GeographicLib  2.1.2
Rhumb.cpp
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1 /**
2  * \file Rhumb.cpp
3  * \brief Implementation for GeographicLib::Rhumb and GeographicLib::RhumbLine
4  * classes
5  *
6  * Copyright (c) Charles Karney (2014-2022) <charles@karney.com> and licensed
7  * under the MIT/X11 License. For more information, see
8  * https://geographiclib.sourceforge.io/
9  **********************************************************************/
10 
11 #include <algorithm>
12 #include <GeographicLib/Rhumb.hpp>
13 
14 #if defined(_MSC_VER)
15 // Squelch warnings about enum-float expressions
16 # pragma warning (disable: 5055)
17 #endif
18 
19 namespace GeographicLib {
20 
21  using namespace std;
22 
23  Rhumb::Rhumb(real a, real f, bool exact)
24  : _ell(a, f)
25  , _exact(exact)
26  , _c2(_ell.Area() / (2 * Math::td))
27  {
28  // Generated by Maxima on 2015-05-15 08:24:04-04:00
29 #if GEOGRAPHICLIB_RHUMBAREA_ORDER == 4
30  static const real coeff[] = {
31  // R[0]/n^0, polynomial in n of order 4
32  691, 7860, -20160, 18900, 0, 56700,
33  // R[1]/n^1, polynomial in n of order 3
34  1772, -5340, 6930, -4725, 14175,
35  // R[2]/n^2, polynomial in n of order 2
36  -1747, 1590, -630, 4725,
37  // R[3]/n^3, polynomial in n of order 1
38  104, -31, 315,
39  // R[4]/n^4, polynomial in n of order 0
40  -41, 420,
41  }; // count = 20
42 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 5
43  static const real coeff[] = {
44  // R[0]/n^0, polynomial in n of order 5
45  -79036, 22803, 259380, -665280, 623700, 0, 1871100,
46  // R[1]/n^1, polynomial in n of order 4
47  41662, 58476, -176220, 228690, -155925, 467775,
48  // R[2]/n^2, polynomial in n of order 3
49  18118, -57651, 52470, -20790, 155925,
50  // R[3]/n^3, polynomial in n of order 2
51  -23011, 17160, -5115, 51975,
52  // R[4]/n^4, polynomial in n of order 1
53  5480, -1353, 13860,
54  // R[5]/n^5, polynomial in n of order 0
55  -668, 5775,
56  }; // count = 27
57 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 6
58  static const real coeff[] = {
59  // R[0]/n^0, polynomial in n of order 6
60  128346268, -107884140, 31126095, 354053700, -908107200, 851350500, 0,
61  2554051500LL,
62  // R[1]/n^1, polynomial in n of order 5
63  -114456994, 56868630, 79819740, -240540300, 312161850, -212837625,
64  638512875,
65  // R[2]/n^2, polynomial in n of order 4
66  51304574, 24731070, -78693615, 71621550, -28378350, 212837625,
67  // R[3]/n^3, polynomial in n of order 3
68  1554472, -6282003, 4684680, -1396395, 14189175,
69  // R[4]/n^4, polynomial in n of order 2
70  -4913956, 3205800, -791505, 8108100,
71  // R[5]/n^5, polynomial in n of order 1
72  1092376, -234468, 2027025,
73  // R[6]/n^6, polynomial in n of order 0
74  -313076, 2027025,
75  }; // count = 35
76 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 7
77  static const real coeff[] = {
78  // R[0]/n^0, polynomial in n of order 7
79  -317195588, 385038804, -323652420, 93378285, 1062161100, -2724321600LL,
80  2554051500LL, 0, 7662154500LL,
81  // R[1]/n^1, polynomial in n of order 6
82  258618446, -343370982, 170605890, 239459220, -721620900, 936485550,
83  -638512875, 1915538625,
84  // R[2]/n^2, polynomial in n of order 5
85  -248174686, 153913722, 74193210, -236080845, 214864650, -85135050,
86  638512875,
87  // R[3]/n^3, polynomial in n of order 4
88  114450437, 23317080, -94230045, 70270200, -20945925, 212837625,
89  // R[4]/n^4, polynomial in n of order 3
90  15445736, -103193076, 67321800, -16621605, 170270100,
91  // R[5]/n^5, polynomial in n of order 2
92  -27766753, 16385640, -3517020, 30405375,
93  // R[6]/n^6, polynomial in n of order 1
94  4892722, -939228, 6081075,
95  // R[7]/n^7, polynomial in n of order 0
96  -3189007, 14189175,
97  }; // count = 44
98 #elif GEOGRAPHICLIB_RHUMBAREA_ORDER == 8
99  static const real coeff[] = {
100  // R[0]/n^0, polynomial in n of order 8
101  71374704821LL, -161769749880LL, 196369790040LL, -165062734200LL,
102  47622925350LL, 541702161000LL, -1389404016000LL, 1302566265000LL, 0,
103  3907698795000LL,
104  // R[1]/n^1, polynomial in n of order 7
105  -13691187484LL, 65947703730LL, -87559600410LL, 43504501950LL,
106  61062101100LL, -184013329500LL, 238803815250LL, -162820783125LL,
107  488462349375LL,
108  // R[2]/n^2, polynomial in n of order 6
109  30802104839LL, -63284544930LL, 39247999110LL, 18919268550LL,
110  -60200615475LL, 54790485750LL, -21709437750LL, 162820783125LL,
111  // R[3]/n^3, polynomial in n of order 5
112  -8934064508LL, 5836972287LL, 1189171080, -4805732295LL, 3583780200LL,
113  -1068242175, 10854718875LL,
114  // R[4]/n^4, polynomial in n of order 4
115  50072287748LL, 3938662680LL, -26314234380LL, 17167059000LL,
116  -4238509275LL, 43418875500LL,
117  // R[5]/n^5, polynomial in n of order 3
118  359094172, -9912730821LL, 5849673480LL, -1255576140, 10854718875LL,
119  // R[6]/n^6, polynomial in n of order 2
120  -16053944387LL, 8733508770LL, -1676521980, 10854718875LL,
121  // R[7]/n^7, polynomial in n of order 1
122  930092876, -162639357, 723647925,
123  // R[8]/n^8, polynomial in n of order 0
124  -673429061, 1929727800,
125  }; // count = 54
126 #else
127 #error "Bad value for GEOGRAPHICLIB_RHUMBAREA_ORDER"
128 #endif
129  static_assert(sizeof(coeff) / sizeof(real) ==
130  ((maxpow_ + 1) * (maxpow_ + 4))/2,
131  "Coefficient array size mismatch for Rhumb");
132  real d = 1;
133  int o = 0;
134  for (int l = 0; l <= maxpow_; ++l) {
135  int m = maxpow_ - l;
136  // R[0] is just an integration constant so it cancels when evaluating a
137  // definite integral. So don't bother computing it. It won't be used
138  // when invoking SinCosSeries.
139  if (l)
140  _rR[l] = d * Math::polyval(m, coeff + o, _ell._n) / coeff[o + m + 1];
141  o += m + 2;
142  d *= _ell._n;
143  }
144  // Post condition: o == sizeof(alpcoeff) / sizeof(real)
145  }
146 
147  const Rhumb& Rhumb::WGS84() {
148  static const Rhumb
149  wgs84(Constants::WGS84_a(), Constants::WGS84_f(), false);
150  return wgs84;
151  }
152 
153  void Rhumb::GenInverse(real lat1, real lon1, real lat2, real lon2,
154  unsigned outmask,
155  real& s12, real& azi12, real& S12) const {
156  real
157  lon12 = Math::AngDiff(lon1, lon2),
158  psi1 = _ell.IsometricLatitude(lat1),
159  psi2 = _ell.IsometricLatitude(lat2),
160  psi12 = psi2 - psi1,
161  h = hypot(lon12, psi12);
162  if (outmask & AZIMUTH)
163  azi12 = Math::atan2d(lon12, psi12);
164  if (outmask & DISTANCE) {
165  real dmudpsi = DIsometricToRectifying(psi2, psi1);
166  s12 = h * dmudpsi * _ell.QuarterMeridian() / Math::qd;
167  }
168  if (outmask & AREA)
169  S12 = _c2 * lon12 *
170  MeanSinXi(psi2 * Math::degree(), psi1 * Math::degree());
171  }
172 
173  RhumbLine Rhumb::Line(real lat1, real lon1, real azi12) const
174  { return RhumbLine(*this, lat1, lon1, azi12); }
175 
176  void Rhumb::GenDirect(real lat1, real lon1, real azi12, real s12,
177  unsigned outmask,
178  real& lat2, real& lon2, real& S12) const
179  { Line(lat1, lon1, azi12).GenPosition(s12, outmask, lat2, lon2, S12); }
180 
181  Math::real Rhumb::DE(real x, real y) const {
182  const EllipticFunction& ei = _ell._ell;
183  real d = x - y;
184  if (x * y <= 0)
185  return d != 0 ? (ei.E(x) - ei.E(y)) / d : 1;
186  // See DLMF: Eqs (19.11.2) and (19.11.4) letting
187  // theta -> x, phi -> -y, psi -> z
188  //
189  // (E(x) - E(y)) / d = E(z)/d - k2 * sin(x) * sin(y) * sin(z)/d
190  //
191  // tan(z/2) = (sin(x)*Delta(y) - sin(y)*Delta(x)) / (cos(x) + cos(y))
192  // = d * Dsin(x,y) * (sin(x) + sin(y))/(cos(x) + cos(y)) /
193  // (sin(x)*Delta(y) + sin(y)*Delta(x))
194  // = t = d * Dt
195  // sin(z) = 2*t/(1+t^2); cos(z) = (1-t^2)/(1+t^2)
196  // Alt (this only works for |z| <= pi/2 -- however, this conditions holds
197  // if x*y > 0):
198  // sin(z) = d * Dsin(x,y) * (sin(x) + sin(y))/
199  // (sin(x)*cos(y)*Delta(y) + sin(y)*cos(x)*Delta(x))
200  // cos(z) = sqrt((1-sin(z))*(1+sin(z)))
201  real sx = sin(x), sy = sin(y), cx = cos(x), cy = cos(y);
202  real Dt = Dsin(x, y) * (sx + sy) /
203  ((cx + cy) * (sx * ei.Delta(sy, cy) + sy * ei.Delta(sx, cx))),
204  t = d * Dt, Dsz = 2 * Dt / (1 + t*t),
205  sz = d * Dsz, cz = (1 - t) * (1 + t) / (1 + t*t);
206  return ((sz != 0 ? ei.E(sz, cz, ei.Delta(sz, cz)) / sz : 1)
207  - ei.k2() * sx * sy) * Dsz;
208  }
209 
210  Math::real Rhumb::DRectifying(real latx, real laty) const {
211  real
212  tbetx = _ell._f1 * Math::tand(latx),
213  tbety = _ell._f1 * Math::tand(laty);
214  return (Math::pi()/2) * _ell._b * _ell._f1 * DE(atan(tbetx), atan(tbety))
215  * Dtan(latx, laty) * Datan(tbetx, tbety) / _ell.QuarterMeridian();
216  }
217 
218  Math::real Rhumb::DIsometric(real latx, real laty) const {
219  real
220  phix = latx * Math::degree(), tx = Math::tand(latx),
221  phiy = laty * Math::degree(), ty = Math::tand(laty);
222  return Dasinh(tx, ty) * Dtan(latx, laty)
223  - Deatanhe(sin(phix), sin(phiy)) * Dsin(phix, phiy);
224  }
225 
226  Math::real Rhumb::SinCosSeries(bool sinp,
227  real x, real y, const real c[], int n) {
228  // N.B. n >= 0 and c[] has n+1 elements 0..n, of which c[0] is ignored.
229  //
230  // Use Clenshaw summation to evaluate
231  // m = (g(x) + g(y)) / 2 -- mean value
232  // s = (g(x) - g(y)) / (x - y) -- average slope
233  // where
234  // g(x) = sum(c[j]*SC(2*j*x), j = 1..n)
235  // SC = sinp ? sin : cos
236  // CS = sinp ? cos : sin
237  //
238  // This function returns only s; m is discarded.
239  //
240  // Write
241  // t = [m; s]
242  // t = sum(c[j] * f[j](x,y), j = 1..n)
243  // where
244  // f[j](x,y) = [ (SC(2*j*x)+SC(2*j*y))/2 ]
245  // [ (SC(2*j*x)-SC(2*j*y))/d ]
246  //
247  // = [ cos(j*d)*SC(j*p) ]
248  // [ +/-(2/d)*sin(j*d)*CS(j*p) ]
249  // (+/- = sinp ? + : -) and
250  // p = x+y, d = x-y
251  //
252  // f[j+1](x,y) = A * f[j](x,y) - f[j-1](x,y)
253  //
254  // A = [ 2*cos(p)*cos(d) -sin(p)*sin(d)*d]
255  // [ -4*sin(p)*sin(d)/d 2*cos(p)*cos(d) ]
256  //
257  // Let b[n+1] = b[n+2] = [0 0; 0 0]
258  // b[j] = A * b[j+1] - b[j+2] + c[j] * I for j = n..1
259  // t = (c[0] * I - b[2]) * f[0](x,y) + b[1] * f[1](x,y)
260  // c[0] is not accessed for s = t[2]
261  real p = x + y, d = x - y,
262  cp = cos(p), cd = cos(d),
263  sp = sin(p), sd = d != 0 ? sin(d)/d : 1,
264  m = 2 * cp * cd, s = sp * sd;
265  // 2x2 matrices stored in row-major order
266  const real a[4] = {m, -s * d * d, -4 * s, m};
267  real ba[4] = {0, 0, 0, 0};
268  real bb[4] = {0, 0, 0, 0};
269  real* b1 = ba;
270  real* b2 = bb;
271  if (n > 0) b1[0] = b1[3] = c[n];
272  for (int j = n - 1; j > 0; --j) { // j = n-1 .. 1
273  swap(b1, b2);
274  // b1 = A * b2 - b1 + c[j] * I
275  b1[0] = a[0] * b2[0] + a[1] * b2[2] - b1[0] + c[j];
276  b1[1] = a[0] * b2[1] + a[1] * b2[3] - b1[1];
277  b1[2] = a[2] * b2[0] + a[3] * b2[2] - b1[2];
278  b1[3] = a[2] * b2[1] + a[3] * b2[3] - b1[3] + c[j];
279  }
280  // Here are the full expressions for m and s
281  // m = (c[0] - b2[0]) * f01 - b2[1] * f02 + b1[0] * f11 + b1[1] * f12;
282  // s = - b2[2] * f01 + (c[0] - b2[3]) * f02 + b1[2] * f11 + b1[3] * f12;
283  if (sinp) {
284  // real f01 = 0, f02 = 0;
285  real f11 = cd * sp, f12 = 2 * sd * cp;
286  // m = b1[0] * f11 + b1[1] * f12;
287  s = b1[2] * f11 + b1[3] * f12;
288  } else {
289  // real f01 = 1, f02 = 0;
290  real f11 = cd * cp, f12 = - 2 * sd * sp;
291  // m = c[0] - b2[0] + b1[0] * f11 + b1[1] * f12;
292  s = - b2[2] + b1[2] * f11 + b1[3] * f12;
293  }
294  return s;
295  }
296 
297  Math::real Rhumb::DConformalToRectifying(real chix, real chiy) const {
298  return 1 + SinCosSeries(true, chix, chiy,
299  _ell.ConformalToRectifyingCoeffs(), tm_maxord);
300  }
301 
302  Math::real Rhumb::DRectifyingToConformal(real mux, real muy) const {
303  return 1 - SinCosSeries(true, mux, muy,
304  _ell.RectifyingToConformalCoeffs(), tm_maxord);
305  }
306 
307  Math::real Rhumb::DIsometricToRectifying(real psix, real psiy) const {
308  if (_exact) {
309  real
310  latx = _ell.InverseIsometricLatitude(psix),
311  laty = _ell.InverseIsometricLatitude(psiy);
312  return DRectifying(latx, laty) / DIsometric(latx, laty);
313  } else {
314  psix *= Math::degree();
315  psiy *= Math::degree();
316  return DConformalToRectifying(gd(psix), gd(psiy)) * Dgd(psix, psiy);
317  }
318  }
319 
320  Math::real Rhumb::DRectifyingToIsometric(real mux, real muy) const {
321  real
322  latx = _ell.InverseRectifyingLatitude(mux/Math::degree()),
323  laty = _ell.InverseRectifyingLatitude(muy/Math::degree());
324  return _exact ?
325  DIsometric(latx, laty) / DRectifying(latx, laty) :
326  Dgdinv(Math::taupf(Math::tand(latx), _ell._es),
327  Math::taupf(Math::tand(laty), _ell._es)) *
328  DRectifyingToConformal(mux, muy);
329  }
330 
331  Math::real Rhumb::MeanSinXi(real psix, real psiy) const {
332  return Dlog(cosh(psix), cosh(psiy)) * Dcosh(psix, psiy)
333  + SinCosSeries(false, gd(psix), gd(psiy), _rR, maxpow_) * Dgd(psix, psiy);
334  }
335 
336  RhumbLine::RhumbLine(const Rhumb& rh, real lat1, real lon1, real azi12)
337  : _rh(rh)
338  , _lat1(Math::LatFix(lat1))
339  , _lon1(lon1)
340  , _azi12(Math::AngNormalize(azi12))
341  {
342  real alp12 = _azi12 * Math::degree();
343  _salp = _azi12 == -Math::hd ? 0 : sin(alp12);
344  _calp = fabs(_azi12) == Math::qd ? 0 : cos(alp12);
345  _mu1 = _rh._ell.RectifyingLatitude(lat1);
346  _psi1 = _rh._ell.IsometricLatitude(lat1);
347  _r1 = _rh._ell.CircleRadius(lat1);
348  }
349 
350  void RhumbLine::GenPosition(real s12, unsigned outmask,
351  real& lat2, real& lon2, real& S12) const {
352  real
353  mu12 = s12 * _calp * Math::qd / _rh._ell.QuarterMeridian(),
354  mu2 = _mu1 + mu12;
355  real psi2, lat2x, lon2x;
356  if (fabs(mu2) <= Math::qd) {
357  if (_calp != 0) {
358  lat2x = _rh._ell.InverseRectifyingLatitude(mu2);
359  real psi12 = _rh.DRectifyingToIsometric( mu2 * Math::degree(),
360  _mu1 * Math::degree()) * mu12;
361  lon2x = _salp * psi12 / _calp;
362  psi2 = _psi1 + psi12;
363  } else {
364  lat2x = _lat1;
365  lon2x = _salp * s12 / (_r1 * Math::degree());
366  psi2 = _psi1;
367  }
368  if (outmask & AREA)
369  S12 = _rh._c2 * lon2x *
370  _rh.MeanSinXi(_psi1 * Math::degree(), psi2 * Math::degree());
371  lon2x = outmask & LONG_UNROLL ? _lon1 + lon2x :
372  Math::AngNormalize(Math::AngNormalize(_lon1) + lon2x);
373  } else {
374  // Reduce to the interval [-180, 180)
375  mu2 = Math::AngNormalize(mu2);
376  // Deal with points on the anti-meridian
377  if (fabs(mu2) > Math::qd) mu2 = Math::AngNormalize(Math::hd - mu2);
378  lat2x = _rh._ell.InverseRectifyingLatitude(mu2);
379  lon2x = Math::NaN();
380  if (outmask & AREA)
381  S12 = Math::NaN();
382  }
383  if (outmask & LATITUDE) lat2 = lat2x;
384  if (outmask & LONGITUDE) lon2 = lon2x;
385  }
386 
387 } // namespace GeographicLib
GeographicLib::Math::real real
Definition: GeodSolve.cpp:31
Header for GeographicLib::Rhumb and GeographicLib::RhumbLine classes.
Math::real IsometricLatitude(real phi) const
Definition: Ellipsoid.cpp:89
Math::real QuarterMeridian() const
Definition: Ellipsoid.cpp:42
Math::real InverseIsometricLatitude(real psi) const
Definition: Ellipsoid.cpp:93
Math::real InverseRectifyingLatitude(real mu) const
Definition: Ellipsoid.cpp:70
Elliptic integrals and functions.
Math::real Delta(real sn, real cn) const
Mathematical functions needed by GeographicLib.
Definition: Math.hpp:76
static T degree()
Definition: Math.hpp:200
static T tand(T x)
Definition: Math.cpp:172
static T atan2d(T y, T x)
Definition: Math.cpp:183
static T AngNormalize(T x)
Definition: Math.cpp:71
static T pi()
Definition: Math.hpp:190
static T NaN()
Definition: Math.cpp:250
static T polyval(int N, const T p[], T x)
Definition: Math.hpp:271
static T AngDiff(T x, T y, T &e)
Definition: Math.cpp:82
@ hd
degrees per half turn
Definition: Math.hpp:144
@ qd
degrees per quarter turn
Definition: Math.hpp:141
Find a sequence of points on a single rhumb line.
Definition: Rhumb.hpp:458
void GenPosition(real s12, unsigned outmask, real &lat2, real &lon2, real &S12) const
Definition: Rhumb.cpp:350
Solve of the direct and inverse rhumb problems.
Definition: Rhumb.hpp:66
RhumbLine Line(real lat1, real lon1, real azi12) const
Definition: Rhumb.cpp:173
Rhumb(real a, real f, bool exact=true)
Definition: Rhumb.cpp:23
friend class RhumbLine
Definition: Rhumb.hpp:69
static const Rhumb & WGS84()
Definition: Rhumb.cpp:147
Namespace for GeographicLib.
Definition: Accumulator.cpp:12
void swap(GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &a, GeographicLib::NearestNeighbor< dist_t, pos_t, distfun_t > &b)