also note that the surface centroid may be nowhere near the intuitive “center” of a spherical triangle. For example, consider the triangle with vertices A=(1,eps,0), B=(0,0,1), C=(1,eps,0) (a quartersphere). The surface centroid of this triangle is at S=(0, 2*eps, 1), which is within a distance of 2*eps of the vertex B. Note that the median from A (the segment connecting A to the midpoint of BC) passes through S, since this is the shortest path connecting the two endpoints. On the other hand, the true centroid is at M=(0, 0.5, 0.5), which when projected onto the surface is a much more reasonable interpretation of the “center” of this triangle. Return the centroid of the planar triangle ABC. This can be normalized to unit length to obtain the “surface centroid” of the corresponding spherical triangle, i.e. the intersection of the three medians. However, note that for large spherical triangles the surface centroid may be nowhere near the intuitive “center” (see example above).

static S2Point
TrueCentroid
(S2Point const &a, S2Point const &b, S2Point const &c)¶ Returns the true centroid of the spherical triangle ABC multiplied by the signed area of spherical triangle ABC. The reasons for multiplying by the signed area are (1) this is the quantity that needs to be summed to compute the centroid of a union or difference of triangles, and (2) it’s actually easier to calculate this way.

static inline double
STtoUV
(double s)¶ Convert an s or t value to the corresponding u or v value. This is a nonlinear transformation from [1,1] to [1,1] that attempts to make the cell sizes more uniform.

static inline double
UVtoST
(double u)¶ The inverse of the STtoUV transformation. Note that it is not always true that UVtoST(STtoUV(x)) == x due to numerical errors.

static inline S2Point
FaceUVtoXYZ
(int face, double u, double v)¶ Convert (face, u, v) coordinates to a direction vector (not necessarily unit length).

static inline bool
FaceXYZtoUV
(int face, S2Point const &p, double *pu, double *pv)¶ If the dot product of p with the given face normal is positive, set the corresponding u and v values (which may lie outside the range [1,1]) and return true. Otherwise return false.

static inline int
XYZtoFaceUV
(S2Point const &p, double *pu, double *pv)¶ Convert a direction vector (not necessarily unit length) to (face, u, v) coordinates.

static inline S2Point
GetUNorm
(int face, double u)¶ Return the righthanded normal (not necessarily unit length) for an edge in the direction of the positive vaxis at the given uvalue on the given face. (This vector is perpendicular to the plane through the sphere origin that contains the given edge.)

static inline S2Point
GetVNorm
(int face, double v)¶ Return the righthanded normal (not necessarily unit length) for an edge in the direction of the positive uaxis at the given vvalue on the given face.

static inline S2Point
GetNorm
(int face)¶ Return the unitlength normal, uaxis, or vaxis for the given face.

template<int
dim
>
classMetric
¶ S2Cell Metrics
The following are various constants that describe the shapes and sizes of cells. They are useful for deciding which cell level to use in order to satisfy a given condition (e.g. that cell vertices must be no further than “x” apart). All of the raw constants are differential quantities; you can use the GetValue(level) method to compute the corresponding length or area on the unit sphere for cells at a given level. The minimum and maximum bounds are valid for cells at all levels, but they may be somewhat conservative for very large cells (e.g. face cells). Defines a cell metric of the given dimension (1 == length, 2 == area).

double
deriv
() const¶ The “deriv” value of a metric is a derivative, and must be multiplied by a length or area in (s,t)space to get a useful value.

double
GetValue
(int level) const¶ Return the value of a metric for cells at the given level. The value is either a length or an area on the unit sphere, depending on the particular metric.

int
GetClosestLevel
(double value) const¶ Return the level at which the metric has approximately the given value. For example, S2::kAvgEdge.GetClosestLevel(0.1) returns the level at which the average cell edge length is approximately 0.1. The return value is always a valid level.

int
GetMinLevel
(double value) const¶ Return the minimum level such that the metric is at most the given value, or S2CellId::kMaxLevel if there is no such level. For example, S2::kMaxDiag.GetMinLevel(0.1) returns the minimum level such that all cell diagonal lengths are 0.1 or smaller. The return value is always a valid level.

int
GetMaxLevel
(double value) const¶ Return the maximum level such that the metric is at least the given value, or zero if there is no such level. For example, S2::kMinWidth.GetMaxLevel(0.1) returns the maximum level such that all cells have a minimum width of 0.1 or larger. The return value is always a valid level.

double

static inline S2Point

class
S2Cap
: public S2Region¶ This class represents a spherical cap, i.e. a portion of a sphere cut off by a plane. The cap is defined by its axis and height. This representation has good numerical accuracy for very small caps (unlike the (axis, mindistancefromorigin) representation), and is also efficient for containment tests (unlike the (axis, angle) representation).
Here are some useful relationships between the cap height (h), the cap opening angle (theta), the maximum chord length from the cap’s center (d), and the radius of cap’s base (a). All formulas assume a unit radius.
h = 1  cos(theta) = 2 sin^2(theta/2) d^2 = 2 h = a^2 + h^2
Caps may be constructed from either an axis and a height, or an axis and an angle. To avoid ambiguity, there are no public constructors except the default constructor.
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator, however it is not a “plain old datatype” (POD) because it has virtual functions.

static inline S2Cap
FromAxisHeight
(S2Point const &axis, double height)¶ Create a cap given its axis and the cap height, i.e. the maximum projected distance along the cap axis from the cap center. ‘axis’ should be a unitlength vector.

static S2Cap
FromAxisAngle
(S2Point const &axis, S1Angle const &angle)¶ Create a cap given its axis and the cap opening angle, i.e. maximum angle between the axis and a point on the cap. ‘axis’ should be a unitlength vector, and ‘angle’ should be nonnegative. If ‘angle’ is 180 degrees or larger, the cap will contain the entire unit sphere.

static inline S2Cap
FromAxisArea
(S2Point const &axis, double area)¶ Create a cap given its axis and its area in steradians. ‘axis’ should be a unitlength vector, and ‘area’ should be between 0 and 4M_PI.

double
height
() const¶

double
area
() const¶

S1Angle
angle
() const¶ Return the cap opening angle in radians, or a negative number for empty caps.

bool
is_valid
() const¶ We allow negative heights (to represent empty caps) but not heights greater than 2.

bool
is_empty
() const¶ Return true if the cap is empty, i.e. it contains no points.

bool
is_full
() const¶ Return true if the cap is full, i.e. it contains all points.

S2Cap
Complement
() const¶ Return the complement of the interior of the cap. A cap and its complement have the same boundary but do not share any interior points. The complement operator is not a bijection, since the complement of a singleton cap (containing a single point) is the same as the complement of an empty cap.

bool
Contains
(S2Cap const &other) const¶ Return true if and only if this cap contains the given other cap (in a set containment sense, e.g. every cap contains the empty cap).

bool
Intersects
(S2Cap const &other) const¶ Return true if and only if this cap intersects the given other cap, i.e. whether they have any points in common.

bool
InteriorIntersects
(S2Cap const &other) const¶ Return true if and only if the interior of this cap intersects the given other cap. (This relationship is not symmetric, since only the interior of this cap is used.)

bool
InteriorContains
(S2Point const &p) const¶ Return true if and only if the given point is contained in the interior of the region (i.e. the region excluding its boundary). ‘p’ should be be a unitlength vector.

void
AddPoint
(S2Point const &p)¶ Increase the cap height if necessary to include the given point. If the cap is empty the axis is set to the given point, but otherwise it is left unchanged. ‘p’ should be a unitlength vector.

void
AddCap
(S2Cap const &other)¶ Increase the cap height if necessary to include “other”. If the current cap is empty it is set to the given other cap.

S2Cap
Expanded
(S1Angle const &distance) const¶ Return a cap that contains all points within a given distance of this cap. Note that any expansion of the empty cap is still empty.

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

static inline S2Cap

class
S2Cell
: public S2Region¶ An S2Cell is an S2Region object that represents a cell. Unlike S2CellIds, it supports efficient containment and intersection tests. However, it is also a more expensive representation (currently 48 bytes rather than 8). This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator, however it is not a “plain old datatype” (POD) because it has virtual functions.

inline int
face
() const¶

inline int
level
() const¶

inline int
orientation
() const¶

inline bool
is_leaf
() const¶

int
GetSizeIJ
() const¶ These are equivalent to the S2CellId methods, but have a more efficient implementation since the level has been precomputed.

double
GetSizeST
() const¶

S2Point
GetVertex
(int k) const¶ Return the kth vertex of the cell (k = 0,1,2,3). Vertices are returned in CCW order. The points returned by GetVertexRaw are not necessarily unit length.

S2Point
GetEdge
(int k) const¶ Return the inwardfacing normal of the great circle passing through the edge from vertex k to vertex k+1 (mod 4). The normals returned by GetEdgeRaw are not necessarily unit length.

bool
Subdivide
(S2Cell children[4]) const¶ If this is not a leaf cell, set children[0..3] to the four children of this cell (in traversal order) and return true. Otherwise returns false. This method is equivalent to the following:
for (pos=0, id=child_begin(); id != child_end(); id = id.next(), ++pos)
children[i] = S2Cell(id);
except that it is more than two times faster.

S2Point
GetCenter
() const¶ Return the direction vector corresponding to the center in (s,t)space of the given cell. This is the point at which the cell is divided into four subcells; it is not necessarily the centroid of the cell in (u,v)space or (x,y,z)space. The point returned by GetCenterRaw is not necessarily unit length.

static double
AverageArea
(int level)¶ Return the average area for cells at the given level.

double
AverageArea
() const¶ Return the average area of cells at this level. This is accurate to within a factor of 1.7 (for S2_QUADRATIC_PROJECTION) and is extremely cheap to compute.

double
ApproxArea
() const¶ Return the approximate area of this cell. This method is accurate to within 3% percent for all cell sizes and accurate to within 0.1% for cells at level 5 or higher (i.e. squares 350km to a side or smaller on the Earth’s surface). It is moderately cheap to compute.

double
ExactArea
() const¶ Return the area of this cell as accurately as possible. This method is more expensive but it is accurate to 6 digits of precision even for leaf cells (whose area is approximately 1e18).

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

inline int

class
S2CellId
¶ An S2CellId is a 64bit unsigned integer that uniquely identifies a cell in the S2 cell decomposition. It has the following format:
id = [face][face_pos] face: a 3bit number (range 0..5) encoding the cube face. face_pos: a 61bit number encoding the position of the center of this cell along the Hilbert curve over this face (see the Wiki pages for details).
Sequentially increasing cell ids follow a continuous spacefilling curve over the entire sphere. They have the following properties:
 The id of a cell at level k consists of a 3bit face number followed by k bit pairs that recursively select one of the four children of each cell. The next bit is always 1, and all other bits are 0. Therefore, the level of a cell is determined by the position of its lowestnumbered bit that is turned on (for a cell at level k, this position is 2(kMaxLevel  k).)
 The id of a parent cell is at the midpoint of the range of ids spanned by its children (or by its descendants at any level).
Leaf cells are often used to represent points on the unit sphere, and this class provides methods for converting directly between these two representations. For cells that represent 2D regions rather than discrete point, it is better to use the S2Cell class.
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator.

static inline S2CellId
Sentinel
()¶ Returns an invalid cell id guaranteed to be larger than any valid cell id. Useful for creating indexes.

static S2CellId
FromFacePosLevel
(int face, uint64 pos, int level)¶ Return a cell given its face (range 0..5), 61bit Hilbert curve position within that face, and level (range 0..kMaxLevel). The given position will be modified to correspond to the Hilbert curve position at the center of the returned cell. This is a static function rather than a constructor in order to give names to the arguments.

static S2CellId
FromPoint
(S2Point const &p)¶ Return the leaf cell containing the given point (a direction vector, not necessarily unit length).

static S2CellId
FromLatLng
(S2LatLng const &ll)¶ Return the leaf cell containing the given normalized S2LatLng.

S2Point
ToPoint
() const¶ Return the direction vector corresponding to the center of the given cell. The vector returned by ToPointRaw is not necessarily unit length.

inline bool
is_valid
() const¶ Return true if id() represents a valid cell.

inline int
face
() const¶ Which cube face this cell belongs to, in the range 0..5.

int
level
() const¶ Return the subdivision level of the cell (range 0..kMaxLevel).

inline int
GetSizeIJ
() const¶ Return the edge length of this cell in (i,j)space.

inline double
GetSizeST
() const¶ Return the edge length of this cell in (s,t)space.

static inline int
GetSizeIJ
(int level)¶ Like the above, but return the size of cells at the given level.

static inline double
GetSizeST
(int level)¶

inline bool
is_leaf
() const¶ Return true if this is a leaf cell (more efficient than checking whether level() == kMaxLevel).

inline bool
is_face
() const¶ Return true if this is a toplevel face cell (more efficient than checking whether level() == 0).

inline int
child_position
(int level) const¶ Return the child position (0..3) of this cell’s ancestor at the given level, relative to its parent. The argument should be in the range
1..kMaxLevel. For example, child_position(1) returns the position of
this cell’s level1 ancestor within its toplevel face cell.

inline S2CellId
range_min
() const¶ Methods that return the range of cell ids that are contained within this cell (including itself). The range isinclusive* (i.e. test using >= and <=) and the return values of both methods are valid leaf cell ids.
These methods should not be used for iteration. If you want to iterate through all the leaf cells, call child_begin(kMaxLevel) and child_end(kMaxLevel) instead.
It would in fact be errorprone to define a range_end() method, because (range_max().id() + 1) is not always a valid cell id, and the iterator would need to be tested using “<” rather that the usual ”!=”.

inline bool
contains
(S2CellId const &other) const¶ Return true if the given cell is contained within this one.

inline bool
intersects
(S2CellId const &other) const¶ Return true if the given cell intersects this one.

inline S2CellId
parent
() const¶ Return the cell at the previous level or at the given level (which must be less than or equal to the current level).

inline S2CellId
child
(int position) const¶ Return the immediate child of this cell at the given traversal order position (in the range 0 to 3). This cell must not be a leaf cell.

inline S2CellId
child_begin
() const¶ Iteratorstyle methods for traversing the immediate children of a cell or all of the children at a given level (greater than or equal to the current level). Note that the end value is exclusive, just like standard STL iterators, and may not even be a valid cell id. You should iterate using code like this:
for(S2CellId c = id.child_begin(); c != id.child_end(); c = c.next()) ...
The convention for advancing the iterator is “c = c.next()” rather than “++c” to avoid possible confusion with incrementing the underlying 64bit cell id.

inline S2CellId
next
() const¶ Return the next/previous cell at the same level along the Hilbert curve. Works correctly when advancing from one face to the next, but doesnot* wrap around from the last face to the first or vice versa.

S2CellId
advance
(int64 steps) const¶ This method advances or retreats the indicated number of steps along the Hilbert curve at the current level, and returns the new position. The position is never advanced past End() or before Begin().

inline S2CellId
next_wrap
() const¶ Like next() and prev(), but these methods wrap around from the last face to the first and vice versa. They shouldnot* be used for iteration in conjunction with child_begin(), child_end(), Begin(), or End(). The input must be a valid cell id.

S2CellId
advance_wrap
(int64 steps) const¶ This method advances or retreats the indicated number of steps along the Hilbert curve at the current level, and returns the new position. The position wraps between the first and last faces as necessary. The input must be a valid cell id.

static inline S2CellId
Begin
(int level)¶ Iteratorstyle methods for traversing all the cells along the Hilbert curve at a given level (across all 6 faces of the cube). Note that the end value is exclusive (just like standard STL iterators), and is not a valid cell id.

void
GetEdgeNeighbors
(S2CellId neighbors[4]) const¶ Return the four cells that are adjacent across the cell’s four edges. Neighbors are returned in the order defined by S2Cell::GetEdge. All neighbors are guaranteed to be distinct.

void
AppendVertexNeighbors
(int level, vector<S2CellId> *output) const¶ Return the neighbors of closest vertex to this cell at the given level, by appending them to “output”. Normally there are four neighbors, but the closest vertex may only have three neighbors if it is one of the 8 cube vertices.
Requires: level < this>level(), so that we can determine which vertex is closest (in particular, level == kMaxLevel is not allowed).

void
AppendAllNeighbors
(int nbr_level, vector<S2CellId> *output) const¶ Append all neighbors of this cell at the given level to “output”. Two cells X and Y are neighbors if their boundaries intersect but their interiors do not. In particular, two cells that intersect at a single point are neighbors.
Requires: nbr_level >= this>level(). Note that for cells adjacent to a face vertex, the same neighbor may be appended more than once.

static S2CellId
FromFaceIJ
(int face, int i, int j)¶ / Lowlevel methods. Return a leaf cell given its cube face (range 0..5) and i and jcoordinates (see s2.h).

int
ToFaceIJOrientation
(int *pi, int *pj, int *orientation) const¶ Return the (face, i, j) coordinates for the leaf cell corresponding to this cell id. Since cells are represented by the Hilbert curve position at the center of the cell, the returned (i,j) for nonleaf cells will be a leaf cell adjacent to the cell center. If “orientation” is nonNULL, also return the Hilbert curve orientation for the current cell.

class
S2CellUnion
: public S2Region¶ An S2CellUnion is a region consisting of cells of various sizes. Typically a cell union is used to approximate some other shape. There is a tradeoff between the accuracy of the approximation and how many cells are used. Unlike polygons, cells have a fixed hierarchical structure. This makes them more suitable for optimizations based on preprocessing.

void
Init
(vector<S2CellId> const &cell_ids)¶ Populates a cell union with the given S2CellIds or 64bit cells ids, and then calls Normalize(). The InitSwap() version takes ownership of the vector data without copying and clears the given vector. These methods may be called multiple times.

void
Init
(vector<uint64> const &cell_ids)¶

void
InitRaw
(vector<S2CellId> const &cell_ids)¶ Like Init(), but does not call Normalize(). The cell unionmust* be normalized before doing any calculations with it, so it is the caller’s responsibility to make sure that the input is normalized. This method is useful when converting cell unions to another representation and back. These methods may be called multiple times.

void
InitRaw
(vector<uint64> const &cell_ids)¶

void
Detach
(vector<S2CellId> *cell_ids)¶ Gives ownership of the vector data to the client without copying, and clears the content of the cell union. The original data in cell_ids is lost if there was any. This is the opposite of InitRawSwap().

int
num_cells
() const¶ Convenience methods for accessing the individual cell ids.

vector<S2CellId> const &
cell_ids
() const¶ Direct access to the underlying vector for STL algorithms.

bool
Normalize
()¶ Normalizes the cell union by discarding cells that are contained by other cells, replacing groups of 4 child cells by their parent cell whenever possible, and sorting all the cell ids in increasing order. Returns true if the number of cells was reduced.
This methodmust* be called before doing any calculations on the cell union, such as Intersects() or Contains().

void
Denormalize
(int min_level, int level_mod, vector<S2CellId> *output) const¶ Replaces “output” with an expanded version of the cell union where any cells whose level is less than “min_level” or where (level  min_level) is not a multiple of “level_mod” are replaced by their children, until either both of these conditions are satisfied or the maximum level is reached.
This method allows a covering generated by S2RegionCoverer using min_level() or level_mod() constraints to be stored as a normalized cell union (which allows various geometric computations to be done) and then converted back to the original list of cell ids that satisfies the desired constraints.

void
Pack
(int excess = 0)¶ If there are more than “excess” elements of the cell_ids() vector that are allocated but unused, reallocate the array to eliminate the excess space. This reduces memory usage when many cell unions need to be held in memory at once.

bool
Contains
(S2CellId const &id) const¶ Return true if the cell union contains the given cell id. Containment is defined with respect to regions, e.g. a cell contains its 4 children. This is a fast operation (logarithmic in the size of the cell union).

bool
Intersects
(S2CellId const &id) const¶ Return true if the cell union intersects the given cell id. This is a fast operation (logarithmic in the size of the cell union).

bool
Contains
(S2CellUnion const *y) const¶ Return true if this cell union contain/intersects the given other cell union.

bool
Intersects
(S2CellUnion const *y) const¶

void
GetUnion
(S2CellUnion const *x, S2CellUnion const *y)¶ Initialize this cell union to the union, intersection, or difference (x  y) of the two given cell unions. Requires: x != this and y != this.

void
GetIntersection
(S2CellUnion const *x, S2CellUnion const *y)¶

void
GetDifference
(S2CellUnion const *x, S2CellUnion const *y)¶

void
GetIntersection
(S2CellUnion const *x, S2CellId const &id)¶ Specialized version of GetIntersection() that gets the intersection of a cell union with the given cell id. This can be useful for “splitting” a cell union into chunks.

void
Expand
(int expand_level)¶ Expands the cell union by adding a “rim” of cells on expand_level around the union boundary.
For each cell c in the union, we add all cells at level expand_level that abut c. There are typically eight of those (four edgeabutting and four sharing a vertex). However, if c is finer than expand_level, we add all cells abutting c.parent(expand_level) as well as c.parent(expand_level) itself, as an expand_level cell rarely abuts a smaller cell.
Note that the size of the output is exponential in “expand_level”. For example, if expand_level == 20 and the input has a cell at level 10, there will be on the order of 4000 adjacent cells in the output. For most applications the Expand(min_radius, max_level_diff) method below is easier to use.

void
Expand
(S1Angle const &min_radius, int max_level_diff)¶ Expand the cell union such that it contains all points whose distance to the cell union is at most “min_radius”, but do not use cells that are more than “max_level_diff” levels higher than the largest cell in the input. The second parameter controls the tradeoff between accuracy and output size when a large region is being expanded by a small amount (e.g. expanding Canada by 1km). For example, if max_level_diff == 4 the region will always be expanded by approximately 1/16 the width of its largest cell. Note that in the worst case, the number of cells in the output can be up to 4(1 + 2* max_level_diff) times larger than the number of cells in the input.

void
InitFromRange
(S2CellId const &min_id, S2CellId const &max_id)¶ Create a cell union that corresponds to a continuous range of cell ids. The output is a normalized collection of cell ids that covers the leaf cells between “min_id” and “max_id” inclusive. Requires: min_id.is_leaf(), max_id.is_leaf(), min_id <= max_id.

double
AverageBasedArea
() const¶ Approximate this cell union’s area by summing the average area of each contained cell’s average area, using the AverageArea method from the S2Cell class. This is equivalent to the number of leaves covered, multiplied by the average area of a leaf. Note that AverageArea does not take into account distortion of cell, and thus may be off by up to a factor of 1.7. NOTE: Since this is proportional to LeafCellsCovered(), it is always better to use the other function if all you care about is the relative average area between objects.

double
ApproxArea
() const¶ Calculates this cell union’s area by summing the approximate area for each contained cell, using the ApproxArea method from the S2Cell class.

double
ExactArea
() const¶ Calculates this cell union’s area by summing the exact area for each contained cell, using the Exact method from the S2Cell class.

virtual S2CellUnion *
Clone
() const¶ S2Region interface (see s2region.h for details):

virtual S2LatLngRect
GetRectBound
() const¶

virtual bool
Contains
(S2Cell const &cell) const¶ This is a fast operation (logarithmic in the size of the cell union).

virtual bool
MayIntersect
(S2Cell const &cell) const¶ This is a fast operation (logarithmic in the size of the cell union).

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

void

class
S2EdgeIndex
¶ This class structures a set S of data edges, so that one can quickly find which edges of S may potentially intersect or touch a query edge.
The set S is assumed to be indexable by a consecutive sequence of integers in the range [0..num_edges()1]. You subclass this class by defining the three virtual functions num_edges(), edge_from(), edge_to(). Then you use it as follows for a query edge (a,b):
MyS2EdgeIndex edge_index; MyS2EdgeIndex::Iterator it(&edge_index); S2Point const* from; S2Point const* to; for (it.GetCandidates(a, b); !it.Done(); it.Next()) { edge_index.GetEdge(it.Index(), &from, &to); ... RobustCrossing(a,b, from,to) ... }
What is this GetEdge()? You don’t want to use edge_from() and edge_to() in your own code: these are virtual functions that will add a lot of overhead. The most efficient way is as above: you define GetEdge() in your S2EdgeIndex subclass that access the edge points as efficiently as possible.
The function GetCandidates initializes the iterator to return a set of candidate edges from S, such that we are sure that any data edge that touches or crosses (a,b) is a candidate.
This class returns all edges until it finds that it is worth it to compute a quad tree on the data set. Chance my have it that you compute the quad tree exactly when it’s too late and all the work is done, If this happens, we only double the total running time.
You can help the class by declaring in advance that you will make a certain number of calls to GetCandidates():
MyS2EdgeIndex::Iterator it(&edge_index) edge_index.PredictAdditionalCalls(n); for (int i = 0; i < n; ++i) { for (it.GetCandidates(v(i), v(i+1)); !it.Done(); it.Next()) { ... RobustCrossing(v(i), v(i+1), it.From(), it.To()) ... } }
Here, we say that we will call GetCandidates() n times. If we have 1000 data edges and n=1000, then we will compute the quad tree immediately instead of waiting till we’ve wasted enough time to justify the cost.
The tradeoff between brute force and quad tree depends on many things, we use a very approximate tradeoff.
See examples in S2Loop.cc and S2Polygon.cc, in particular, look at the optimization that allows one to use the EdgeCrosser.
TODO(user): Get a better API without the clumsy GetCandidates().
Maybe edge_index.GetIterator()?

class
Iterator
¶ An iterator on data edges that may cross a query edge (a,b). Create the iterator, call GetCandidates(), then Done()/Next() repeatedly.
The current edge in the iteration has index Index(), goes between From() and To().

void
GetCandidates
(S2Point const &a, S2Point const &b)¶ Initializes the iterator to iterate over a set of candidates that may cross the edge (a,b).

int
Index
() const¶ Index of the current edge in the iteration.

bool
Done
() const¶ True if there is no more candidate.

void
Next
()¶ Iterate to the next available candidate.

void

void
Reset
()¶ Empties the index in case it already contained something.

void
ComputeIndex
()¶ Computes the index if not yet done and tells if the index has been computed.

bool
IsIndexComputed
() const¶

void
PredictAdditionalCalls
(int n)¶ If the index hasn’t been computed yet, looks at how much work has gone into iterating using the brute force method, and how much more work is planned as defined by ‘cost’. If it were to have been cheaper to use a quad tree from the beginning, then compute it now. This guarantees that we will never use more than twice the time we would have used had we known in advance exactly how many edges we would have wanted to test. It is the theoretical best.
The value ‘n’ is the number of iterators we expect to request from this edge index.

virtual int
num_edges
() const = 0¶ Overwrite these functions to give access to the underlying data. The function num_edges() returns the number of edges in the index, while edge_from(index) and edge_to(index) return the “from” and “to” endpoints of the edge at the given index.

class

class
S2EdgeUtil
¶ This class contains various utility functions related to edges. It collects together common code that is needed to implement polygonal geometry such as polylines, loops, and general polygons.

class
EdgeCrosser
¶ This class allows a vertex chain v0, v1, v2, ... to be efficiently tested for intersection with a given fixed edge AB.

inline int
RobustCrossing
(S2Point const *d)¶ This method is equivalent to calling the S2EdgeUtil::RobustCrossing() function (defined below) on the edges AB and CD. It returns +1 if there is a crossing, 1 if there is no crossing, and 0 if two points from different edges are the same. Returns 0 or 1 if either edge is degenerate. As a side effect, it saves vertex D to be used as the next vertex C.

inline bool
EdgeOrVertexCrossing
(S2Point const *d)¶ This method is equivalent to the S2EdgeUtil::EdgeOrVertexCrossing() method defined below. It is similar to RobustCrossing, but handles cases where two vertices are identical in a way that makes it easy to implement pointinpolygon containment tests.

inline int

class
RectBounder
¶ This class computes a bounding rectangle that contains all edges defined by a vertex chain v0, v1, v2, ... All vertices must be unit length. Note that the bounding rectangle of an edge can be larger than the bounding rectangle of its endpoints, e.g. consider an edge that passes through the north pole.

void
AddPoint
(S2Point const *b)¶ This method is called to add each vertex to the chain. ‘b’ must point to fixed storage that persists through the next call to AddPoint. This means that if you don’t store all of your points for the lifetime of the bounder, you must at least store the last two points and alternate which one you use for the next point.

S2LatLngRect
GetBound
() const¶ Return the bounding rectangle of the edge chain that connects the vertices defined so far.

void

class
LongitudePruner
¶ The purpose of this class is to find edges that intersect a given longitude interval. It can be used as an efficient rejection test when attempting to find edges that intersect a given region. It accepts a vertex chain v0, v1, v2, ... and returns a boolean value indicating whether each edge intersects the specified longitude interval.

static bool
SimpleCrossing
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &d)¶ Return true if edge AB crosses CD at a point that is interior to both edges. Properties:
 SimpleCrossing(b,a,c,d) == SimpleCrossing(a,b,c,d)
 SimpleCrossing(c,d,a,b) == SimpleCrossing(a,b,c,d)

static int
RobustCrossing
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &d)¶ Like SimpleCrossing, except that points that lie exactly on a line are arbitrarily classified as being on one side or the other (according to the rules of S2::RobustCCW). It returns +1 if there is a crossing, 1 if there is no crossing, and 0 if any two vertices from different edges are the same. Returns 0 or 1 if either edge is degenerate. Properties of RobustCrossing:
 RobustCrossing(b,a,c,d) == RobustCrossing(a,b,c,d)
 RobustCrossing(c,d,a,b) == RobustCrossing(a,b,c,d)
 RobustCrossing(a,b,c,d) == 0 if a==c, a==d, b==c, b==d
 RobustCrossing(a,b,c,d) <= 0 if a==b or c==d
Note that if you want to check an edge against achain* of other edges, it is much more efficient to use an EdgeCrosser (above).

static bool
VertexCrossing
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &d)¶ Given two edges AB and CD where at least two vertices are identical (i.e. RobustCrossing(a,b,c,d) == 0), this function defines whether the two edges “cross” in a such a way that pointinpolygon containment tests can be implemented by counting the number of edge crossings. The basic rule is that a “crossing” occurs if AB is encountered after CD during a CCW sweep around the shared vertex starting from a fixed reference point.
Note that according to this rule, if AB crosses CD then in general CD does not cross AB. However, this leads to the correct result when counting polygon edge crossings. For example, suppose that A,B,C are three consecutive vertices of a CCW polygon. If we now consider the edge crossings of a segment BP as P sweeps around B, the crossing number changes parity exactly when BP crosses BA or BC.
Useful properties of VertexCrossing (VC):
 VC(a,a,c,d) == VC(a,b,c,c) == false
 VC(a,b,a,b) == VC(a,b,b,a) == true
 VC(a,b,c,d) == VC(a,b,d,c) == VC(b,a,c,d) == VC(b,a,d,c)
 If exactly one of a,b equals one of c,d, then exactly one of VC(a,b,c,d) and VC(c,d,a,b) is true
It is an error to call this method with 4 distinct vertices.

static bool
EdgeOrVertexCrossing
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &d)¶ A convenience function that calls RobustCrossing() to handle cases where all four vertices are distinct, and VertexCrossing() to handle cases where two or more vertices are the same. This defines a crossing function such that pointinpolygon containment tests can be implemented by simply counting edge crossings.

static S2Point
GetIntersection
(S2Point const &a, S2Point const &b, S2Point const &c, S2Point const &d)¶ Given two edges AB and CD such that RobustCrossing() is true, return their intersection point. Useful properties of GetIntersection (GI):
 GI(b,a,c,d) == GI(a,b,d,c) == GI(a,b,c,d)
 GI(c,d,a,b) == GI(a,b,c,d)
The returned intersection point X is guaranteed to be close to the edges AB and CD, but if the edges intersect at a very small angle then X may not be close to the true mathematical intersection point P. See the description of “kIntersectionTolerance” below for details.

static double
GetDistanceFraction
(S2Point const &x, S2Point const &a, S2Point const &b)¶ Given a point X and an edge AB, return the distance ratio AX / (AX + BX). If X happens to be on the line segment AB, this is the fraction “t” such that X == Interpolate(A, B, t). Requires that A and B are distinct.

static S2Point
Interpolate
(double t, S2Point const &a, S2Point const &b)¶ Return the point X along the line segment AB whose distance from A is the given fraction “t” of the distance AB. Does NOT require that “t” be between 0 and 1. Note that all distances are measured on the surface of the sphere, so this is more complicated than just computing (1t)*a + t*b and normalizing the result.

static S2Point
InterpolateAtDistance
(S1Angle const &ax, S2Point const &a, S2Point const &b)¶ Like Interpolate(), except that the parameter “ax” represents the desired distance from A to the result X rather than a fraction between 0 and 1.

static S1Angle
GetDistance
(S2Point const &x, S2Point const &a, S2Point const &b)¶ Return the minimum distance from X to any point on the edge AB. All arguments should be unit length. The result is very accurate for small distances but may have some numerical error if the distance is large (approximately Pi/2 or greater). The case A == B is handled correctly.

class

class
S2LatLng
¶ This class represents a point on the unit sphere as a pair of latitudelongitude coordinates. Like the rest of the “geometry” package, the intent is to represent spherical geometry as a mathematical abstraction, so functions that are specifically related to the Earth’s geometry (e.g. easting/northing conversions) should be put elsewhere.
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator.

static inline S2LatLng
FromRadians
(double lat_radians, double lng_radians)¶ Convenience functions – shorter than calling S1Angle::Radians(), etc.

static inline S2LatLng
FromUnsignedE6
(uint32 lat_e6, uint32 lng_e6)¶ Convenience functions – to use when args have been fixed32s in protos.
The arguments are static_cast into int32, so very large unsigned values are treated as negative numbers.

static inline S1Angle
Latitude
(S2Point const &p)¶ Methods to compute the latitude and longitude of a point separately.

Vector2_d const &
coords
() const¶

inline bool
is_valid
() const¶ Return true if the latitude is between 90 and 90 degrees inclusive and the longitude is between 180 and 180 degrees inclusive.

S2LatLng
Normalized
() const¶ Clamps the latitude to the range [90, 90] degrees, and adds or subtracts a multiple of 360 degrees to the longitude if necessary to reduce it to the range [180, 180].

S1Angle
GetDistance
(S2LatLng const &o) const¶ Return the distance (measured along the surface of the sphere) to the given S2LatLng. This is mathematically equivalent to:
S1Angle::Radians(ToPoint().Angle(o.ToPoint()))
but this implementation is slightly more efficient. Both S2LatLngs must be normalized.

void
ToStringInDegrees
(string *s) const¶

static inline S2LatLng

class
S2LatLngRect
: public S2Region¶ An S2LatLngRect represents a closed latitudelongitude rectangle. It is capable of representing the empty and full rectangles as well as single points.
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator, however it is not a “plain old datatype” (POD) because it has virtual functions.

static S2LatLngRect
FromCenterSize
(S2LatLng const ¢er, S2LatLng const &size)¶ Construct a rectangle of the given size centered around the given point. “center” needs to be normalized, but “size” does not. The latitude interval of the result is clamped to [90,90] degrees, and the longitude interval of the result is Full() if and only if the longitude size is 360 degrees or more. Examples of clamping (in degrees):
center=(80,170), size=(40,60) > lat=[60,90], lng=[140,160] center=(10,40), size=(210,400) > lat=[90,90], lng=[180,180] center=(90,180), size=(20,50) > lat=[90,80], lng=[155,155]

static S2LatLngRect
FromPoint
(S2LatLng const &p)¶ Construct a rectangle containing a single (normalized) point.

static S2LatLngRect
FromPointPair
(S2LatLng const &p1, S2LatLng const &p2)¶ Construct the minimal bounding rectangle containing the two given normalized points. This is equivalent to starting with an empty rectangle and calling AddPoint() twice. Note that it is different than the S2LatLngRect(lo, hi) constructor, where the first point is always used as the lowerleft corner of the resulting rectangle.

R1Interval const &
lat
() const¶

S1Interval const &
lng
() const¶

static inline S2LatLngRect
Empty
()¶ The canonical empty and full rectangles.

static inline S2LatLngRect
Full
()¶

static R1Interval
FullLat
()¶ The full allowable range of latitudes and longitudes.

static S1Interval
FullLng
()¶

inline bool
is_valid
() const¶ Return true if the rectangle is valid, which essentially just means that the latitude bounds do not exceed Pi/2 in absolute value and the longitude bounds do not exceed Pi in absolute value. Also, if either the latitude or longitude bound is empty then both must be.

inline bool
is_empty
() const¶ Return true if the rectangle is empty, i.e. it contains no points at all.

inline bool
is_full
() const¶ Return true if the rectangle is full, i.e. it contains all points.

inline bool
is_point
() const¶ Return true if the rectangle is a point, i.e. lo() == hi()

bool
is_inverted
() const¶ Return true if
lng_.lo() > lng_.hi()
, i.e. the rectangle crosses the 180 degree longitude line.

S2LatLng
GetVertex
(int k) const¶ Return the kth vertex of the rectangle (k = 0,1,2,3) in CCW order.

S2LatLng
GetCenter
() const¶ Return the center of the rectangle in latitudelongitude space (in general this is not the center of the region on the sphere).

S2LatLng
GetSize
() const¶ Return the width and height of this rectangle in latitudelongitude space. Empty rectangles have a negative width and height.

double
Area
() const¶ Returns the surface area of this rectangle on the unit sphere.

bool
Contains
(S2LatLng const &ll) const¶ More efficient version of Contains() that accepts a S2LatLng rather than an S2Point. The argument must be normalized.

bool
InteriorContains
(S2Point const &p) const¶ Return true if and only if the given point is contained in the interior of the region (i.e. the region excluding its boundary). The point ‘p’ does not need to be normalized.

bool
InteriorContains
(S2LatLng const &ll) const¶ More efficient version of InteriorContains() that accepts a S2LatLng rather than an S2Point. The argument must be normalized.

bool
Contains
(S2LatLngRect const &other) const¶ Return true if and only if the rectangle contains the given other rectangle.

bool
InteriorContains
(S2LatLngRect const &other) const¶ Return true if and only if the interior of this rectangle contains all points of the given other rectangle (including its boundary).

bool
Intersects
(S2LatLngRect const &other) const¶ Return true if this rectangle and the given other rectangle have any points in common.

bool
Intersects
(S2Cell const &cell) const¶ Returns true if this rectangle intersects the given cell. (This is an exact test and may be fairly expensive, see also MayIntersect below.)

bool
InteriorIntersects
(S2LatLngRect const &other) const¶ Return true if and only if the interior of this rectangle intersects any point (including the boundary) of the given other rectangle.

void
AddPoint
(S2Point const &p)¶ Increase the size of the bounding rectangle to include the given point. The rectangle is expanded by the minimum amount possible. The S2LatLng argument must be normalized.

S2LatLngRect
Expanded
(S2LatLng const &margin) const¶ Return a rectangle that contains all points whose latitude distance from this rectangle is at most margin.lat(), and whose longitude distance from this rectangle is at most margin.lng(). In particular, latitudes are clamped while longitudes are wrapped. Note that any expansion of an empty interval remains empty, and both components of the given margin must be nonnegative. “margin” does not need to be normalized.
NOTE: If you are trying to grow a rectangle by a certaindistance* on the sphere (e.g. 5km), use the ConvolveWithCap() method instead.

S2LatLngRect
Union
(S2LatLngRect const &other) const¶ Return the smallest rectangle containing the union of this rectangle and the given rectangle.

S2LatLngRect
Intersection
(S2LatLngRect const &other) const¶ Return the smallest rectangle containing the intersection of this rectangle and the given rectangle. Note that the region of intersection may consist of two disjoint rectangles, in which case a single rectangle spanning both of them is returned.

S2LatLngRect
ConvolveWithCap
(S1Angle const &angle) const¶ Return a rectangle that contains the convolution of this rectangle with a cap of the given angle. This expands the rectangle by a fixed distance (as opposed to growing the rectangle in latitudelongitude space). The returned rectangle includes all points whose minimum distance to the original rectangle is at most the given angle.

S1Angle
GetDistance
(S2LatLngRect const &other) const¶ Returns the minimum distance (measured along the surface of the sphere) to the given S2LatLngRect. Both S2LatLngRects must be nonempty.

S1Angle
GetDistance
(S2LatLng const &p) const¶ Returns the minimum distance (measured along the surface of the sphere) from a given point to the rectangle (both its boundary and its interior). The latlng must be valid.

S1Angle
GetDirectedHausdorffDistance
(S2LatLngRect const &other) const¶ Returns the (directed or undirected) Hausdorff distance (measured along the surface of the sphere) to the given S2LatLngRect. The directed Hausdorff distance from rectangle A to rectangle B is given by
h(A, B) = max_{p in A} min_{q in B} d(p, q).
The Hausdorff distance between rectangle A and rectangle B is given by
H(A, B) = max{h(A, B), h(B, A)}.

S1Angle
GetHausdorffDistance
(S2LatLngRect const &other) const¶

bool
ApproxEquals
(S2LatLngRect const &other, double max_error = 1e15) const¶ Return true if the latitude and longitude intervals of the two rectangles are the same up to the given tolerance (see r1interval.h and s1interval.h for details).

virtual S2LatLngRect *
Clone
() const¶ S2Region interface (see s2region.h for details):

virtual S2LatLngRect
GetRectBound
() const¶

virtual bool
MayIntersect
(S2Cell const &cell) const¶ This test is cheap but is NOT exact. Use Intersects() if you want a more accurate and more expensive test. Note that when this method is used by an S2RegionCoverer, the accuracy isn’t all that important since if a cell may intersect the region then it is subdivided, and the accuracy of this method goes up as the cells get smaller.

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

static S2LatLngRect

class
S2LoopIndex
: public S2EdgeIndex¶ Indexing structure to efficiently compute intersections.

virtual S2Point const *
edge_from
(int index) const¶ There is no need to overwrite Reset(), as the only data that’s used to implement this class is all contained in the loop data. void Reset();

virtual int
num_edges
() const¶

virtual S2Point const *

class
S2Loop
: public S2Region¶ An S2Loop represents a simple spherical polygon. It consists of a single chain of vertices where the first vertex is implicitly connected to the last. All loops are defined to have a CCW orientation, i.e. the interior of the polygon is on the left side of the edges. This implies that a clockwise loop enclosing a small area is interpreted to be a CCW loop enclosing a very large area.
Loops are not allowed to have any duplicate vertices (whether adjacent or not), and nonadjacent edges are not allowed to intersect. Loops must have at least 3 vertices. Although these restrictions are not enforced in optimized code, you may get unexpected results if they are violated.
Point containment is defined such that if the sphere is subdivided into faces (loops), every point is contained by exactly one face. This implies that loops do not necessarily contain all (or any) of their vertices.
TODO(user): When doing operations on two loops, always create the edgeindex for the bigger of the two. Same for polygons.

void
Init
(vector<S2Point> const &vertices)¶ Initialize a loop connecting the given vertices. The last vertex is implicitly connected to the first. All points should be unit length. Loops must have at least 3 vertices.

bool
IsValid
() const¶ Check whether this loop is valid. Note that in debug mode, validity is checked at loop creation time, so IsValid() should always return true.

static bool
IsValid
(vector<S2Point> const &vertices, int max_adjacent)¶ These two versions are deprecated and ignore max_adjacent. DEPRECATED.

bool
IsValid
(int max_adjacent) const¶ DEPRECATED.

int
depth
() const¶ The depth of a loop is defined as its nesting level within its containing polygon. “Outer shell” loops have depth 0, holes within those loops have depth 1, shells within those holes have depth 2, etc. This field is only used by the S2Polygon implementation.

void
set_depth
(int depth)¶

bool
is_hole
() const¶ Return true if this loop represents a hole in its containing polygon.

int
sign
() const¶ The sign of a loop is 1 if the loop represents a hole in its containing polygon, and +1 otherwise.

int
num_vertices
() const¶

S2Point const &
vertex
(int i) const¶ For convenience, we make two entire copies of the vertex list available: vertex(n..2*n1) is mapped to vertex(0..n1), where n == num_vertices().

bool
IsNormalized
() const¶ Return true if the loop area is at most 2*Pi. Degenerate loops are handled consistently with S2::RobustCCW(), i.e., if a loop can be expressed as the union of degenerate or nearlydegenerate CCW triangles, then it will always be considered normalized.

void
Normalize
()¶ Invert the loop if necessary so that the area enclosed by the loop is at most 2*Pi.

void
Invert
()¶ Reverse the order of the loop vertices, effectively complementing the region represented by the loop.

double
GetArea
() const¶ Return the area of the loop interior, i.e. the region on the left side of the loop. The return value is between 0 and 4*Pi. (Note that the return value is not affected by whether this loop is a “hole” or a “shell”.)

S2Point
GetCentroid
() const¶ Return the true centroid of the loop multiplied by the area of the loop (see s2.h for details on centroids). The result is not unit length, so you may want to normalize it. Also note that in general, the centroid may not be contained by the loop.
We prescale by the loop area for two reasons: (1) it is cheaper to compute this way, and (2) it makes it easier to compute the centroid of more complicated shapes (by splitting them into disjoint regions and adding their centroids).
Note that the return value is not affected by whether this loop is a “hole” or a “shell”.

double
GetTurningAngle
() const¶ Return the sum of the turning angles at each vertex. The return value is positive if the loop is counterclockwise, negative if the loop is clockwise, and zero if the loop is a great circle. Degenerate and nearlydegenerate loops are handled consistently with S2::RobustCCW(). So for example, if a loop has zero area (i.e., it is a very small CCW loop) then the turning angle will always be negative.
This quantity is also called the “geodesic curvature” of the loop.

bool
Contains
(S2Loop const *b) const¶ Return true if the region contained by this loop is a superset of the region contained by the given other loop.

bool
Intersects
(S2Loop const *b) const¶ Return true if the region contained by this loop intersects the region contained by the given other loop.

bool
ContainsNested
(S2Loop const *b) const¶ Given two loops of a polygon (see s2polygon.h for requirements), return true if A contains B. This version of Contains() is much cheaper since it does not need to check whether the boundaries of the two loops cross.

int
ContainsOrCrosses
(S2Loop const *b) const¶ Return +1 if A contains B (i.e. the interior of B is a subset of the interior of A), 1 if the boundaries of A and B cross, and 0 otherwise. Requires that A does not properly contain the complement of B, i.e. A and B do not contain each other’s boundaries. This method is used for testing whether multiloop polygons contain each other.
WARNING: there is a bug in this function  it does not detect loop crossings in certain situations involving shared edges. CL 2926852 works around this bug for polygon intersection, but it probably effects other tests. TODO: fix ContainsOrCrosses() and revert CL 2926852.

bool
BoundaryEquals
(S2Loop const *b) const¶ Return true if two loops have the same boundary. This is true if and only if the loops have the same vertices in the same cyclic order. (For testing purposes.)

bool
BoundaryApproxEquals
(S2Loop const *b, double max_error = 1e15) const¶ Return true if two loops have the same boundary except for vertex perturbations. More precisely, the vertices in the two loops must be in the same cyclic order, and corresponding vertex pairs must be separated by no more than “max_error”. (For testing purposes.)

bool
BoundaryNear
(S2Loop const *b, double max_error = 1e15) const¶ Return true if the two loop boundaries are within “max_error” of each other along their entire lengths. The two loops may have different numbers of vertices. More precisely, this method returns true if the two loops have parameterizations a:[0,1] > S^2, b:[0,1] > S^2 such that distance(a(t), b(t)) <= max_error for all t. You can think of this as testing whether it is possible to drive two cars all the way around the two loops such that no car ever goes backward and the cars are always within “max_error” of each other. (For testing purposes.)

virtual S2Loop *
Clone
() const¶ S2Region interface (see s2region.h for details): GetRectBound() is guaranteed to return exact results, while GetCapBound() is conservative.

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

virtual bool
DecodeWithinScope
(Decoder *const decoder)¶

void

class
S2PointRegion
: public S2Region¶ An S2PointRegion is a region that contains a single point. It is more expensive than the raw S2Point type and is useful mainly for completeness.

virtual S2PointRegion *
Clone
() const¶ S2Region interface (see s2region.h for details):

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

virtual S2PointRegion *

class
S2Polygon
: public S2Region¶ An S2Polygon is an S2Region object that represents a polygon. A polygon consists of zero or more loops representing “shells” and “holes”. All loops should be oriented CCW, i.e. the shell or hole is on the left side of the loop. Loops may be specified in any order. A point is defined to be inside the polygon if it is contained by an odd number of loops.
Polygons have the following restrictions:
 Loops may not cross, i.e. the boundary of a loop may not intersect both the interior and exterior of any other loop.
 Loops may not share edges, i.e. if a loop contains an edge AB, then no other loop may contain AB or BA.
 No loop may cover more than half the area of the sphere. This ensures that no loop properly contains the complement of any other loop, even if the loops are from different polygons. (Loops that represent exact hemispheres are allowed.)
Loops may share vertices, however no vertex may appear twice in a single loop (see s2loop.h).

void
Init
(vector<S2Loop *> *loops)¶ Initialize a polygon by taking ownership of the given loops and clearing the given vector. This method figures out the loop nesting hierarchy and then reorders the loops by following a preorder traversal. This implies that each loop is immediately followed by its descendants in the nesting hierarchy. (See also GetParent and GetLastDescendant.)

void
Release
(vector<S2Loop *> *loops)¶ Release ownership of the loops of this polygon, and appends them to “loops” if nonNULL. Resets the polygon to be empty.

void
Copy
(S2Polygon const *src)¶ Makes a deep copy of the given source polygon. Requires that the destination polygon is empty.

static bool
IsValid
(const vector<S2Loop *> &loops)¶ Return true if the given loops form a valid polygon. Assumes that all of the given loops have already been validated.

bool
IsValid
() const¶ Return true if this is a valid polygon. Note that in debug mode, validity is checked at polygon creation time, so IsValid() should always return true.

bool
IsValid
(bool check_loops, int max_adjacent) const¶ DEPRECATED.

int
num_loops
() const¶

int
num_vertices
() const¶ Total number of vertices in all loops.

int
GetParent
(int k) const¶ Return the index of the parent of loop k, or 1 if it has no parent.

int
GetLastDescendant
(int k) const¶ Return the index of the last loop that is contained within loop k. Returns num_loops()  1 if k < 0. Note that loops are indexed according to a preorder traversal of the nesting hierarchy, so the immediate children of loop k can be found by iterating over loops (k+1)..GetLastDescendant(k) and selecting those whose depth is equal to (loop(k)>depth() + 1).

double
GetArea
() const¶ Return the area of the polygon interior, i.e. the region on the left side of an odd number of loops. The return value is between 0 and 4*Pi.

S2Point
GetCentroid
() const¶ Return the true centroid of the polygon multiplied by the area of the polygon (see s2.h for details on centroids). The result is not unit length, so you may want to normalize it. Also note that in general, the centroid may not be contained by the polygon.
We prescale by the polygon area for two reasons: (1) it is cheaper to compute this way, and (2) it makes it easier to compute the centroid of more complicated shapes (by splitting them into disjoint regions and adding their centroids).

bool
Contains
(S2Polygon const *b) const¶ Return true if this polygon contains the given other polygon, i.e. if polygon A contains all points contained by polygon B.

bool
ApproxContains
(S2Polygon const *b, S1Angle vertex_merge_radius) const¶ Returns true if this polgyon (A) approximately contains the given other polygon (B). This is true if it is possible to move the vertices of B no further than “vertex_merge_radius” such that A contains the modified B.
For example, the empty polygon will contain any polygon whose maximum width is no more than vertex_merge_radius.

bool
Intersects
(S2Polygon const *b) const¶ Return true if this polygon intersects the given other polygon, i.e. if there is a point that is contained by both polygons.

void
InitToIntersection
(S2Polygon const *a, S2Polygon const *b)¶ Initialize this polygon to the intersection, union, or difference (A  B) of the given two polygons. The “vertex_merge_radius” determines how close two vertices must be to be merged together and how close a vertex must be to an edge in order to be spliced into it (see S2PolygonBuilder for details). By default, the merge radius is just large enough to compensate for errors that occur when computing intersection points between edges (S2EdgeUtil::kIntersectionTolerance).
If you are going to convert the resulting polygon to a lowerprecision format, it is necessary to increase the merge radius in order to get a valid result after rounding (i.e. no duplicate vertices, etc). For example, if you are going to convert them to geostore::PolygonProto format, then S1Angle::E7(1) is a good value for “vertex_merge_radius”.

void
InitToSimplified
(S2Polygon const *a, S1Angle tolerance)¶ Initializes this polygon to a polygon that contains fewer vertices and is within tolerance of the polygon a, with some caveats.
 If there is a very small island in the original polygon, it may disappear completely. Thus some parts of the original polygon may not be close to the simplified polygon. Those parts are small, though, and arguably don’t need to be kept.
 However, if there are dense islands, they may all disappear, instead of replacing them by a big simplified island.
 For small tolerances (compared to the polygon size), it may happen that the simplified polygon has more vertices than the original, if the first step of the simplification creates too many selfintersections. One can construct irrealistic cases where that happens to an extreme degree.

void
IntersectWithPolyline
(S2Polyline const *in, vector<S2Polyline *> *out) const¶ Intersect this polygon with the polyline “in” and append the resulting zero or more polylines to “out” (which must be empty). The polylines are appended in the order they would be encountered by traversing “in” from beginning to end. Note that the output may include polylines with only one vertex, but there will not be any zerovertex polylines.
This is equivalent to calling IntersectWithPolylineSloppy() with the “vertex_merge_radius” set to S2EdgeUtil::kIntersectionTolerance.

void
SubtractFromPolyline
(S2Polyline const *in, vector<S2Polyline *> *out) const¶ Same as IntersectWithPolyline, but subtracts this polygon from the given polyline.

static S2Polygon *
DestructiveUnion
(vector<S2Polygon *> *polygons)¶ Return a polygon which is the union of the given polygons. Clears the vector and deletes the polygons!

static S2Polygon *
DestructiveUnionSloppy
(vector<S2Polygon *> *polygons, S1Angle vertex_merge_radius)¶

void
InitToCellUnionBorder
(S2CellUnion const &cells)¶ Initialize this polygon to the outline of the given cell union. In principle this polygon should exactly contain the cell union and this polygon’s inverse should not intersect the cell union, but rounding issues may cause this not to be the case. Does not work correctly if the union covers more than half the sphere.

bool
IsNormalized
() const¶ Return true if every loop of this polygon shares at most one vertex with its parent loop. Every polygon has a unique normalized form. Normalized polygons are useful for testing since it is easy to compare whether two polygons represent the same region.

bool
BoundaryEquals
(S2Polygon const *b) const¶ Return true if two polygons have the same boundary. More precisely, this method requires that both polygons have loops with the same cyclic vertex order and the same nesting hierarchy.

bool
BoundaryApproxEquals
(S2Polygon const *b, double max_error = 1e15) const¶ Return true if two polygons have the same boundary except for vertex perturbations. Both polygons must have loops with the same cyclic vertex order and the same nesting hierarchy, but the vertex locations are allowed to differ by up to “max_error”.

bool
BoundaryNear
(S2Polygon const *b, double max_error = 1e15) const¶ Return true if two polygons have boundaries that are within “max_error” of each other along their entire lengths. More precisely, there must be a bijection between the two sets of loops such that for each pair of loops, “a_loop>BoundaryNear(b_loop)” is true.

S2Point
Project
(S2Point const &point) const¶ If the point is not contained by the polygon returns a point on the polygon closest to the given point. Otherwise returns the point itself. The polygon must not be empty.

virtual S2Polygon *
Clone
() const¶ S2Region interface (see s2region.h for details): GetRectBound() guarantees that it will return exact bounds. GetCapBound() does not.

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

virtual bool
DecodeWithinScope
(Decoder *const decoder)¶

class
S2PolygonBuilderOptions
¶ This is a simple class for assembling polygons out of edges. It requires that no two edges cross. It can handle both directed and undirected edges, and optionally it can also remove duplicate edge pairs (consisting of two identical edges or an edge and its reverse edge). This is useful for computing seamless unions of polygons that have been cut into pieces.
Here are some of the situations this class was designed to handle:
 Computing the union of disjoint polygons that may share part of their boundaries. For example, reassembling a lake that has been split into two loops by a state boundary.
 Constructing polygons from input data that does not follow S2 conventions, i.e. where loops may have repeated vertices, or distinct loops may share edges, or shells and holes have opposite or unspecified orientations.
 Computing the symmetric difference of a set of polygons whose edges intersect only at vertices. This can be used to implement a limited form of polygon intersection or subtraction as well as unions.
 As a tool for implementing other polygon operations by generating a collection of directed edges and then assembling them into loops.

static inline S2PolygonBuilderOptions
DIRECTED_XOR
()¶ These are the options that should be used for assembling wellbehaved input data into polygons. All edges should be directed such that “shells” and “holes” have opposite orientations (typically CCW shells and clockwise holes), unless it is known that shells and holes do not share any edges.

static inline S2PolygonBuilderOptions
UNDIRECTED_XOR
()¶ These are the options that should be used for assembling polygons that do not follow the conventions above, e.g. where edge directions may vary within a single loop, or shells and holes are not oppositely oriented and may share edges.

static inline S2PolygonBuilderOptions
UNDIRECTED_UNION
()¶ These are the options that should be used for assembling edges where the desired output is a collection of loops rather than a polygon, and edges may occur more than once. Edges are treated as undirected and are not XORed together.

bool
undirected_edges
() const¶ Default value: false.
If “undirected_edges” is false, then the input is assumed to consist of edges that can be assembled into oriented loops without reversing any of the edges. Otherwise, “undirected_edges” should be set to true.

void
set_undirected_edges
(bool undirected_edges)¶

bool
xor_edges
() const¶ Default value: true.
If “xor_edges” is true, then any duplicate edge pairs are removed. This is useful for computing the union of a collection of polygons whose interiors are disjoint but whose boundaries may share some common edges (e.g. computing the union of South Africa, Lesotho, and Swaziland).
Note that for directed edges, a “duplicate edge pair” consists of an edge and its corresponding reverse edge. This means that either (a) “shells” and “holes” must have opposite orientations, or (b) shells and holes do not share edges. Otherwise undirected_edges() should be specified.
There are only two reasons to turn off xor_edges():
 AssemblePolygon() will be called, and you want to assert that there are no duplicate edge pairs in the input.
 AssembleLoops() will be called, and you want to keep abutting loops separate in the output rather than merging their regions together (e.g. assembling loops for Kansas City, KS and Kansas City, MO simultaneously).

void
set_xor_edges
(bool xor_edges)¶

bool
validate
() const¶ Default value: false.
If true, IsValid() is called on all loops and polygons before constructing them. If any loop is invalid (e.g. selfintersecting), it is rejected and returned as a set of “unused edges”. Any remaining valid loops are kept. If the entire polygon is invalid (e.g. two loops intersect), then all loops are rejected and returned as unused edges.

void
set_validate
(bool validate)¶

S1Angle
vertex_merge_radius
() const¶ Default value: 0.
If set to a positive value, all vertex pairs that are separated by less than this distance will be merged together. Note that vertices can move arbitrarily far if there is a long chain of vertices separated by less than this distance.
This method is useful for assembling polygons out of input data where vertices and/or edges may not be perfectly aligned.

double
edge_splice_fraction
() const¶ Default value: 0.866 (approximately sqrt(3)/2).
The edge splice radius is automatically set to this fraction of the vertex merge radius. If the edge splice radius is positive, then all vertices that are closer than this distance to an edge are spliced into that edge. Note that edges can move arbitrarily far if there is a long chain of vertices in just the right places.
You can turn off edge splicing by setting this value to zero. This will save some time if you don’t need this feature, or you don’t want vertices to be spliced into nearby edges for some reason.
Note that the edge splice fraction must be less than sqrt(3)/2 in order to avoid infinite loops in the merge algorithm. The default value is very close to this maximum and therefore results in the maximum amount of edge splicing for a given vertex merge radius.
The only reason to reduce the edge splice fraction is if you want to limit changes in edge direction due to splicing. The direction of an edge can change by up to asin(edge_splice_fraction) due to each splice. Thus by default, edges are allowed to change direction by up to 60 degrees per splice. However, note that most direction changes are much smaller than this: the worst case occurs only if the vertex being spliced is just outside the vertex merge radius from one of the edge endpoints.

void
set_edge_splice_fraction
(double edge_splice_fraction)¶

class
S2PolygonBuilder
¶ 
S2PolygonBuilderOptions const &
options
() const¶

bool
AddEdge
(S2Point const &v0, S2Point const &v1)¶ Add the given edge to the polygon builder. This method should be used for input data that may not follow S2 polygon conventions. Note that edges are not allowed to cross each other. Also note that as a convenience, edges where v0 == v1 are ignored.
Returns true if an edge was added, and false if an edge was erased (due to XORing) or not added (if both endpoints were the same).

void
AddLoop
(S2Loop const *loop)¶ Add all edges in the given loop. If the sign() of the loop is negative (i.e. this loop represents a hole), the reverse edges are added instead. This implies that “shells” are CCW and “holes” are CW, as required for the directed edges convention described above.
This method does not take ownership of the loop.

void
AddPolygon
(S2Polygon const *polygon)¶ Add all loops in the given polygon. Shells and holes are added with opposite orientations as described for AddLoop().
This method does not take ownership of the polygon.

typedef vector<pair<S2Point, S2Point>>
EdgeList
¶ This type is used to return any edges that could not be assembled.

bool
AssembleLoops
(vector<S2Loop *> *loops, EdgeList *unused_edges)¶ Assembles the given edges into as many noncrossing loops as possible. When there is a choice about how to assemble the loops, then CCW loops are preferred. Returns true if all edges were assembled. If “unused_edges” is not NULL, it is initialized to the set of edges that could not be assembled into loops.
Note that if xor_edges() is false and duplicate edge pairs may be present, then undirected_edges() should be specified unless all loops can be assembled in a counterclockwise direction. Otherwise this method may not be able to assemble all loops due to its preference for CCW loops.
This method resets the S2PolygonBuilder state so that it can be reused.

bool
AssemblePolygon
(S2Polygon *polygon, EdgeList *unused_edges)¶ Like AssembleLoops, but normalizes all the loops so that they enclose less than half the sphere, and then assembles the loops into a polygon.
For this method to succeed, there should be no duplicate edges in the input. If this is not known to be true, then the “xor_edges” option should be set (which is true by default).
Note that S2Polygons cannot represent arbitrary regions on the sphere, because of the limitation that no loop encloses more than half of the sphere. For example, an S2Polygon cannot represent a 100km wide band around the equator. In such cases, this method will return the complement of the expected region. So for example if all the world’s coastlines were assembled, the output S2Polygon would represent the land area (irrespective of the input edge or loop orientations).

void
set_debug_matrix
(Matrix3x3_d const &m)¶ This function is only for debugging. If it is called, then points will be transformed by the inverse of the given matrix before being printed as latlng coordinates in degrees. “m” should be orthonormal.

S2PolygonBuilderOptions const &

class
S2Polyline
: public S2Region¶ An S2Polyline represents a sequence of zero or more vertices connected by straight edges (geodesics). Edges of length 0 and 180 degrees are not allowed, i.e. adjacent vertices should not be identical or antipodal.

void
Init
(vector<S2Point> const &vertices)¶ Initialize a polyline that connects the given vertices. Empty polylines are allowed. Adjacent vertices should not be identical or antipodal. All vertices should be unit length.

void
Init
(vector<S2LatLng> const &vertices)¶ Convenience initialization function that accepts latitudelongitude coordinates rather than S2Points.

static bool
IsValid
(vector<S2Point> const &vertices)¶ Return true if the given vertices form a valid polyline.

int
num_vertices
() const¶

S2Point
GetCentroid
() const¶ Return the true centroid of the polyline multiplied by the length of the polyline (see s2.h for details on centroids). The result is not unit length, so you may want to normalize it.
Prescaling by the polyline length makes it easy to compute the centroid of several polylines (by simply adding up their centroids).

S2Point
Interpolate
(double fraction) const¶ Return the point whose distance from vertex 0 along the polyline is the given fraction of the polyline’s total length. Fractions less than zero or greater than one are clamped. The return value is unit length. This cost of this function is currently linear in the number of vertices. The polyline must not be empty.

S2Point
GetSuffix
(double fraction, int *next_vertex) const¶ Like Interpolate(), but also return the index of the next polyline vertex after the interpolated point P. This allows the caller to easily construct a given suffix of the polyline by concatenating P with the polyline vertices starting at “next_vertex”. Note that P is guaranteed to be different than
vertex(*next_vertex)
, so this will never result in a duplicate vertex.The polyline must not be empty. Note that if “fraction” >= 1.0, then “next_vertex” will be set to num_vertices() (indicating that no vertices from the polyline need to be appended). The value of “next_vertex” is always between 1 and num_vertices().
This method can also be used to construct a prefix of the polyline, by taking the polyline vertices up to “next_vertex  1” and appending the returned point P if it is different from the last vertex (since in this case there is no guarantee of distinctness).

double
UnInterpolate
(S2Point const &point, int next_vertex) const¶ The inverse operation of GetSuffix/Interpolate. Given a point on the polyline, returns the ratio of the distance to the point from the beginning of the polyline over the length of the polyline. The return value is always betwen 0 and 1 inclusive. See GetSuffix() for the meaning of “next_vertex”.
The polyline should not be empty. If it has fewer than 2 vertices, the return value is zero.

S2Point
Project
(S2Point const &point, int *next_vertex) const¶ Given a point, returns a point on the polyline that is closest to the given point. See GetSuffix() for the meaning of “next_vertex”, which is chosen here w.r.t. the projected point as opposed to the interpolated point in GetSuffix().
The polyline must be nonempty.

bool
IsOnRight
(S2Point const &point) const¶ Returns true if the point given is on the right hand side of the polyline, using a naive definition of “righthandsideness” where the point is on the RHS of the polyline iff the point is on the RHS of the line segment in the polyline which it is closest to.
The polyline must have at least 2 vertices.

bool
Intersects
(S2Polyline const *line) const¶ Return true if this polyline intersects the given polyline. If the polylines share a vertex they are considered to be intersecting. When a polyline endpoint is the only intersection with the other polyline, the function may return true or false arbitrarily.
The running time is quadratic in the number of vertices.

void
Reverse
()¶ Reverse the order of the polyline vertices.

void
SubsampleVertices
(S1Angle const &tolerance, vector<int> *indices) const¶ Return a subsequence of vertex indices such that the polyline connecting these vertices is never further than “tolerance” from the original polyline. The first and last vertices are always preserved.
Some useful properties of the algorithm:
 It runs in linear time.
 The output is always a valid polyline. In particular, adjacent output vertices are never identical or antipodal.
 The method is not optimal, but it tends to produce 23% fewer vertices than the DouglasPeucker algorithm with the same tolerance.
 The output isparametrically* equivalent to the original polyline to within the given tolerance. For example, if a polyline backtracks on itself and then proceeds onwards, the backtracking will be preserved (to within the given tolerance). This is different than the DouglasPeucker algorithm used in maps/util/geoutilinl.h, which only guarantees geometric equivalence.

bool
ApproxEquals
(S2Polyline const *b, double max_error = 1e15) const¶ Return true if two polylines have the same number of vertices, and corresponding vertex pairs are separated by no more than “max_error”. (For testing purposes.)

bool
NearlyCoversPolyline
(S2Polyline const &covered, S1Angle const &max_error) const¶ Return true if “covered” is within “max_error” of a contiguous subpath of this polyline over its entire length. Specifically, this method returns true if this polyline has parameterization a:[0,1] > S^2, “covered” has parameterization b:[0,1] > S^2, and there is a nondecreasing function f:[0,1] > [0,1] such that distance(a(f(t)), b(t)) <= max_error for all t.
You can think of this as testing whether it is possible to drive a car along “covered” and a car along some subpath of this polyline such that no car ever goes backward, and the cars are always within “max_error” of each other.

virtual S2Polyline *
Clone
() const¶ S2Region interface (see s2region.h for details):

virtual S2LatLngRect
GetRectBound
() const¶

virtual bool
VirtualContainsPoint
(S2Point const &p) const¶ Polylines do not have a Contains(S2Point) method, because “containment” is not numerically welldefined except at the polyline vertices.

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

void

typedef Vector2_d
R2Point
¶ TODO: Create an r2.h and move this definition into it.

class
S2R2Rect
: public S2Region¶ This class is a stopgap measure that allows some of the S2 spherical geometry machinery to be applied to planar geometry. An S2R2Rect represents a closed axisaligned rectangle in the (x,y) plane, but it also happens to be a subtype of S2Region, which means that you can use an S2RegionCoverer to approximate it as a collection of S2CellIds.
With respect to the S2Cell decomposition, an S2R2Rect is interpreted as a region of (s,t)space on face 0. In particular, the rectangle [0,1]x[0,1] corresponds to the S2CellId that covers all of face 0. This means that only rectangles that are subsets of [0,1]x[0,1] can be approximated using the S2RegionCoverer interface.
The S2R2Rect class is also a convenient way to find the (s,t)region covered by a given S2CellId (see the FromCell and FromCellId methods).
TODO: If the geometry library is extended to have better support for planar geometry, then this class should no longer be necessary.
This class is intended to be copied by value as desired. It uses the default copy constructor and assignment operator, however it is not a “plain old datatype” (POD) because it has virtual functions.

static S2R2Rect
FromCell
(S2Cell const &cell)¶ Construct a rectangle that corresponds to the boundary of the given cell is (s,t)space. Such rectangles are always a subset of [0,1]x[0,1].

static S2R2Rect
FromCenterSize
(R2Point const ¢er, R2Point const &size)¶ Construct a rectangle from a center point and size in each dimension. Both components of size should be nonnegative, i.e. this method cannot be used to create an empty rectangle.

static S2R2Rect
FromPoint
(R2Point const &p)¶ Convenience method to construct a rectangle containing a single point.

static S2R2Rect
FromPointPair
(R2Point const &p1, R2Point const &p2)¶ Convenience method to construct the minimal bounding rectangle containing the two given points. This is equivalent to starting with an empty rectangle and calling AddPoint() twice. Note that it is different than the S2R2Rect(lo, hi) constructor, where the first point is always used as the lowerleft corner of the resulting rectangle.

R1Interval const &
x
() const¶ Accessor methods.

R1Interval const &
y
() const¶

static inline S2R2Rect
Empty
()¶ The canonical empty rectangle. Use is_empty() to test for empty rectangles, since they have more than one representation.

inline bool
is_valid
() const¶ Return true if the rectangle is valid, which essentially just means that if the bound for either axis is empty then both must be.

inline bool
is_empty
() const¶ Return true if the rectangle is empty, i.e. it contains no points at all.

R2Point
GetVertex
(int k) const¶ Return the kth vertex of the rectangle (k = 0,1,2,3) in CCW order. Vertex 0 is in the lowerleft corner.

R2Point
GetCenter
() const¶ Return the center of the rectangle in (x,y)space (in general this is not the center of the region on the sphere).

R2Point
GetSize
() const¶ Return the width and height of this rectangle in (x,y)space. Empty rectangles have a negative width and height.

bool
Contains
(R2Point const &p) const¶ Return true if the rectangle contains the given point. Note that rectangles are closed regions, i.e. they contain their boundary.

bool
InteriorContains
(R2Point const &p) const¶ Return true if and only if the given point is contained in the interior of the region (i.e. the region excluding its boundary).

bool
Contains
(S2R2Rect const &other) const¶ Return true if and only if the rectangle contains the given other rectangle.

bool
InteriorContains
(S2R2Rect const &other) const¶ Return true if and only if the interior of this rectangle contains all points of the given other rectangle (including its boundary).

bool
Intersects
(S2R2Rect const &other) const¶ Return true if this rectangle and the given other rectangle have any points in common.

bool
InteriorIntersects
(S2R2Rect const &other) const¶ Return true if and only if the interior of this rectangle intersects any point (including the boundary) of the given other rectangle.

void
AddPoint
(R2Point const &p)¶ Increase the size of the bounding rectangle to include the given point. The rectangle is expanded by the minimum amount possible.

S2R2Rect
Expanded
(R2Point const &margin) const¶ Return a rectangle that contains all points whose xdistance from this rectangle is at most margin.x(), and whose ydistance from this rectangle is at most margin.y(). Note that any expansion of an empty interval remains empty, and both components of the given margin must be nonnegative.

S2R2Rect
Union
(S2R2Rect const &other) const¶ Return the smallest rectangle containing the union of this rectangle and the given rectangle.

S2R2Rect
Intersection
(S2R2Rect const &other) const¶ Return the smallest rectangle containing the intersection of this rectangle and the given rectangle.

bool
ApproxEquals
(S2R2Rect const &other, double max_error = 1e15) const¶ Return true if the x and yintervals of the two rectangles are the same up to the given tolerance (see r1interval.h for details).

static S2Point
ToS2Point
(R2Point const &p)¶ Return the unitlength S2Point corresponding to the given point “p” in the (s,t)plane. “p” need not be restricted to the range [0,1]x[0,1].

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

static S2R2Rect

class
S2Region
¶ An S2Region represents a twodimensional region over the unit sphere. It is an abstract interface with various concrete subtypes.
The main purpose of this interface is to allow complex regions to be approximated as simpler regions. So rather than having a wide variety of virtual methods that are implemented by all subtypes, the interface is restricted to methods that are useful for computing approximations.

virtual S2Region *
Clone
() const = 0¶ Return a deep copy of this region. If you want to narrow the result to a specific known region type, use down_cast<T*> from basictypes.h. Subtypes return pointers to that subtype from their Clone() methods.

virtual S2Cap
GetCapBound
() const = 0¶ Return a bounding spherical cap. This is not guaranteed to be exact.

virtual S2LatLngRect
GetRectBound
() const = 0¶ Return a bounding latitudelongitude rectangle that contains the region. The bounds are not guaranteed to be tight.

virtual bool
Contains
(S2Cell const &cell) const = 0¶ If this method returns true, the region completely contains the given cell. Otherwise, either the region does not contain the cell or the containment relationship could not be determined.

virtual bool
MayIntersect
(S2Cell const &cell) const = 0¶ If this method returns false, the region does not intersect the given cell. Otherwise, either region intersects the cell, or the intersection relationship could not be determined.

virtual bool
VirtualContainsPoint
(S2Point const &p) const = 0¶ Return true if and only if the given point is contained by the region. The point ‘p’ is generally required to be unit length, although some subtypes may relax this restriction. NOTE: If you will be calling this function on one specific subtype only, or if performance is a consideration, please use the nonvirtual method Contains(S2Point const& p) declared below!

virtual void
Encode
(Encoder *const encoder) const = 0¶ Use encoder to generate a serialized representation of this region. Assumes that encoder can be enlarged using calls to Ensure(int).
The representation chosen is left up to the subclasses but it should satisfy the following constraints:
 It should encode a version number.
 It should be deserializable using the corresponding Decode method.
 Performance, not space, should be the chief consideration. Encode() and Decode() should be implemented such that the combination is equivalent to calling Clone().

virtual bool
Decode
(Decoder *const decoder) = 0¶ Reconstruct a region from the serialized representation generated by Encode(). Note that since this method is virtual, it requires that a Region object of the appropriate concrete type has already been constructed. It is not possible to decode regions of unknown type.
Whenever the Decode method is changed to deal with new serialized representations, it should be done so in a manner that allows for older versions to be decoded i.e. the version number in the serialized representation should be used to decide how to decode the data.
Returns true on success.

virtual bool
DecodeWithinScope
(Decoder *const decoder)¶ Provide the same functionality as Decode, except that decoded regions are allowed to point directly into the Decoder’s memory buffer rather than copying the data. This method can be much faster for regions that have a lot of data (such as polygons), but the decoded region is only valid within the scope (lifetime) of the Decoder’s memory buffer. Default implementation just calls Decode.

virtual S2Region *

class
S2RegionCoverer
¶ An S2RegionCoverer is a class that allows arbitrary regions to be approximated as unions of cells (S2CellUnion). This is useful for implementing various sorts of search and precomputation operations.
Typical usage:
S2RegionCoverer coverer; coverer.set_max_cells(5); S2Cap cap = S2Cap::FromAxisAngle(...); vector<S2CellId> covering; coverer.GetCovering(cap, &covering);
This yields a vector of at most 5 cells that is guaranteed to cover the given cap (a discshaped region on the sphere).
The approximation algorithm is not optimal but does a pretty good job in practice. The output does not always use the maximum number of cells allowed, both because this would not always yield a better approximation, and because max_cells() is a limit on how much work is done exploring the possible covering as well as a limit on the final output size.
One can also generate interior coverings, which are sets of cells which are entirely contained within a region. Interior coverings can be empty, even for nonempty regions, if there are no cells that satisfy the provided constraints and are contained by the region. Note that for performance reasons, it is wise to specify a max_level when computing interior coverings  otherwise for regions with small or zero area, the algorithm may spend a lot of time subdividing cells all the way to leaf level to try to find contained cells.

void
set_min_level
(int min_level)¶ Set the minimum and maximum cell level to be used. The default is to use all cell levels. Requires: max_level() >= min_level().
To find the cell level corresponding to a given physical distance, use the S2Cell metrics defined in s2.h. For example, to find the cell level that corresponds to an average edge length of 10km, use:
int level = S2::kAvgEdge.GetClosestLevel( geostore::S2Earth::KmToRadians(length_km));
Note: min_level() takes priority over max_cells(), i.e. cells below the given level will never be used even if this causes a large number of cells to be returned.

void
set_max_level
(int max_level)¶

int
min_level
() const¶

int
max_level
() const¶

void
set_level_mod
(int level_mod)¶ If specified, then only cells where (level  min_level) is a multiple of “level_mod” will be used (default 1). This effectively allows the branching factor of the S2CellId hierarchy to be increased. Currently the only parameter values allowed are 1, 2, or 3, corresponding to branching factors of 4, 16, and 64 respectively.

int
level_mod
() const¶

void
set_max_cells
(int max_cells)¶ Sets the maximum desired number of cells in the approximation (defaults to kDefaultMaxCells). Note the following:
 For any setting of max_cells(), up to 6 cells may be returned if that is the minimum number of cells required (e.g. if the region intersects all six face cells). Up to 3 cells may be returned even for very tiny convex regions if they happen to be located at the intersection of three cube faces.
 For any setting of max_cells(), an arbitrary number of cells may be returned if min_level() is too high for the region being approximated.
 If max_cells() is less than 4, the area of the covering may be arbitrarily large compared to the area of the original region even if the region is convex (e.g. an S2Cap or S2LatLngRect).
Accuracy is measured by dividing the area of the covering by the area of the original region. The following table shows the median and worst case values for this area ratio on a test case consisting of 100,000 spherical caps of random size (generated using s2regioncoverer_unittest):
max_cells: 3 4 5 6 8 12 20 100 1000 median ratio: 5.33 3.32 2.73 2.34 1.98 1.66 1.42 1.11 1.01 worst case: 215518 14.41 9.72 5.26 3.91 2.75 1.92 1.20 1.02

int
max_cells
() const¶

void
GetCovering
(S2Region const ®ion, vector<S2CellId> *covering)¶ Return a vector of cell ids that covers (GetCovering) or is contained within (GetInteriorCovering) the given region and satisfies the various restrictions specified above.

void
GetCellUnion
(S2Region const ®ion, S2CellUnion *covering)¶ Return a normalized cell union that covers (GetCellUnion) or is contained within (GetInteriorCellUnion) the given region and satisfies the restrictionsEXCEPT* for min_level() and level_mod(). These criteria cannot be satisfied using a cell union because cell unions are automatically normalized by replacing four child cells with their parent whenever possible. (Note that the list of cell ids passed to the cell union constructor does in fact satisfy all the given restrictions.)

void
GetInteriorCellUnion
(S2Region const ®ion, S2CellUnion *interior)¶

void

class
S2RegionIntersection
: public S2Region¶ An S2RegionIntersection represents the intersection of a set of regions. It is convenient for computing a covering of the intersection of a set of regions.

void
Release
(vector<S2Region *> *regions)¶ Release ownership of the regions of this union, and appends them to “regions” if nonNULL. Resets the region to be empty.

int
num_regions
() const¶ Accessor methods.

virtual S2RegionIntersection *
Clone
() const¶ S2Region interface (see s2region.h for details):

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

void

class
S2RegionUnion
: public S2Region¶ An S2RegionUnion represents a union of possibly overlapping regions. It is convenient for computing a covering of a set of regions.

void
Release
(vector<S2Region *> *regions)¶ Release ownership of the regions of this union, and appends them to “regions” if nonNULL. Resets the region to be empty.

void
Add
(S2Region *region)¶ Add the given region to the union. This method can be called repeatedly as an alternative to Init(). Takes ownership of the pointer.

int
num_regions
() const¶ Accessor methods.

virtual S2RegionUnion *
Clone
() const¶ S2Region interface (see s2region.h for details):

virtual S2LatLngRect
GetRectBound
() const¶

virtual void
Encode
(Encoder *const encoder) const¶

virtual bool
Decode
(Decoder *const decoder)¶

void