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Intersect.hpp
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/**
* \file Intersect.hpp
* \brief Header for GeographicLib::Intersect class
*
* Copyright (c) Charles Karney (2023-2024) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_INTERSECT_HPP)
#define GEOGRAPHICLIB_INTERSECT_HPP 1
#include <vector>
#include <set>
#include <GeographicLib/Math.hpp>
#include <GeographicLib/Geodesic.hpp>
#include <GeographicLib/GeodesicLine.hpp>
namespace GeographicLib {
/**
* \brief %Geodesic intersections
*
* Find the intersections of two geodesics \e X and \e Y. Four calling
* sequences are supported.
* - The geodesics are defined by a position (latitude and longitude) and an
* azimuth. In this case the \e closest intersection is found.
* - The geodesics are defined by two endpoints. The intersection of the two
* segments is found. If they don't intersect, then the closest
* intersection is returned.
* - The geodesics are defined as an intersection point, a single position
* and two azimuths. In this case, the next closest intersection is found.
* - The geodesics are defined as in the first case and all intersection
* within a specified distance are returned.
* .
* In all cases the position of the intersection is given by the signed
* displacements \e x and \e y along the geodesics from the starting point
* (the first point in the case of a geodesic segment). The closest
* itersection is defined as the one that minimizes the L1 distance,
* Intersect::Dist([<i>x</i>, <i>y</i>) = |<i>x</i>| + |<i>y</i>|.
*
* The routines also optionally return a coincidence indicator \e c. This is
* typically 0. However if the geodesics lie on top of one another at the
* point of intersection, then \e c is set to +1, if they are parallel, and
* &minus;1, if they are antiparallel.
*
* Example of use:
* \include example-Intersect.cpp
*
* <a href="IntersectTool.1.html">IntersectTool</a> is a command-line utility
* providing access to the functionality of this class.
*
* This solution for intersections is described in
* - C. F. F. Karney,<br>
* <a href="https://doi.org/10.1061/JSUED2.SUENG-1483">
* Geodesic intersections</a>,
* J. Surveying Eng. <b>150</b>(3), 04024005:1--9 (2024);
* preprint
* <a href="https://arxiv.org/abs/2308.00495">arxiv:2308.00495</a>.
* .
* It is based on the work of
* - S. Baseldga and J. C. Martinez-Llario,
* <a href="https://doi.org/10.1007/s11200-017-1020-z">
* Intersection and point-to-line solutions for geodesics
* on the ellipsoid</a>,
* Stud. Geophys. Geod. <b>62</b>, 353--363 (2018);
* DOI: <a href="https://doi.org/10.1007/s11200-017-1020-z">
* 10.1007/s11200-017-1020-z</a>.
**********************************************************************/
class GEOGRAPHICLIB_EXPORT Intersect {
private:
typedef Math::real real;
public:
/**
* The type used to hold the two displacement along the geodesics. This is
* just a std::pair with \e x = \e first and \e y = \e second.
**********************************************************************/
typedef std::pair<Math::real, Math::real> Point;
/**
* The minimum capabilities for GeodesicLine objects which are passed to
* this class.
**********************************************************************/
static const unsigned LineCaps = Geodesic::LATITUDE | Geodesic::LONGITUDE |
Geodesic::AZIMUTH | Geodesic::REDUCEDLENGTH | Geodesic::GEODESICSCALE |
Geodesic::DISTANCE_IN;
private:
static const int numit_ = 100;
const Geodesic _geod;
real _a, _f, // equatorial radius, flattening
_rR, // authalic radius
_d, // pi*_rR
_eps, // criterion for intersection + coincidence
_tol, // convergence for Newton in Solve1
_delta, // for equality tests, safety margin for tiling
_t1, // min distance between intersections
_t2, // furthest dist to closest intersection
_t3, // 1/2 furthest min dist to next intersection
_t4, // capture radius for spherical sol in Solve0
_t5, // longest shortest geodesic
_d1, // tile spacing for Closest
_d2, // tile spacing for Next
_d3; // tile spacing for All
// The L1 distance
static Math::real d1(Math::real x, Math::real y)
{ using std::fabs; return fabs(x) + fabs(y); }
// An internal version of Point with a little more functionality
class XPoint {
public:
real x, y;
int c;
XPoint(Math::real x, Math::real y, int c = 0)
: x(x), y(y), c(c)
{}
XPoint()
: x(Math::NaN()), y(Math::NaN()), c(0)
{}
XPoint(const Point& p)
: x(p.first), y(p.second), c(0)
{}
XPoint& operator+=(const XPoint& p) {
x += p.x; y += p.y;
if (p.c) c = p.c; // pass along a nonzero c from either operand
return *this;
}
XPoint operator+(const XPoint& p) const {
XPoint t = *this; t += p; return t;
}
Math::real Dist() const { return d1(x, y); }
Math::real Dist(const XPoint& p) const { return d1(x - p.x, y - p.y); }
Point data() const { return Point(x, y); }
};
// Comparing XPoints for insertions into sets, but ensure that close
// XPoints test equal.
class GEOGRAPHICLIB_EXPORT SetComp {
private:
const real _delta;
public:
SetComp(Math::real delta) : _delta(delta) {}
bool eq(const XPoint& p, const XPoint& q) const {
return d1(p.x - q.x, p.y - q.y) <= _delta;
}
bool operator()(const XPoint& p, const XPoint& q) const {
return !eq(p, q) && ( p.x != q.x ? p.x < q.x : p.y < q.y );
}
};
SetComp _comp;
// For ranking XPoints by closeness
class RankPoint {
private:
const real _x, _y;
public:
RankPoint(const Point& p0) : _x(p0.first), _y(p0.second) {}
RankPoint(const XPoint& p0) : _x(p0.x), _y(p0.y) {}
bool operator()(const XPoint& p, const XPoint& q) const {
real dp = d1(p.x - _x, p.y - _y),
dq = d1(q.x - _x, q.y - _y);
return dp != dq ? (dp < dq) :
(p.x != q.x ? (p.x < q.x) : (p.y < q.y));
}
};
// The spherical solution
XPoint Spherical(const GeodesicLine& lineX, const GeodesicLine& lineY,
const XPoint& p) const;
// The basic algorithm
XPoint Basic(const GeodesicLine& lineX, const GeodesicLine& lineY,
const XPoint& p0) const;
// The closest intersecton
XPoint ClosestInt(const GeodesicLine& lineX, const GeodesicLine& lineY,
const XPoint& p0) const;
// The next intersecton
XPoint NextInt(const GeodesicLine& lineX, const GeodesicLine& lineY) const;
// Segment intersecton
XPoint SegmentInt(const GeodesicLine& lineX, const GeodesicLine& lineY,
int& segmode) const;
// All intersectons
std::vector<XPoint>
AllInt0(const GeodesicLine& lineX, const GeodesicLine& lineY,
Math::real maxdist, const XPoint& p0) const;
std::vector<Point>
AllInternal(const GeodesicLine& lineX, const GeodesicLine& lineY,
Math::real maxdist, const Point& p0,
std::vector<int>& c, bool cp) const;
// Find {semi-,}conjugate point which is close to s3. Optional m12, M12,
// M21 use {semi-,}conjugacy relative to point 2
Math::real ConjugateDist(const GeodesicLine& line, Math::real s3, bool semi,
Math::real m12 = 0, Math::real M12 = 1,
Math::real M21 = 1) const;
Math::real distpolar(Math::real lat1, Math::real* lat2 = nullptr) const;
Math::real polarb(Math::real* lata = nullptr, Math::real* latb = nullptr)
const;
Math::real conjdist(Math::real azi, Math::real* ds = nullptr,
Math::real* sp = nullptr, Math::real* sm = nullptr)
const;
Math::real distoblique(Math::real* azi = nullptr, Math::real* sp = nullptr,
Math::real* sm = nullptr) const;
// p is intersection point on coincident lines orientation = c; p0 is
// origin point. Change p to center point wrt p0, i.e, abs((p-p0)_x) =
// abs((p-p0)_y)
static XPoint fixcoincident(const XPoint& p0, const XPoint& p);
static XPoint fixcoincident(const XPoint& p0, const XPoint& p, int c);
static XPoint fixsegment(Math::real sx, Math::real sy, const XPoint& p);
static int segmentmode(Math::real sx, Math::real sy, const XPoint& p) {
return (p.x < 0 ? -1 : p.x <= sx ? 0 : 1) * 3
+ (p.y < 0 ? -1 : p.y <= sy ? 0 : 1);
}
mutable long long _cnt0, _cnt1, _cnt2, _cnt3, _cnt4;
public:
/** \name Constructor
**********************************************************************/
///@{
/**
* Constructor for an ellipsoid with
*
* @param[in] geod a Geodesic object. This sets the parameters \e a and \e
* f for the ellipsoid.
* @exception GeographicErr if the eccentricity of the elliposdoid is too
* large.
*
* \note This class has been validated for -1/4 &le; \e f &le; 1/5. It may
* give satisfactory results slightly outside this range; however
* sufficient far outside the range, some internal checks will fail and an
* exception thrown.
*
* \note If |<i>f</i>| > 1/50, then the Geodesic object should be
* constructed with \e exact = true.
**********************************************************************/
Intersect(const Geodesic& geod);
///@}
/** \name Finding intersections
**********************************************************************/
///@{
/**
* Find the closest intersection point, with each geodesic specified by
* position and azimuth.
*
* @param[in] latX latitude of starting point for geodesic \e X (degrees).
* @param[in] lonX longitude of starting point for geodesic \e X (degrees).
* @param[in] aziX azimuth at starting point for geodesic \e X (degrees).
* @param[in] latY latitude of starting point for geodesic \e Y (degrees).
* @param[in] lonY longitude of starting point for geodesic \e Y (degrees).
* @param[in] aziY azimuth at starting point for geodesic \e Y (degrees).
* @param[in] p0 an optional offset for the starting points (meters),
* default = [0,0].
* @param[out] c optional pointer to an integer coincidence indicator.
* @return \e p the intersection point closest to \e p0.
*
* The returned intersection minimizes Intersect::Dist(\e p, \e p0).
**********************************************************************/
Point Closest(Math::real latX, Math::real lonX, Math::real aziX,
Math::real latY, Math::real lonY, Math::real aziY,
const Point& p0 = Point(0, 0), int* c = nullptr) const;
/**
* Find the closest intersection point, with each geodesic given as a
* GeodesicLine.
*
* @param[in] lineX geodesic \e X.
* @param[in] lineY geodesic \e Y.
* @param[in] p0 an optional offset for the starting points (meters),
* default = [0,0].
* @param[out] c optional pointer to an integer coincidence indicator.
* @return \e p the intersection point closest to \e p0.
*
* The returned intersection minimizes Intersect::Dist(\e p, \e p0).
*
* \note \e lineX and \e lineY should be created with minimum capabilities
* Intersect::LineCaps. The methods for creating a GeodesicLine include
* all these capabilities by default.
**********************************************************************/
Point Closest(const GeodesicLine& lineX, const GeodesicLine& lineY,
const Point& p0 = Point(0, 0), int* c = nullptr) const;
/**
* Find the intersection of two geodesic segments defined by their
* endpoints.
*
* @param[in] latX1 latitude of first point for segment \e X (degrees).
* @param[in] lonX1 longitude of first point for segment \e X (degrees).
* @param[in] latX2 latitude of second point for segment \e X (degrees).
* @param[in] lonX2 longitude of second point for segment \e X (degrees).
* @param[in] latY1 latitude of first point for segment \e Y (degrees).
* @param[in] lonY1 longitude of first point for segment \e Y (degrees).
* @param[in] latY2 latitude of second point for segment \e Y (degrees).
* @param[in] lonY2 longitude of second point for segment \e Y (degrees).
* @param[out] segmode an indicator equal to zero if the segments
* intersect (see below).
* @param[out] c optional pointer to an integer coincidence indicator.
* @return \e p the intersection point if the segments intersect, otherwise
* the intersection point closest to the midpoints of the two
* segments.
*
* \warning The results are only well defined if there's a \e unique
* shortest geodesic between the endpoints of the two segments.
*
* \e segmode codes up information about the closest intersection in the
* case where the segments intersect. Let <i>x</i><sub>12</sub> be the
* length of the segment \e X and \e x = <i>p</i>.first, the position of
* the intersection on segment \e X. Define
* - \e k<sub><i>x</i></sub> = &minus;1, if \e x < 0,
* - \e k<sub><i>x</i></sub> = 0,
* if 0 &le; \e x &le; <i>x</i><sub>12</sub>,
* - \e k<sub><i>x</i></sub> = 1, if <i>x</i><sub>12</sub> < \e x.
* .
* and similarly for segment \e Y. Then
* \e segmode = 3 \e k<sub><i>x</i></sub> + \e k<sub><i>y</i></sub>.
**********************************************************************/
Point Segment(Math::real latX1, Math::real lonX1,
Math::real latX2, Math::real lonX2,
Math::real latY1, Math::real lonY1,
Math::real latY2, Math::real lonY2,
int& segmode, int* c = nullptr) const;
/**
* Find the intersection of two geodesic segments each defined by a
* GeodesicLine.
*
* @param[in] lineX segment \e X.
* @param[in] lineY segment \e Y.
* @param[out] segmode an indicator equal to zero if the segments
* intersect (see below).
* @param[out] c optional pointer to an integer coincidence indicator.
* @return \e p the intersection point if the segments intersect, otherwise
* the intersection point closest to the midpoints of the two
* segments.
*
* \warning \e lineX and \e lineY must represent shortest geodesics, e.g.,
* they can be created by Geodesic::InverseLine. The results are only well
* defined if there's a \e unique shortest geodesic between the endpoints
* of the two segments.
*
* \note \e lineX and \e lineY should be created with minimum capabilities
* Intersect::LineCaps. The methods for creating a GeodesicLine include
* all these capabilities by default.
*
* See previous definition of Intersect::Segment for more information on \e
* segmode.
**********************************************************************/
Point Segment(const GeodesicLine& lineX, const GeodesicLine& lineY,
int& segmode, int* c = nullptr) const;
/**
* Find the next closest intersection point to a given intersection,
* specified by position and two azimuths.
*
* @param[in] latX latitude of starting points for geodesics \e X and \e Y
* (degrees).
* @param[in] lonX longitude of starting points for geodesics \e X and \e Y
* (degrees).
* @param[in] aziX azimuth at starting point for geodesic \e X (degrees).
* @param[in] aziY azimuth at starting point for geodesic \e Y (degrees).
* @param[out] c optional pointer to an integer coincidence indicator.
* @return \e p the next closest intersection point.
*
* The returned intersection minimizes Intersect::Dist(\e p) (excluding \e
* p = [0,0]).
*
* \note Equidistant closest intersections are surprisingly common. If
* this may be a problem, use Intersect::All with a sufficiently large \e
* maxdist to capture close intersections.
**********************************************************************/
Point Next(Math::real latX, Math::real lonX,
Math::real aziX, Math::real aziY, int* c = nullptr) const;
/**
* Find the next closest intersection point to a given intersection,
* with each geodesic specified a GeodesicLine.
*
* @param[in] lineX geodesic \e X.
* @param[in] lineY geodesic \e Y.
* @param[out] c optional pointer to an integer coincidence indicator.
* @return \e p the next closest intersection point.
*
* \warning \e lineX and \e lineY must both have the same starting point,
* i.e., the distance between [<i>lineX</i>.Latitude(),
* <i>lineX</i>.Longitude()] and [<i>lineY</i>.Latitude(),
* <i>lineY</i>.Longitude()] must be zero.
*
* \note \e lineX and \e lineY should be created with minimum capabilities
* Intersect::LineCaps. The methods for creating a GeodesicLine include
* all these capabilities by default.
*
* \note Equidistant closest intersections are surprisingly common. If
* this may be a problem, use Intersect::All with a sufficiently large \e
* maxdist to capture close intersections.
**********************************************************************/
Point Next(const GeodesicLine& lineX, const GeodesicLine& lineY,
int* c = nullptr) const;
///@}
/** \name Finding all intersections
**********************************************************************/
///@{
/**
* Find all intersections within a certain distance, with each geodesic
* specified by position and azimuth.
*
* @param[in] latX latitude of starting point for geodesic \e X (degrees).
* @param[in] lonX longitude of starting point for geodesic \e X (degrees).
* @param[in] aziX azimuth at starting point for geodesic \e X (degrees).
* @param[in] latY latitude of starting point for geodesic \e Y (degrees).
* @param[in] lonY longitude of starting point for geodesic \e Y (degrees).
* @param[in] aziY azimuth at starting point for geodesic \e Y (degrees).
* @param[in] maxdist the maximum distance for the returned intersections
* (meters).
* @param[out] c vector of coincidences.
* @param[in] p0 an optional offset for the starting points (meters),
* default = [0,0].
* @return \e plist a vector for the intersections closest to \e p0.
*
* Each intersection point satisfies Intersect::Dist(\e p, \e p0) &le; \e
* maxdist. The vector of returned intersections is sorted on the distance
* from \e p0.
**********************************************************************/
std::vector<Point> All(Math::real latX, Math::real lonX, Math::real aziX,
Math::real latY, Math::real lonY, Math::real aziY,
Math::real maxdist, std::vector<int>& c,
const Point& p0 = Point(0, 0))
const;
/**
* Find all intersections within a certain distance, with each geodesic
* specified by position and azimuth. Don't return vector of
* coincidences.
*
* @param[in] latX latitude of starting point for geodesic \e X (degrees).
* @param[in] lonX longitude of starting point for geodesic \e X (degrees).
* @param[in] aziX azimuth at starting point for geodesic \e X (degrees).
* @param[in] latY latitude of starting point for geodesic \e Y (degrees).
* @param[in] lonY longitude of starting point for geodesic \e Y (degrees).
* @param[in] aziY azimuth at starting point for geodesic \e Y (degrees).
* @param[in] maxdist the maximum distance for the returned intersections
* (meters).
* @param[in] p0 an optional offset for the starting points (meters),
* default = [0,0].
* @return \e plist a vector for the intersections closest to \e p0.
*
* Each intersection point satisfies Intersect::Dist(\e p, \e p0) &le; \e
* maxdist. The vector of returned intersections is sorted on the distance
* from \e p0.
**********************************************************************/
std::vector<Point> All(Math::real latX, Math::real lonX, Math::real aziX,
Math::real latY, Math::real lonY, Math::real aziY,
Math::real maxdist, const Point& p0 = Point(0, 0))
const;
/**
* Find all intersections within a certain distance, with each geodesic
* specified by a GeodesicLine.
*
* @param[in] lineX geodesic \e X.
* @param[in] lineY geodesic \e Y.
* @param[in] maxdist the maximum distance for the returned intersections
* (meters).
* @param[out] c vector of coincidences.
* @param[in] p0 an optional offset for the starting points (meters),
* default = [0,0].
* @return \e plist a vector for the intersections closest to \e p0.
*
* Each intersection point satisfies Intersect::Dist(\e p, \e p0) &le; \e
* maxdist. The vector of returned intersections is sorted on the distance
* from \e p0.
*
* \note \e lineX and \e lineY should be created with minimum capabilities
* Intersect::LineCaps. The methods for creating a GeodesicLine include
* all these capabilities by default.
**********************************************************************/
std::vector<Point> All(const GeodesicLine& lineX, const GeodesicLine& lineY,
Math::real maxdist, std::vector<int>& c,
const Point& p0 = Point(0, 0))
const;
/**
* Find all intersections within a certain distance, with each geodesic
* specified by a GeodesicLine. Don't return vector or coincidences.
*
* @param[in] lineX geodesic \e X.
* @param[in] lineY geodesic \e Y.
* @param[in] maxdist the maximum distance for the returned intersections
* (meters).
* @param[in] p0 an optional offset for the starting points (meters),
* default = [0,0].
* @return \e plist a vector for the intersections closest to \e p0.
*
* Each intersection point satisfies Intersect::Dist(\e p, \e p0) &le; \e
* maxdist. The vector of returned intersections is sorted on the distance
* from \e p0.
*
* \note \e lineX and \e lineY should be created with minimum capabilities
* Intersect::LineCaps. The methods for creating a GeodesicLine include
* all these capabilities by default.
**********************************************************************/
std::vector<Point> All(const GeodesicLine& lineX, const GeodesicLine& lineY,
Math::real maxdist, const Point& p0 = Point(0, 0))
const;
///@}
/** \name Diagnostic counters
**********************************************************************/
///@{
/**
* @return the cumulative number of invocations of **h**.
*
* This is a count of the number of times the spherical triangle needs to
* be solved. Each involves a call to Geodesic::Inverse and this is a good
* metric for the overall cost. This counter is set to zero by the
* constructor.
*
* \warning The counter is a mutable variable and so is not thread safe.
**********************************************************************/
long long NumInverse() const { return _cnt0; }
/**
* @return the cumulative number of invocations of **b**.
*
* This is a count of the number of invocations of the basic algorithm,
* which is used by all the intersection methods. This counter is set to
* zero by the constructor.
*
* \warning The counter is a mutable variable and so is not thread safe.
**********************************************************************/
long long NumBasic() const { return _cnt1; }
/**
* @return the number of times intersection point was changed in
* Intersect::Closest and Intersect::Next.
*
* If this counter is incremented by just 1 in Intersect::Closest, then the
* initial result of the basic algorithm was eventually accepted. This
* counter is set to zero by the constructor.
*
* \note This counter is also incremented by Intersect::Segment, which
* calls Intersect::Closest.
*
* \warning The counter is a mutable variable and so is not thread safe.
**********************************************************************/
long long NumChange() const { return _cnt2; }
/**
* @return the number of times a corner point is checked in
* Intersect::Segment.
*
* This counter is set to zero by the constructor.
*
* \warning The counter is a mutable variable and so is not thread safe.
**********************************************************************/
long long NumCorner() const { return _cnt3; }
/**
* @return the number of times a corner point is returned by
* Intersect::Segment.
*
* This counter is set to zero by the constructor.
*
* \note A conjecture is that a corner point never results in an
* intersection that overrides the intersection closest to the midpoints of
* the segments; i.e., NumCorner() always returns 0.
*
* \warning The counter is a mutable variable and so is not thread safe.
**********************************************************************/
long long NumOverride() const { return _cnt4; }
///@}
/** \name Insepctor function
**********************************************************************/
///@{
/**
* @return \e geod the Geodesic object used in the constructor.
*
* This can be used to query Geodesic::EquatorialRadius(),
* Geodesic::Flattening(), Geodesic::Exact(), and
* Geodesic::EllipsoidArea().
**********************************************************************/
const Geodesic& GeodesicObject() const { return _geod; }
///@}
/**
* The L1 distance.
*
* @param[in] p the position along geodesics \e X and \e Y.
* @param[in] p0 [optional] the reference position, default = [0, 0].
* @return the L1 distance of \e p from \e p0, i.e.,
* |<i>p</i><sub><i>x</i></sub> &minus; <i>p0</i><sub><i>x</i></sub>| +
* |<i>p</i><sub><i>y</i></sub> &minus; <i>p0</i><sub><i>y</i></sub>|.
**********************************************************************/
static Math::real Dist(const Point& p, const Point& p0 = Point(0, 0)) {
using std::fabs;
return fabs(p.first - p0.first) + fabs(p.second - p0.second);
}
};
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_INTERSECT_HPP
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition Constants.hpp:67
real
GeographicLib::Math::real real
Definition GeodSolve.cpp:28
GeodesicLine.hpp
Header for GeographicLib::GeodesicLine class.
Geodesic.hpp
Header for GeographicLib::Geodesic class.
Math.hpp
Header for GeographicLib::Math class.
GeographicLib::Geodesic
Geodesic calculations
Definition Geodesic.hpp:175
GeographicLib::Intersect
Geodesic intersections
Definition Intersect.hpp:71
GeographicLib::Intersect::GeodesicObject
const Geodesic & GeodesicObject() const
Definition Intersect.hpp:563
GeographicLib::Intersect::NumChange
long long NumChange() const
Definition Intersect.hpp:528
GeographicLib::Intersect::NumBasic
long long NumBasic() const
Definition Intersect.hpp:514
GeographicLib::Intersect::NumInverse
long long NumInverse() const
Definition Intersect.hpp:504
GeographicLib::Intersect::Dist
static Math::real Dist(const Point &p, const Point &p0=Point(0, 0))
Definition Intersect.hpp:576
GeographicLib::Intersect::Point
std::pair< Math::real, Math::real > Point
Definition Intersect.hpp:79
GeographicLib::Intersect::NumOverride
long long NumOverride() const
Definition Intersect.hpp:550
GeographicLib::Intersect::NumCorner
long long NumCorner() const
Definition Intersect.hpp:537
GeographicLib::Math::real
double real
Definition Math.hpp:115
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12