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SphericalEngine.cpp
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/**
* \file SphericalEngine.cpp
* \brief Implementation for GeographicLib::SphericalEngine class
*
* Copyright (c) Charles Karney (2011-2022) <karney@alum.mit.edu> and licensed
* under the MIT/X11 License. For more information, see
* https://geographiclib.sourceforge.io/
*
* The general sum is\verbatim
V(r, theta, lambda) = sum(n = 0..N) sum(m = 0..n)
q^(n+1) * (C[n,m] * cos(m*lambda) + S[n,m] * sin(m*lambda)) * P[n,m](t)
\endverbatim
* where <tt>t = cos(theta)</tt>, <tt>q = a/r</tt>. In addition write <tt>u =
* sin(theta)</tt>.
*
* <tt>P[n,m]</tt> is a normalized associated Legendre function of degree
* <tt>n</tt> and order <tt>m</tt>. Here the formulas are given for full
* normalized functions (usually denoted <tt>Pbar</tt>).
*
* Rewrite outer sum\verbatim
V(r, theta, lambda) = sum(m = 0..N) * P[m,m](t) * q^(m+1) *
[Sc[m] * cos(m*lambda) + Ss[m] * sin(m*lambda)]
\endverbatim
* where the inner sums are\verbatim
Sc[m] = sum(n = m..N) q^(n-m) * C[n,m] * P[n,m](t)/P[m,m](t)
Ss[m] = sum(n = m..N) q^(n-m) * S[n,m] * P[n,m](t)/P[m,m](t)
\endverbatim
* Evaluate sums via Clenshaw method. The overall framework is similar to
* Deakin with the following changes:
* - Clenshaw summation is used to roll the computation of
* <tt>cos(m*lambda)</tt> and <tt>sin(m*lambda)</tt> into the evaluation of
* the outer sum (rather than independently computing an array of these
* trigonometric terms).
* - Scale the coefficients to guard against overflow when <tt>N</tt> is large.
* .
* For the general framework of Clenshaw, see
* http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html
*
* Let\verbatim
S = sum(k = 0..N) c[k] * F[k](x)
F[n+1](x) = alpha[n](x) * F[n](x) + beta[n](x) * F[n-1](x)
\endverbatim
* Evaluate <tt>S</tt> with\verbatim
y[N+2] = y[N+1] = 0
y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]
S = c[0] * F[0] + y[1] * F[1] + beta[1] * F[0] * y[2]
\endverbatim
* \e IF <tt>F[0](x) = 1</tt> and <tt>beta(0,x) = 0</tt>, then <tt>F[1](x) =
* alpha(0,x)</tt> and we can continue the recursion for <tt>y[k]</tt> until
* <tt>y[0]</tt>, giving\verbatim
S = y[0]
\endverbatim
*
* Evaluating the inner sum\verbatim
l = n-m; n = l+m
Sc[m] = sum(l = 0..N-m) C[l+m,m] * q^l * P[l+m,m](t)/P[m,m](t)
F[l] = q^l * P[l+m,m](t)/P[m,m](t)
\endverbatim
* Holmes + Featherstone, Eq. (11), give\verbatim
P[n,m] = sqrt((2*n-1)*(2*n+1)/((n-m)*(n+m))) * t * P[n-1,m] -
sqrt((2*n+1)*(n+m-1)*(n-m-1)/((n-m)*(n+m)*(2*n-3))) * P[n-2,m]
\endverbatim
* thus\verbatim
alpha[l] = t * q * sqrt(((2*n+1)*(2*n+3))/
((n-m+1)*(n+m+1)))
beta[l+1] = - q^2 * sqrt(((n-m+1)*(n+m+1)*(2*n+5))/
((n-m+2)*(n+m+2)*(2*n+1)))
\endverbatim
* In this case, <tt>F[0] = 1</tt> and <tt>beta[0] = 0</tt>, so the <tt>Sc[m]
* = y[0]</tt>.
*
* Evaluating the outer sum\verbatim
V = sum(m = 0..N) Sc[m] * q^(m+1) * cos(m*lambda) * P[m,m](t)
+ sum(m = 0..N) Ss[m] * q^(m+1) * cos(m*lambda) * P[m,m](t)
F[m] = q^(m+1) * cos(m*lambda) * P[m,m](t) [or sin(m*lambda)]
\endverbatim
* Holmes + Featherstone, Eq. (13), give\verbatim
P[m,m] = u * sqrt((2*m+1)/((m>1?2:1)*m)) * P[m-1,m-1]
\endverbatim
* also, we have\verbatim
cos((m+1)*lambda) = 2*cos(lambda)*cos(m*lambda) - cos((m-1)*lambda)
\endverbatim
* thus\verbatim
alpha[m] = 2*cos(lambda) * sqrt((2*m+3)/(2*(m+1))) * u * q
= cos(lambda) * sqrt( 2*(2*m+3)/(m+1) ) * u * q
beta[m+1] = -sqrt((2*m+3)*(2*m+5)/(4*(m+1)*(m+2))) * u^2 * q^2
* (m == 0 ? sqrt(2) : 1)
\endverbatim
* Thus\verbatim
F[0] = q [or 0]
F[1] = cos(lambda) * sqrt(3) * u * q^2 [or sin(lambda)]
beta[1] = - sqrt(15/4) * u^2 * q^2
\endverbatim
*
* Here is how the various components of the gradient are computed
*
* Differentiate wrt <tt>r</tt>\verbatim
d q^(n+1) / dr = (-1/r) * (n+1) * q^(n+1)
\endverbatim
* so multiply <tt>C[n,m]</tt> by <tt>n+1</tt> in inner sum and multiply the
* sum by <tt>-1/r</tt>.
*
* Differentiate wrt <tt>lambda</tt>\verbatim
d cos(m*lambda) = -m * sin(m*lambda)
d sin(m*lambda) = m * cos(m*lambda)
\endverbatim
* so multiply terms by <tt>m</tt> in outer sum and swap sine and cosine
* variables.
*
* Differentiate wrt <tt>theta</tt>\verbatim
dV/dtheta = V' = -u * dV/dt = -u * V'
\endverbatim
* here <tt>'</tt> denotes differentiation wrt <tt>theta</tt>.\verbatim
d/dtheta (Sc[m] * P[m,m](t)) = Sc'[m] * P[m,m](t) + Sc[m] * P'[m,m](t)
\endverbatim
* Now <tt>P[m,m](t) = const * u^m</tt>, so <tt>P'[m,m](t) = m * t/u *
* P[m,m](t)</tt>, thus\verbatim
d/dtheta (Sc[m] * P[m,m](t)) = (Sc'[m] + m * t/u * Sc[m]) * P[m,m](t)
\endverbatim
* Clenshaw recursion for <tt>Sc[m]</tt> reads\verbatim
y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]
\endverbatim
* Substituting <tt>alpha[k] = const * t</tt>, <tt>alpha'[k] = -u/t *
* alpha[k]</tt>, <tt>beta'[k] = c'[k] = 0</tt> gives\verbatim
y'[k] = alpha[k] * y'[k+1] + beta[k+1] * y'[k+2] - u/t * alpha[k] * y[k+1]
\endverbatim
*
* Finally, given the derivatives of <tt>V</tt>, we can compute the components
* of the gradient in spherical coordinates and transform the result into
* cartesian coordinates.
**********************************************************************/
#include <GeographicLib/SphericalEngine.hpp>
#include <GeographicLib/CircularEngine.hpp>
#include <GeographicLib/Utility.hpp>
#if defined(_MSC_VER)
// Squelch warnings about potentially uninitialized local variables
# pragma warning (disable: 4701)
#endif
namespace GeographicLib {
using namespace std;
vector<Math::real>& SphericalEngine::sqrttable() {
static vector<real> sqrttable(0);
return sqrttable;
}
template<bool gradp, SphericalEngine::normalization norm, int L>
Math::real SphericalEngine::Value(const coeff c[], const real f[],
real x, real y, real z, real a,
real& gradx, real& grady, real& gradz)
{
static_assert(L > 0, "L must be positive");
static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
int N = c[0].nmx(), M = c[0].mmx();
real
p = hypot(x, y),
cl = p != 0 ? x / p : 1, // cos(lambda); at pole, pick lambda = 0
sl = p != 0 ? y / p : 0, // sin(lambda)
r = hypot(z, p),
t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
u = r != 0 ? fmax(p / r, eps()) : 1, // sin(theta); but avoid the pole
q = a / r;
real
q2 = Math::sq(q),
uq = u * q,
uq2 = Math::sq(uq),
tu = t / u;
// Initialize outer sum
real vc = 0, vc2 = 0, vs = 0, vs2 = 0; // v [N + 1], v [N + 2]
// vr, vt, vl and similar w variable accumulate the sums for the
// derivatives wrt r, theta, and lambda, respectively.
real vrc = 0, vrc2 = 0, vrs = 0, vrs2 = 0; // vr[N + 1], vr[N + 2]
real vtc = 0, vtc2 = 0, vts = 0, vts2 = 0; // vt[N + 1], vt[N + 2]
real vlc = 0, vlc2 = 0, vls = 0, vls2 = 0; // vl[N + 1], vl[N + 2]
int k[L];
const vector<real>& root( sqrttable() );
for (int m = M; m >= 0; --m) { // m = M .. 0
// Initialize inner sum
real
wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
for (int l = 0; l < L; ++l)
k[l] = c[l].index(N, m) + 1;
for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
real w, A, Ax, B, R; // alpha[l], beta[l + 1]
switch (norm) {
case FULL:
w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
Ax = q * w * root[2 * n + 3];
A = t * Ax;
B = - q2 * root[2 * n + 5] /
(w * root[n - m + 2] * root[n + m + 2]);
break;
case SCHMIDT:
w = root[n - m + 1] * root[n + m + 1];
Ax = q * (2 * n + 1) / w;
A = t * Ax;
B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
break;
default: break; // To suppress warning message from Visual Studio
}
R = c[0].Cv(--k[0]);
for (int l = 1; l < L; ++l)
R += c[l].Cv(--k[l], n, m, f[l]);
R *= scale();
w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
if (gradp) {
w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
}
if (m) {
R = c[0].Sv(k[0]);
for (int l = 1; l < L; ++l)
R += c[l].Sv(k[l], n, m, f[l]);
R *= scale();
w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
if (gradp) {
w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
}
}
}
// Now Sc[m] = wc, Ss[m] = ws
// Sc'[m] = wtc, Ss'[m] = wtc
if (m) {
real v, A, B; // alpha[m], beta[m + 1]
switch (norm) {
case FULL:
v = root[2] * root[2 * m + 3] / root[m + 1];
A = cl * v * uq;
B = - v * root[2 * m + 5] / (root[8] * root[m + 2]) * uq2;
break;
case SCHMIDT:
v = root[2] * root[2 * m + 1] / root[m + 1];
A = cl * v * uq;
B = - v * root[2 * m + 3] / (root[8] * root[m + 2]) * uq2;
break;
default: break; // To suppress warning message from Visual Studio
}
v = A * vc + B * vc2 + wc ; vc2 = vc ; vc = v;
v = A * vs + B * vs2 + ws ; vs2 = vs ; vs = v;
if (gradp) {
// Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
wtc += m * tu * wc; wts += m * tu * ws;
v = A * vrc + B * vrc2 + wrc; vrc2 = vrc; vrc = v;
v = A * vrs + B * vrs2 + wrs; vrs2 = vrs; vrs = v;
v = A * vtc + B * vtc2 + wtc; vtc2 = vtc; vtc = v;
v = A * vts + B * vts2 + wts; vts2 = vts; vts = v;
v = A * vlc + B * vlc2 + m*ws; vlc2 = vlc; vlc = v;
v = A * vls + B * vls2 - m*wc; vls2 = vls; vls = v;
}
} else {
real A, B, qs;
switch (norm) {
case FULL:
A = root[3] * uq; // F[1]/(q*cl) or F[1]/(q*sl)
B = - root[15]/2 * uq2; // beta[1]/q
break;
case SCHMIDT:
A = uq;
B = - root[3]/2 * uq2;
break;
default: break; // To suppress warning message from Visual Studio
}
qs = q / scale();
vc = qs * (wc + A * (cl * vc + sl * vs ) + B * vc2);
if (gradp) {
qs /= r;
// The components of the gradient in spherical coordinates are
// r: dV/dr
// theta: 1/r * dV/dtheta
// lambda: 1/(r*u) * dV/dlambda
vrc = - qs * (wrc + A * (cl * vrc + sl * vrs) + B * vrc2);
vtc = qs * (wtc + A * (cl * vtc + sl * vts) + B * vtc2);
vlc = qs / u * ( A * (cl * vlc + sl * vls) + B * vlc2);
}
}
}
if (gradp) {
// Rotate into cartesian (geocentric) coordinates
gradx = cl * (u * vrc + t * vtc) - sl * vlc;
grady = sl * (u * vrc + t * vtc) + cl * vlc;
gradz = t * vrc - u * vtc ;
}
return vc;
}
template<bool gradp, SphericalEngine::normalization norm, int L>
CircularEngine SphericalEngine::Circle(const coeff c[], const real f[],
real p, real z, real a) {
static_assert(L > 0, "L must be positive");
static_assert(norm == FULL || norm == SCHMIDT, "Unknown normalization");
int N = c[0].nmx(), M = c[0].mmx();
real
r = hypot(z, p),
t = r != 0 ? z / r : 0, // cos(theta); at origin, pick theta = pi/2
u = r != 0 ? fmax(p / r, eps()) : 1, // sin(theta); but avoid the pole
q = a / r;
real
q2 = Math::sq(q),
tu = t / u;
CircularEngine circ(M, gradp, norm, a, r, u, t);
int k[L];
const vector<real>& root( sqrttable() );
for (int m = M; m >= 0; --m) { // m = M .. 0
// Initialize inner sum
real
wc = 0, wc2 = 0, ws = 0, ws2 = 0, // w [N - m + 1], w [N - m + 2]
wrc = 0, wrc2 = 0, wrs = 0, wrs2 = 0, // wr[N - m + 1], wr[N - m + 2]
wtc = 0, wtc2 = 0, wts = 0, wts2 = 0; // wt[N - m + 1], wt[N - m + 2]
for (int l = 0; l < L; ++l)
k[l] = c[l].index(N, m) + 1;
for (int n = N; n >= m; --n) { // n = N .. m; l = N - m .. 0
real w, A, Ax, B, R; // alpha[l], beta[l + 1]
switch (norm) {
case FULL:
w = root[2 * n + 1] / (root[n - m + 1] * root[n + m + 1]);
Ax = q * w * root[2 * n + 3];
A = t * Ax;
B = - q2 * root[2 * n + 5] /
(w * root[n - m + 2] * root[n + m + 2]);
break;
case SCHMIDT:
w = root[n - m + 1] * root[n + m + 1];
Ax = q * (2 * n + 1) / w;
A = t * Ax;
B = - q2 * w / (root[n - m + 2] * root[n + m + 2]);
break;
default: break; // To suppress warning message from Visual Studio
}
R = c[0].Cv(--k[0]);
for (int l = 1; l < L; ++l)
R += c[l].Cv(--k[l], n, m, f[l]);
R *= scale();
w = A * wc + B * wc2 + R; wc2 = wc; wc = w;
if (gradp) {
w = A * wrc + B * wrc2 + (n + 1) * R; wrc2 = wrc; wrc = w;
w = A * wtc + B * wtc2 - u*Ax * wc2; wtc2 = wtc; wtc = w;
}
if (m) {
R = c[0].Sv(k[0]);
for (int l = 1; l < L; ++l)
R += c[l].Sv(k[l], n, m, f[l]);
R *= scale();
w = A * ws + B * ws2 + R; ws2 = ws; ws = w;
if (gradp) {
w = A * wrs + B * wrs2 + (n + 1) * R; wrs2 = wrs; wrs = w;
w = A * wts + B * wts2 - u*Ax * ws2; wts2 = wts; wts = w;
}
}
}
if (!gradp)
circ.SetCoeff(m, wc, ws);
else {
// Include the terms Sc[m] * P'[m,m](t) and Ss[m] * P'[m,m](t)
wtc += m * tu * wc; wts += m * tu * ws;
circ.SetCoeff(m, wc, ws, wrc, wrs, wtc, wts);
}
}
return circ;
}
void SphericalEngine::RootTable(int N) {
// Need square roots up to max(2 * N + 5, 15).
vector<real>& root( sqrttable() );
int L = max(2 * N + 5, 15) + 1, oldL = int(root.size());
if (oldL >= L)
return;
root.resize(L);
for (int l = oldL; l < L; ++l)
root[l] = sqrt(real(l));
}
void SphericalEngine::coeff::readcoeffs(istream& stream, int& N, int& M,
vector<real>& C,
vector<real>& S,
bool truncate) {
if (truncate) {
if (!((N >= M && M >= 0) || (N == -1 && M == -1)))
// The last condition is that M = -1 implies N = -1.
throw GeographicErr("Bad requested degree and order " +
Utility::str(N) + " " + Utility::str(M));
}
int nm[2];
Utility::readarray<int, int, false>(stream, nm, 2);
int N0 = nm[0], M0 = nm[1];
if (!((N0 >= M0 && M0 >= 0) || (N0 == -1 && M0 == -1)))
// The last condition is that M0 = -1 implies N0 = -1.
throw GeographicErr("Bad degree and order " +
Utility::str(N0) + " " + Utility::str(M0));
N = truncate ? min(N, N0) : N0;
M = truncate ? min(M, M0) : M0;
C.resize(SphericalEngine::coeff::Csize(N, M));
S.resize(SphericalEngine::coeff::Ssize(N, M));
int skip = (SphericalEngine::coeff::Csize(N0, M0) -
SphericalEngine::coeff::Csize(N0, M )) * sizeof(double);
if (N == N0) {
Utility::readarray<double, real, false>(stream, C);
if (skip) stream.seekg(streamoff(skip), ios::cur);
Utility::readarray<double, real, false>(stream, S);
if (skip) stream.seekg(streamoff(skip), ios::cur);
} else {
for (int m = 0, k = 0; m <= M; ++m) {
Utility::readarray<double, real, false>(stream, &C[k], N + 1 - m);
stream.seekg((N0 - N) * sizeof(double), ios::cur);
k += N + 1 - m;
}
if (skip) stream.seekg(streamoff(skip), ios::cur);
for (int m = 1, k = 0; m <= M; ++m) {
Utility::readarray<double, real, false>(stream, &S[k], N + 1 - m);
stream.seekg((N0 - N) * sizeof(double), ios::cur);
k += N + 1 - m;
}
if (skip) stream.seekg(streamoff(skip), ios::cur);
}
return;
}
/// \cond SKIP
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<true, SphericalEngine::FULL, 1>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<false, SphericalEngine::FULL, 1>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 1>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 1>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<true, SphericalEngine::FULL, 2>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<false, SphericalEngine::FULL, 2>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 2>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 2>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<true, SphericalEngine::FULL, 3>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<false, SphericalEngine::FULL, 3>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<true, SphericalEngine::SCHMIDT, 3>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template Math::real GEOGRAPHICLIB_EXPORT
SphericalEngine::Value<false, SphericalEngine::SCHMIDT, 3>
(const coeff[], const real[], real, real, real, real, real&, real&, real&);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<true, SphericalEngine::FULL, 1>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<false, SphericalEngine::FULL, 1>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 1>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 1>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<true, SphericalEngine::FULL, 2>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<false, SphericalEngine::FULL, 2>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 2>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 2>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<true, SphericalEngine::FULL, 3>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<false, SphericalEngine::FULL, 3>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<true, SphericalEngine::SCHMIDT, 3>
(const coeff[], const real[], real, real, real);
template CircularEngine GEOGRAPHICLIB_EXPORT
SphericalEngine::Circle<false, SphericalEngine::SCHMIDT, 3>
(const coeff[], const real[], real, real, real);
/// \endcond
} // namespace GeographicLib
CircularEngine.hpp
Header for GeographicLib::CircularEngine class.
GEOGRAPHICLIB_EXPORT
#define GEOGRAPHICLIB_EXPORT
Definition Constants.hpp:67
SphericalEngine.hpp
Header for GeographicLib::SphericalEngine class.
Utility.hpp
Header for GeographicLib::Utility class.
GeographicLib::CircularEngine
Spherical harmonic sums for a circle.
Definition CircularEngine.hpp:52
GeographicLib::GeographicErr
Exception handling for GeographicLib.
Definition Constants.hpp:316
GeographicLib::Math::sq
static T sq(T x)
Definition Math.hpp:221
GeographicLib::Math::real
double real
Definition Math.hpp:110
GeographicLib::SphericalEngine::coeff
Package up coefficients for SphericalEngine.
Definition SphericalEngine.hpp:99
GeographicLib::SphericalEngine::coeff::N
int N() const
Definition SphericalEngine.hpp:173
GeographicLib::SphericalEngine::coeff::Sv
Math::real Sv(int k) const
Definition SphericalEngine.hpp:204
GeographicLib::SphericalEngine::coeff::Csize
static int Csize(int N, int M)
Definition SphericalEngine.hpp:238
GeographicLib::SphericalEngine::coeff::Ssize
static int Ssize(int N, int M)
Definition SphericalEngine.hpp:249
GeographicLib::SphericalEngine::coeff::readcoeffs
static void readcoeffs(std::istream &stream, int &N, int &M, std::vector< real > &C, std::vector< real > &S, bool truncate=false)
Definition SphericalEngine.cpp:384
GeographicLib::SphericalEngine::coeff::nmx
int nmx() const
Definition SphericalEngine.hpp:177
GeographicLib::SphericalEngine::coeff::mmx
int mmx() const
Definition SphericalEngine.hpp:181
GeographicLib::SphericalEngine::coeff::Cv
Math::real Cv(int k) const
Definition SphericalEngine.hpp:197
GeographicLib::SphericalEngine::Value
static Math::real Value(const coeff c[], const real f[], real x, real y, real z, real a, real &gradx, real &grady, real &gradz)
Definition SphericalEngine.cpp:152
GeographicLib::SphericalEngine::SCHMIDT
@ SCHMIDT
Definition SphericalEngine.hpp:84
GeographicLib::SphericalEngine::FULL
@ FULL
Definition SphericalEngine.hpp:77
GeographicLib::SphericalEngine::RootTable
static void RootTable(int N)
Definition SphericalEngine.cpp:373
GeographicLib::SphericalEngine::Circle
static CircularEngine Circle(const coeff c[], const real f[], real p, real z, real a)
Definition SphericalEngine.cpp:296
GeographicLib::Utility::readarray
static void readarray(std::istream &str, IntT array[], size_t num)
Definition Utility.hpp:321
GeographicLib::Utility::str
static std::string str(T x, int p=-1)
Definition Utility.hpp:161
GeographicLib
Namespace for GeographicLib.
Definition Accumulator.cpp:12
std
Definition NearestNeighbor.hpp:806