GRASS GIS 8 Programmer's Manual 8.3.2(2024)-exported
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area_poly1.c
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1/*!
2 * \file lib/gis/area_poly1.c
3 *
4 * \brief GIS Library - Polygon area calculation routines.
5 *
6 * (C) 2001-2013 by the GRASS Development Team
7 *
8 * This program is free software under the GNU General Public License
9 * (>=v2). Read the file COPYING that comes with GRASS for details.
10 *
11 * \author Original author CERL
12 */
13
14#include <math.h>
15#include <grass/gis.h>
16#include "pi.h"
17
18#define TWOPI M_PI + M_PI
19
20static struct state {
21 double QA, QB, QC;
22 double QbarA, QbarB, QbarC, QbarD;
23
24 double AE; /** a^2(1-e^2) */
25
26 double Qp; /** Q at the north pole */
27
28 double E; /** Area of the earth */
29} state;
30
31static struct state *st = &state;
32
33static double Q(double x)
34{
35 double sinx, sinx2;
36
37 sinx = sin(x);
38 sinx2 = sinx * sinx;
39
40 return sinx * (1 + sinx2 * (st->QA + sinx2 * (st->QB + sinx2 * st->QC)));
41}
42
43static double Qbar(double x)
44{
45 double cosx, cosx2;
46
47 cosx = cos(x);
48 cosx2 = cosx * cosx;
49
50 return cosx *
51 (st->QbarA +
52 cosx2 * (st->QbarB + cosx2 * (st->QbarC + cosx2 * st->QbarD)));
53}
54
55/*!
56 * \brief Begin area calculations.
57 *
58 * This initializes the polygon area calculations for the
59 * ellipsoid with semi-major axis <i>a</i> (in meters) and ellipsoid
60 * eccentricity squared <i>e2</i>.
61 *
62 * \param a semi-major axis
63 * \param e2 ellipsoid eccentricity squared
64 */
65
66void G_begin_ellipsoid_polygon_area(double a, double e2)
67{
68 double e4, e6;
69
70 e4 = e2 * e2;
71 e6 = e4 * e2;
72
73 st->AE = a * a * (1 - e2);
74
75 st->QA = (2.0 / 3.0) * e2;
76 st->QB = (3.0 / 5.0) * e4;
77 st->QC = (4.0 / 7.0) * e6;
78
79 st->QbarA = -1.0 - (2.0 / 3.0) * e2 - (3.0 / 5.0) * e4 - (4.0 / 7.0) * e6;
80 st->QbarB = (2.0 / 9.0) * e2 + (2.0 / 5.0) * e4 + (4.0 / 7.0) * e6;
81 st->QbarC = -(3.0 / 25.0) * e4 - (12.0 / 35.0) * e6;
82 st->QbarD = (4.0 / 49.0) * e6;
83
84 st->Qp = Q(M_PI_2);
85 st->E = 4 * M_PI * st->Qp * st->AE;
86 if (st->E < 0.0)
87 st->E = -st->E;
88}
89
90/*!
91 * \brief Area of lat-long polygon.
92 *
93 * Returns the area in square meters of the polygon described by the
94 * <i>n</i> pairs of <i>lat,long</i> vertices for latitude-longitude
95 * grids.
96 *
97 * <b>Note:</b> This routine computes the area of a polygon on the
98 * ellipsoid. The sides of the polygon are rhumb lines and, in general,
99 * not geodesics. Each side is actually defined by a linear relationship
100 * between latitude and longitude, i.e., on a rectangular/equidistant
101 * cylindrical/Plate Carr{'e}e grid, the side would appear as a
102 * straight line. For two consecutive vertices of the polygon,
103 * (lat_1, long1) and (lat_2,long_2), the line joining them (i.e., the
104 * polygon's side) is defined by:
105 *
106 \verbatim
107 lat_2 - lat_1
108 lat = lat_1 + (long - long_1) * ---------------
109 long_2 - long_1
110 \endverbatim
111 *
112 * where long_1 < long < long_2.
113 * The values of QbarA, etc., are determined by the integration of
114 * the Q function. Into www.integral-calculator.com, paste this
115 * expression :
116 *
117 \verbatim
118 sin(x)+ (2/3)e^2(sin(x))^3 + (3/5)e^4(sin(x))^5 + (4/7)e^6(sin(x))^7
119 \endverbatim
120 *
121 * and you'll get their values. (Last checked 30 Oct 2013).
122 *
123 * This function correctly computes (within the limits of the series
124 * approximation) the area of a quadrilateral on the ellipsoid when
125 * two of its sides run along meridians and the other two sides run
126 * along parallels of latitude.
127 *
128 * \param lon array of longitudes
129 * \param lat array of latitudes
130 * \param n number of lat,lon pairs
131 *
132 * \return area in square meters
133 */
134double G_ellipsoid_polygon_area(const double *lon, const double *lat, int n)
135{
136 double x1, y1, x2, y2, dx, dy;
137 double Qbar1, Qbar2;
138 double area;
139 double thresh =
140 1e-6; /* threshold for dy, should be between 1e-4 and 1e-7 */
141
142 x2 = Radians(lon[n - 1]);
143 y2 = Radians(lat[n - 1]);
144 Qbar2 = Qbar(y2);
145
146 area = 0.0;
147
148 while (--n >= 0) {
149 x1 = x2;
150 y1 = y2;
151 Qbar1 = Qbar2;
152
153 x2 = Radians(*lon++);
154 y2 = Radians(*lat++);
155 Qbar2 = Qbar(y2);
156
157 if (x1 > x2)
158 while (x1 - x2 > M_PI)
159 x2 += TWOPI;
160 else if (x2 > x1)
161 while (x2 - x1 > M_PI)
162 x1 += TWOPI;
163
164 dx = x2 - x1;
165 dy = y2 - y1;
166
167 if (fabs(dy) > thresh) {
168 /* account for different latitudes y1, y2 */
169 area += dx * (st->Qp - (Qbar2 - Qbar1) / dy);
170 /* original:
171 * area += dx * st->Qp - (dx / dy) * (Qbar2 - Qbar1);
172 */
173 }
174 else {
175 /* latitudes y1, y2 are (nearly) identical */
176 /* if y2 becomes similar to y1, i.e. y2 -> y1
177 * Qbar2 - Qbar1 -> 0 and dy -> 0
178 * (Qbar2 - Qbar1) / dy -> ?
179 * (Qbar2 - Qbar1) / dy should approach Q((y1 + y2) / 2)
180 * Metz 2017
181 */
182 area += dx * (st->Qp - Q((y1 + y2) / 2));
183 }
184 }
185 if ((area *= st->AE) < 0.0)
186 area = -area;
187
188 /* kludge - if polygon circles the south pole the area will be
189 * computed as if it cirlced the north pole. The correction is
190 * the difference between total surface area of the earth and
191 * the "north pole" area.
192 */
193 if (area > st->E)
194 area = st->E;
195 if (area > st->E / 2)
196 area = st->E - area;
197
198 return area;
199}
#define TWOPI
Definition area_poly1.c:18
void G_begin_ellipsoid_polygon_area(double a, double e2)
Begin area calculations.
Definition area_poly1.c:66
double G_ellipsoid_polygon_area(const double *lon, const double *lat, int n)
Area of lat-long polygon.
Definition area_poly1.c:134
#define Radians(x)
Definition pi.h:6