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SphericalEngine.cpp File Reference

Implementation for GeographicLib::SphericalEngine class. More...

`#include <GeographicLib/SphericalEngine.hpp>`
`#include <GeographicLib/CircularEngine.hpp>`
`#include <GeographicLib/Utility.hpp>`

Go to the source code of this file.

## Namespaces

namespace | GeographicLib
| Namespace for GeographicLib.

## Detailed Description

Implementation for GeographicLib::SphericalEngine class.

Copyright (c) Charles Karney (2011-2022) karne.nosp@m.y@al.nosp@m.um.mi.nosp@m.t.ed.nosp@m.u and licensed under the MIT/X11 License. For more information, see https://geographiclib.sourceforge.io/

The general sum is

     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)

where `t = cos(theta)`, `q = a/r`. In addition write `u = sin(theta)`.

`P[n,m]` is a normalized associated Legendre function of degree `n` and order `m`. Here the formulas are given for full normalized functions (usually denoted `Pbar`).

Rewrite outer sum

     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)]

where the inner sums are

       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)

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 `cos(m*lambda)` and `sin(m*lambda)` 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 `N` is large.

For the general framework of Clenshaw, see http://mathworld.wolfram.com/ClenshawRecurrenceFormula.html

Let

        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)

Evaluate `S` with

        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]

IF `F[0](x) = 1` and `beta(0,x) = 0`, then `F[1](x) = alpha(0,x)` and we can continue the recursion for `y[k]` until `y[0]`, giving

        S = y[0]

Evaluating the inner sum

     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)

Holmes + Featherstone, Eq. (11), give

       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]

thus

       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)))

In this case, `F[0] = 1` and `beta[0] = 0`, so the `Sc[m] = y[0]`.

Evaluating the outer sum

     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)]

Holmes + Featherstone, Eq. (13), give

       P[m,m] = u * sqrt((2*m+1)/((m>1?2:1)*m)) * P[m-1,m-1]

also, we have

       cos((m+1)*lambda) = 2*cos(lambda)*cos(m*lambda) - cos((m-1)*lambda)

thus

       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)

Thus

     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

Here is how the various components of the gradient are computed

Differentiate wrt `r`

       d q^(n+1) / dr = (-1/r) * (n+1) * q^(n+1)

so multiply `C[n,m]` by `n+1` in inner sum and multiply the sum by `-1/r`.

Differentiate wrt `lambda`

       d cos(m*lambda) = -m * sin(m*lambda)
       d sin(m*lambda) =  m * cos(m*lambda)

so multiply terms by `m` in outer sum and swap sine and cosine variables.

Differentiate wrt `theta`

      dV/dtheta = V' = -u * dV/dt = -u * V'

here `'` denotes differentiation wrt `theta`.

       d/dtheta (Sc[m] * P[m,m](t)) = Sc'[m] * P[m,m](t) + Sc[m] * P'[m,m](t)

Now `P[m,m](t) = const * u^m`, so `P'[m,m](t) = m * t/u * P[m,m](t)`, thus

       d/dtheta (Sc[m] * P[m,m](t)) = (Sc'[m] + m * t/u * Sc[m]) * P[m,m](t)

Clenshaw recursion for `Sc[m]` reads

        y[k] = alpha[k] * y[k+1] + beta[k+1] * y[k+2] + c[k]

Substituting `alpha[k] = const * t`, `alpha'[k] = -u/t * alpha[k]`, `beta'[k] = c'[k] = 0` gives

        y'[k] = alpha[k] * y'[k+1] + beta[k+1] * y'[k+2] - u/t * alpha[k] * y[k+1]

Finally, given the derivatives of `V`, we can compute the components of the gradient in spherical coordinates and transform the result into cartesian coordinates.

Definition in file SphericalEngine.cpp.