EQ2EMu/EQ2/source/depends/boost_1_72_0/libs/geometry/doc/other/maxima/geod.mac

/*
Copyright (c) 2018 Adeel Ahmad, Islamabad, Pakistan.

Contributed and/or modified by Adeel Ahmad, as part of Google Summer of Code 2018 program.

Use, modification and distribution is subject to the Boost Software License,
Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at
http://www.boost.org/LICENSE_1_0.txt)

This file is converted from GeographicLib, https://geographiclib.sourceforge.io
GeographicLib is originally written by Charles Karney.

Author: Charles Karney (2008-2017)

Last updated version of GeographicLib: 1.49

Original copyright notice:

Copyright (c) Charles Karney (2009-2015) <charles@karney.com> and
licensed under the MIT/X11 License. For more information, see
https://geographiclib.sourceforge.io

Compute the series expansions for the ellipsoidal geodesic problem.

References:

   Charles F. F. Karney,
   Algorithms for geodesics, J. Geodesy 87, 43-55 (2013),
   https://doi.org/10.1007/s00190-012-0578-z
   Addenda: https://geographiclib.sourceforge.io/geod-addenda.html

The code below contains minor modifications to conform with
Boost Geometry style guidelines.

To run the code, start Maxima and enter

  load("geod.mac")$
*/

taylordepth:5$
ataylor(expr,var,ord):=expand(ratdisrep(taylor(expr,var,0,ord)))$
jtaylor(expr,var1,var2,ord):=block([zz],expand(subst([zz=1],
ratdisrep(taylor(subst([var1=zz*var1,var2=zz*var2],expr),zz,0,ord)))))$

computeI1(maxpow):=block([sintegrand,sintegrandexp,s,sigma,tau1,k2,eps],
  sintegrand:sqrt(1+k2*sin(sigma)^2),
  sintegrandexp:ataylor(
      (1-eps)*subst([k2=4*eps/(1-eps)^2],sintegrand),
      eps,maxpow),
  s:trigreduce(integrate(sintegrandexp,sigma)),
  s:s-subst(sigma=0,s),
  A1:expand(subst(sigma=2*%pi,s)/(2*%pi)),
  tau1:ataylor(s/A1,eps,maxpow),
  for i:1 thru maxpow do C1[i]:coeff(tau1,sin(2*i*sigma)),
  if expand(tau1-sigma-sum(C1[i]*sin(2*i*sigma),i,1,maxpow)) # 0
  then error("left over terms in B1"),
  A1:A1/(1-eps),
  'done)$

codeA1(maxpow):=block([tab2:"    ",tab3:"        "],
print("// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
static inline CT evaluate_series_A1(CT eps) {
    CT eps2 = math::sqr(eps);
    CT t;
    switch (SeriesOrder/2) {"),
  for n:0 thru entier(maxpow/2) do block([
    q:horner(ataylor(subst([eps=sqrt(eps2)],A1*(1-eps)-1),eps2,n)),
    linel:1200],
    print(concat(tab2,"case ",string(n),":")),
    print(concat(tab3,"t = ",string(q),";")),
    print(concat(tab3,"break;"))),
  print("    }
    return (t + eps) / (1 - eps);
}"),
'done)$

computeI2(maxpow):=block([sintegrand,sintegrandexp,s,sigma,tau1,k2,eps],
  sintegrand:1/sqrt(1+k2*sin(sigma)^2),
  sintegrandexp:ataylor(
      (1+eps)*subst([k2=4*eps/(1-eps)^2],sintegrand),
      eps,maxpow),
  s:trigreduce(integrate(sintegrandexp,sigma)),
  s:s-subst(sigma=0,s),
  A2:expand(subst(sigma=2*%pi,s)/(2*%pi)),
  tau1:ataylor(s/A2,eps,maxpow),
  for i:1 thru maxpow do C2[i]:coeff(tau1,sin(2*i*sigma)),
  if expand(tau1-sigma-sum(C2[i]*sin(2*i*sigma),i,1,maxpow)) # 0
  then error("left over terms in B2"),
  A2:A2/(1+eps),
  'done)$

codeA2(maxpow):=block([tab2:"    ",tab3:"        "],
print("// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
CT evaluate_series_A2(CT const& eps)
{
    CT const eps2 = math::sqr(eps);
    CT t;
    switch (SeriesOrder/2) {"),
  for n:0 thru entier(maxpow/2) do block([
    q:horner(ataylor(subst([eps=sqrt(eps2)],A2*(1+eps)-1),eps2,n)),
    linel:1200],
    print(concat(tab2,"case ",string(n),":")),
    print(concat(tab3,"t = ",string(q),";")),
    print(concat(tab3,"break;"))),
  print("    }
    return (t - eps) / (1 + eps);
}"),
'done)$

computeI3(maxpow):=block([int,intexp,dlam,eta,del,eps,nu,f,z,n],
  maxpow:maxpow-1,
  int:subst([k2=4*eps/(1-eps)^2],
    (2-f)/(1+(1-f)*sqrt(1+k2*sin(sigma)^2))),
  int:subst([f=2*n/(1+n)],int),
  intexp:jtaylor(int,n,eps,maxpow),
  dlam:trigreduce(integrate(intexp,sigma)),
  dlam:dlam-subst(sigma=0,dlam),
  A3:expand(subst(sigma=2*%pi,dlam)/(2*%pi)),
  eta:jtaylor(dlam/A3,n,eps,maxpow),
  A3:jtaylor(A3,n,eps,maxpow),
  for i:1 thru maxpow do C3[i]:coeff(eta,sin(2*i*sigma)),
  if expand(eta-sigma-sum(C3[i]*sin(2*i*sigma),i,1,maxpow)) # 0
  then error("left over terms in B3"),
  'done)$

codeA3(maxpow):=block([tab2:"    ",tab3:"        "],
print("// The scale factor A3 = mean value of (d/dsigma)I3
static inline void evaluate_series_A3(CT const& n, CT c[])
{
    switch (SeriesOrder) {"),
  for nn:0 thru maxpow do block(
    [q:if nn=0 then 0 else
    jtaylor(subst([n=n],A3),n,eps,nn-1),
    linel:1200],
    print(concat(tab2,"case ",string(nn),":")),
    for i : 0 thru nn-1 do
    print(concat(tab3,"c[",i,"] = ",
        string(horner(coeff(q,eps,i))),";")),
    print(concat(tab3,"break;"))),
  print("    }
}"),
'done)$

codeC1(maxpow):=block([tab2:"    ",tab3:"        "],
  print("// The coefficients C1[l] in the Fourier expansion of B1
static inline evaluate_coeffs_C1(CT eps, CT c[]) {
    CT eps2 = math::sqr(eps);
    CT d = eps;
    switch (SeriesOrder) {"),
  for n:0 thru maxpow do (
    print(concat(tab2,"case ",string(n),":")),
    for m:1 thru n do block([q:d*horner(
        subst([eps=sqrt(eps2)],ataylor(C1[m],eps,n)/eps^m)),
      linel:1200],
      if m>1 then print(concat(tab3,"d *= eps;")),
      print(concat(tab3,"c[",string(m),"] = ",string(q),";"))),
    print(concat(tab3,"break;"))),
  print("    }
}"),
'done)$

revertI1(maxpow):=block([tau,eps,tauacc:1,sigacc:0],
  for n:1 thru maxpow do (
    tauacc:trigreduce(ataylor(
          -sum(C1[j]*sin(2*j*tau),j,1,maxpow-n+1)*tauacc/n,
          eps,maxpow)),
    sigacc:sigacc+expand(diff(tauacc,tau,n-1))),
  for i:1 thru maxpow do C1p[i]:coeff(sigacc,sin(2*i*tau)),
  if expand(sigacc-sum(C1p[i]*sin(2*i*tau),i,1,maxpow)) # 0
  then error("left over terms in B1p"),
  'done)$

codeC1p(maxpow):=block([tab2:"    ",tab3:"        "],
  print("// The coefficients C1p[l] in the Fourier expansion of B1p
static inline evaluate_coeffs_C1p(CT eps, CT c[])
{
    CT const eps2 = math::sqr(eps);
    CT d = eps;
    switch (SeriesOrder) {"),
  for n:0 thru maxpow do (
    print(concat(tab2,"case ",string(n),":")),
    for m:1 thru n do block([q:d*horner(
        subst([eps=sqrt(eps2)],ataylor(C1p[m],eps,n)/eps^m)),
      linel:1200],
      if m>1 then print(concat(tab3,"d *= eps;")),
      print(concat(tab3,"c[",string(m),"] = ",string(q),";"))),
    print(concat(tab3,"break;"))),
  print("    }
}"),
'done)$

codeC2(maxpow):=block([tab2:"    ",tab3:"        "],
print("// The coefficients C2[l] in the Fourier expansion of B2
static inline void evaluate_coeffs_C2(CT const& eps, CT c[])
{
    CT const eps2 = math::sqr(eps);
    CT d = eps;
    switch (SeriesOrder) {"),
  for n:0 thru maxpow do (
    print(concat(tab2,"case ",string(n),":")),
    for m:1 thru n do block([q:d*horner(
        subst([eps=sqrt(eps2)],ataylor(C2[m],eps,n)/eps^m)),
      linel:1200],
      if m>1 then print(concat(tab3,"d *= eps;")),
      print(concat(tab3,"c[",string(m),"] = ",string(q),";"))),
    print(concat(tab3,"break;"))),
print("    }
}"),
'done)$

codeC3(maxpow):=block([tab2:"    ",tab3:"        "],
print("// The coefficients C3[l] in the Fourier expansion of B3
static inline void evaluate_coeffs_C3(CT const& n, CT c[])
{
    const CT n2 = math::sqr(n);
    switch (SeriesOrder) {"),
  for nn:0 thru maxpow do block([c],
    print(concat(tab2,"case ",string(nn),":")),
    c:0,
    for m:1 thru nn-1 do block(
      [q:if nn = 0 then 0 else
      jtaylor(subst([n=n],C3[m]),_n,eps,nn-1),
      linel:1200],
      for j:m thru nn-1 do (
        print(concat(tab3,"c[",c,"] = ",
            string(horner(coeff(q,eps,j))),";")),
        c:c+1)
    ),
    print(concat(tab3,"break;"))),
  print("    }
}"),
'done)$

maxpow:8$

computeI1(maxpow)$
computeI2(maxpow)$
computeI3(maxpow)$

revertI1(maxpow)$
codeA1(maxpow)$
codeA2(maxpow)$
codeA3(maxpow)$

codeC1(maxpow)$
codeC2(maxpow)$
codeC3(maxpow)$

codeC1p(maxpow)$