1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
/**
* \file PolygonArea.hpp
* \brief Header for GeographicLib::PolygonAreaT class
*
* Copyright (c) Charles Karney (2010-2023) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
**********************************************************************/
#if !defined(GEOGRAPHICLIB_POLYGONAREA_HPP)
#define GEOGRAPHICLIB_POLYGONAREA_HPP 1
#include <GeographicLib/Geodesic.hpp>
#include <GeographicLib/GeodesicExact.hpp>
#include <GeographicLib/Rhumb.hpp>
#include <GeographicLib/Accumulator.hpp>
namespace GeographicLib {
/**
* \brief Polygon areas
*
* This computes the area of a polygon whose edges are geodesics using the
* method given in Section 6 of
* - C. F. F. Karney,
* <a href="https://doi.org/10.1007/s00190-012-0578-z">
* Algorithms for geodesics</a>,
* J. Geodesy <b>87</b>, 43--55 (2013);
* DOI: <a href="https://doi.org/10.1007/s00190-012-0578-z">
* 10.1007/s00190-012-0578-z</a>;
* addenda:
* <a href="https://geographiclib.sourceforge.io/geod-addenda.html">
* geod-addenda.html</a>.
*
* Arbitrarily complex polygons are allowed. In the case self-intersecting
* of polygons the area is accumulated "algebraically", e.g., the areas of
* the 2 loops in a figure-8 polygon will partially cancel.
*
* This class lets you add vertices and edges one at a time to the polygon.
* The sequence must start with a vertex and thereafter vertices and edges
* can be added in any order. Any vertex after the first creates a new edge
* which is the \e shortest geodesic from the previous vertex. In some
* cases there may be two or many such shortest geodesics and the area is
* then not uniquely defined. In this case, either add an intermediate
* vertex or add the edge \e as an edge (by defining its direction and
* length).
*
* The area and perimeter are accumulated at two times the standard floating
* point precision to guard against the loss of accuracy with many-sided
* polygons. At any point you can ask for the perimeter and area so far.
* There's an option to treat the points as defining a polyline instead of a
* polygon; in that case, only the perimeter is computed.
*
* This is a templated class to allow it to be used with Geodesic,
* GeodesicExact, and Rhumb. GeographicLib::PolygonArea,
* GeographicLib::PolygonAreaExact, and GeographicLib::PolygonAreaRhumb are
* typedefs for these cases.
*
* For GeographicLib::PolygonArea (edges defined by Geodesic), an upper bound
* on the error is about 0.1 m<sup>2</sup> per vertex. However this is a
* wildly pessimistic estimate in most cases. A more realistic estimate of
* the error is given by a test involving 10<sup>7</sup> approximately
* regular polygons on the WGS84 ellipsoid. The centers and the orientations
* of the polygons were uniformly distributed, the number of vertices was
* log-uniformly distributed in [3, 300], and the center to vertex distance
* log-uniformly distributed in [0.1 m, 9000 km].
*
* Using double precision (the standard precision for GeographicLib), the
* maximum error in the perimeter was 200 nm, and the maximum error in the
* area was<pre>
* 0.0013 m^2 for perimeter < 10 km
* 0.0070 m^2 for perimeter < 100 km
* 0.070 m^2 for perimeter < 1000 km
* 0.11 m^2 for all perimeters
* </pre>
* The errors are given in terms of the perimeter, because it is expected
* that the errors depend mainly on the number of edges and the edge lengths.
*
* Using long doubles (GEOGRPAHICLIB_PRECISION = 3), the maximum error in the
* perimeter was 200 pm, and the maximum error in the area was<pre>
* 0.7 mm^2 for perim < 10 km
* 3.2 mm^2 for perimeter < 100 km
* 21 mm^2 for perimeter < 1000 km
* 45 mm^2 for all perimeters
* </pre>
*
* @tparam GeodType the geodesic class to use.
*
* Example of use:
* \include example-PolygonArea.cpp
*
* <a href="Planimeter.1.html">Planimeter</a> is a command-line utility
* providing access to the functionality of PolygonAreaT.
**********************************************************************/
template<class GeodType = Geodesic>
class PolygonAreaT {
private:
typedef Math::real real;
GeodType _earth;
real _area0; // Full ellipsoid area
bool _polyline; // Assume polyline (don't close and skip area)
unsigned _mask;
unsigned _num;
int _crossings;
Accumulator<> _areasum, _perimetersum;
real _lat0, _lon0, _lat1, _lon1;
static int transit(real lon1, real lon2);
// an alternate version of transit to deal with longitudes in the direct
// problem.
static int transitdirect(real lon1, real lon2);
void Remainder(Accumulator<>& a) const { a.remainder(_area0); }
void Remainder(real& a) const {
using std::remainder;
a = remainder(a, _area0);
}
template<typename T>
void AreaReduce(T& area, int crossings, bool reverse, bool sign) const;
public:
/**
* Constructor for PolygonAreaT.
*
* @param[in] earth the Geodesic object to use for geodesic calculations.
* @param[in] polyline if true that treat the points as defining a polyline
* instead of a polygon (default = false).
**********************************************************************/
PolygonAreaT(const GeodType& earth, bool polyline = false)
: _earth(earth)
, _area0(_earth.EllipsoidArea())
, _polyline(polyline)
, _mask(GeodType::LATITUDE | GeodType::LONGITUDE | GeodType::DISTANCE |
(_polyline ? GeodType::NONE :
GeodType::AREA | GeodType::LONG_UNROLL))
{ Clear(); }
/**
* Clear PolygonAreaT, allowing a new polygon to be started.
**********************************************************************/
void Clear() {
_num = 0;
_crossings = 0;
_areasum = 0;
_perimetersum = 0;
_lat0 = _lon0 = _lat1 = _lon1 = Math::NaN();
}
/**
* Add a point to the polygon or polyline.
*
* @param[in] lat the latitude of the point (degrees).
* @param[in] lon the longitude of the point (degrees).
*
* \e lat should be in the range [−90°, 90°].
**********************************************************************/
void AddPoint(real lat, real lon);
/**
* Add an edge to the polygon or polyline.
*
* @param[in] azi azimuth at current point (degrees).
* @param[in] s distance from current point to next point (meters).
*
* This does nothing if no points have been added yet. Use
* PolygonAreaT::CurrentPoint to determine the position of the new vertex.
**********************************************************************/
void AddEdge(real azi, real s);
/**
* Return the results so far.
*
* @param[in] reverse if true then clockwise (instead of counter-clockwise)
* traversal counts as a positive area.
* @param[in] sign if true then return a signed result for the area if
* the polygon is traversed in the "wrong" direction instead of returning
* the area for the rest of the earth.
* @param[out] perimeter the perimeter of the polygon or length of the
* polyline (meters).
* @param[out] area the area of the polygon (meters<sup>2</sup>); only set
* if \e polyline is false in the constructor.
* @return the number of points.
*
* More points can be added to the polygon after this call.
**********************************************************************/
unsigned Compute(bool reverse, bool sign,
real& perimeter, real& area) const;
/**
* Return the results assuming a tentative final test point is added;
* however, the data for the test point is not saved. This lets you report
* a running result for the perimeter and area as the user moves the mouse
* cursor. Ordinary floating point arithmetic is used to accumulate the
* data for the test point; thus the area and perimeter returned are less
* accurate than if PolygonAreaT::AddPoint and PolygonAreaT::Compute are
* used.
*
* @param[in] lat the latitude of the test point (degrees).
* @param[in] lon the longitude of the test point (degrees).
* @param[in] reverse if true then clockwise (instead of counter-clockwise)
* traversal counts as a positive area.
* @param[in] sign if true then return a signed result for the area if
* the polygon is traversed in the "wrong" direction instead of returning
* the area for the rest of the earth.
* @param[out] perimeter the approximate perimeter of the polygon or length
* of the polyline (meters).
* @param[out] area the approximate area of the polygon
* (meters<sup>2</sup>); only set if polyline is false in the
* constructor.
* @return the number of points.
*
* \e lat should be in the range [−90°, 90°].
**********************************************************************/
unsigned TestPoint(real lat, real lon, bool reverse, bool sign,
real& perimeter, real& area) const;
/**
* Return the results assuming a tentative final test point is added via an
* azimuth and distance; however, the data for the test point is not saved.
* This lets you report a running result for the perimeter and area as the
* user moves the mouse cursor. Ordinary floating point arithmetic is used
* to accumulate the data for the test point; thus the area and perimeter
* returned are less accurate than if PolygonAreaT::AddEdge and
* PolygonAreaT::Compute are used.
*
* @param[in] azi azimuth at current point (degrees).
* @param[in] s distance from current point to final test point (meters).
* @param[in] reverse if true then clockwise (instead of counter-clockwise)
* traversal counts as a positive area.
* @param[in] sign if true then return a signed result for the area if
* the polygon is traversed in the "wrong" direction instead of returning
* the area for the rest of the earth.
* @param[out] perimeter the approximate perimeter of the polygon or length
* of the polyline (meters).
* @param[out] area the approximate area of the polygon
* (meters<sup>2</sup>); only set if polyline is false in the
* constructor.
* @return the number of points.
**********************************************************************/
unsigned TestEdge(real azi, real s, bool reverse, bool sign,
real& perimeter, real& area) const;
/** \name Inspector functions
**********************************************************************/
///@{
/**
* @return \e a the equatorial radius of the ellipsoid (meters). This is
* the value inherited from the Geodesic object used in the constructor.
**********************************************************************/
Math::real EquatorialRadius() const { return _earth.EquatorialRadius(); }
/**
* @return \e f the flattening of the ellipsoid. This is the value
* inherited from the Geodesic object used in the constructor.
**********************************************************************/
Math::real Flattening() const { return _earth.Flattening(); }
/**
* Report the previous vertex added to the polygon or polyline.
*
* @param[out] lat the latitude of the point (degrees).
* @param[out] lon the longitude of the point (degrees).
*
* If no points have been added, then NaNs are returned. Otherwise, \e lon
* will be in the range [−180°, 180°].
**********************************************************************/
void CurrentPoint(real& lat, real& lon) const
{ lat = _lat1; lon = _lon1; }
/**
* Report the number of points currently in the polygon or polyline.
*
* @return the number of points.
*
* If no points have been added, then 0 is returned.
**********************************************************************/
unsigned NumberPoints() const { return _num; }
/**
* Report whether the current object is a polygon or a polyline.
*
* @return true if the object is a polyline.
**********************************************************************/
bool Polyline() const { return _polyline; }
///@}
};
/**
* @relates PolygonAreaT
*
* Polygon areas using Geodesic. This should be used if the flattening is
* small.
**********************************************************************/
typedef PolygonAreaT<Geodesic> PolygonArea;
/**
* @relates PolygonAreaT
*
* \deprecated Polygon areas using GeodesicExact. Instead use PolygonArea
* with a Geodesic object specified with \e exact = true.
**********************************************************************/
typedef PolygonAreaT<GeodesicExact> PolygonAreaExact;
/**
* @relates PolygonAreaT
*
* Polygon areas using Rhumb.
**********************************************************************/
typedef PolygonAreaT<Rhumb> PolygonAreaRhumb;
} // namespace GeographicLib
#endif // GEOGRAPHICLIB_POLYGONAREA_HPP