// The coefficients C2[l] in the Fourier expansion of B2
public static void C2f(double eps, double[] c)
{
double[] coeff =
{
// C2[1]/eps^1, polynomial in eps2 of order 2
1, 2, 16, 32,
// C2[2]/eps^2, polynomial in eps2 of order 2
35, 64, 384, 2048,
// C2[3]/eps^3, polynomial in eps2 of order 1
15, 80, 768,
// C2[4]/eps^4, polynomial in eps2 of order 1
7, 35, 512,
// C2[5]/eps^5, polynomial in eps2 of order 0
63, 1280,
// C2[6]/eps^6, polynomial in eps2 of order 0
77, 2048,
};
double eps2 = GeoMath.Square(eps),
d = eps;
int o = 0;
for (int l = 1; l <= nC2_; ++l)
{ // l is index of C2[l]
int m = (nC2_ - l) / 2; // order of polynomial in eps^2
c[l] = d * GeoMath.PolyVal(m, coeff, o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
}
// The coefficients C1p[l] in the Fourier expansion of B1p
public static void C1pf(double eps, double[] c)
{
double[] coeff =
{
// C1p[1]/eps^1, polynomial in eps2 of order 2
205, -432, 768, 1536,
// C1p[2]/eps^2, polynomial in eps2 of order 2
4005, -4736, 3840, 12288,
// C1p[3]/eps^3, polynomial in eps2 of order 1
-225, 116, 384,
// C1p[4]/eps^4, polynomial in eps2 of order 1
-7173, 2695, 7680,
// C1p[5]/eps^5, polynomial in eps2 of order 0
3467, 7680,
// C1p[6]/eps^6, polynomial in eps2 of order 0
38081, 61440,
};
double eps2 = GeoMath.Square(eps);
double d = eps;;
int o = 0;
for (int l = 1; l <= nC1p_; ++l)
{ // l is index of C1p[l]
int m = (nC1p_ - l) / 2; // order of polynomial in eps^2
c[l] = d * GeoMath.PolyVal(m, coeff, o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
}
// The coefficients C1[l] in the Fourier expansion of B1
public static void C1f(double eps, double[] c)
{
double[] coeff =
{
// C1[1]/eps^1, polynomial in eps2 of order 2
-1, 6, -16, 32,
// C1[2]/eps^2, polynomial in eps2 of order 2
-9, 64, -128, 2048,
// C1[3]/eps^3, polynomial in eps2 of order 1
9, -16, 768,
// C1[4]/eps^4, polynomial in eps2 of order 1
3, -5, 512,
// C1[5]/eps^5, polynomial in eps2 of order 0
-7, 1280,
// C1[6]/eps^6, polynomial in eps2 of order 0
-7, 2048,
};
double eps2 = GeoMath.Square(eps);
double d = eps;
int o = 0;
for (int l = 1; l <= nC1_; ++l)
{ // l is index of C1p[l]
int m = (nC1_ - l) / 2; // order of polynomial in eps^2
c[l] = d * GeoMath.PolyVal(m, coeff, o, eps2) / coeff[o + m + 1];
o += m + 2;
d *= eps;
}
}
// The static const coefficient arrays in the following functions are
// generated by Maxima and give the coefficients of the Taylor expansions for
// the geodesics. The convention on the order of these coefficients is as
// follows:
//
// ascending order in the trigonometric expansion,
// then powers of eps in descending order,
// finally powers of n in descending order.
//
// (For some expansions, only a subset of levels occur.) For each polynomial
// of order n at the lowest level, the (n+1) coefficients of the polynomial
// are followed by a divisor which is applied to the whole polynomial. In
// this way, the coefficients are expressible with no round off error. The
// sizes of the coefficient arrays are:
//
// A1m1f, A2m1f = floor(N/2) + 2
// C1f, C1pf, C2f, A3coeff = (N^2 + 7*N - 2*floor(N/2)) / 4
// C3coeff = (N - 1) * (N^2 + 7*N - 2*floor(N/2)) / 8
// C4coeff = N * (N + 1) * (N + 5) / 6
//
// where N = GEOGRAPHICLIB_GEODESIC_ORDER
// = nA1 = nA2 = nC1 = nC1p = nA3 = nC4
// The scale factor A1-1 = mean value of (d/dsigma)I1 - 1
public static double A1m1f(double eps)
{
double[] coeff =
{
// (1-eps)*A1-1, polynomial in eps2 of order 3
1, 4, 64, 0, 256,
};
int m = nA1_ / 2;
double t = GeoMath.PolyVal(m, coeff, 0, GeoMath.Square(eps)) / coeff[m + 1];
return((t + eps) / (1 - eps));
}
// The scale factor A2-1 = mean value of (d/dsigma)I2 - 1
public static double A2m1f(double eps)
{
double[] coeff =
{
// (eps+1)*A2-1, polynomial in eps2 of order 3
-11, -28, -192, 0, 256,
}; // count = 5
int m = nA2_ / 2;
double t = GeoMath.PolyVal(m, coeff, 0, GeoMath.Square(eps)) / coeff[m + 1];
return((t - eps) / (1 + eps));
}
public static double[] GenerateC4(double n)
{
double[] coeff =
{
97, 15015, // C4[0], coeff of eps^5, polynomial in n of order 0
1088, 156, 45045, // C4[0], coeff of eps^4, polynomial in n of order 1
-224, -4784, 1573, 45045, // C4[0], coeff of eps^3, polynomial in n of order 2
-10656, 14144, -4576, -858, 45045, // C4[0], coeff of eps^2, polynomial in n of order 3
64, 624, -4576, 6864, -3003,15015, // C4[0], coeff of eps^1, polynomial in n of order 4
100, 208, 572, 3432, -12012,30030, 45045, // C4[0], coeff of eps^0, polynomial in n of order 5
1, 9009, // C4[1], coeff of eps^5, polynomial in n of order 0
-2944, 468,135135, // C4[1], coeff of eps^4, polynomial in n of order 1
5792, 1040, -1287,135135, // C4[1], coeff of eps^3, polynomial in n of order 2
5952, -11648, 9152, -2574, 135135, // C4[1], coeff of eps^2, polynomial in n of order 3
-64, -624, 4576, -6864, 3003,135135, // C4[1], coeff of eps^1, polynomial in n of order 4
8, 10725, // C4[2], coeff of eps^5, polynomial in n of order 0
1856, -936,225225, // C4[2], coeff of eps^4, polynomial in n of order 1
-8448, 4992, -1144,225225, // C4[2], coeff of eps^3, polynomial in n of order 2
-1440, 4160, -4576, 1716, 225225, // C4[2], coeff of eps^2, polynomial in n of order 3
-136, 63063, // C4[3], coeff of eps^5, polynomial in n of order 0
1024, -208,105105, // C4[3], coeff of eps^4, polynomial in n of order 1
3584, -3328, 1144,315315, // C4[3], coeff of eps^3, polynomial in n of order 2
-128, 135135, // C4[4], coeff of eps^5, polynomial in n of order 0
-2560, 832,405405, // C4[4], coeff of eps^4, polynomial in n of order 1
128, 99099 // C4[5], coeff of eps^5, polynomial in n of order 0
};
var C4x = new double[nC4x_];
int o = 0, k = 0;
for (int l = 0; l < nC4_; ++l) // l is index of C4[l]
{
for (int j = nC4_ - 1; j >= l; --j) // coeff of eps^j
{
int m = nC4_ - j - 1; // order of polynomial in n
C4x[k++] = GeoMath.PolyVal(m, coeff, o, n) / coeff[o + m + 1];
o += m + 2;
}
}
return(C4x);
}
public static double[] GenerateC3(double n)
{
double[] coeff =
{
3, 128, // C3[1], coeff of eps^5, polynomial in n of order 0
2, 5,128, // C3[1], coeff of eps^4, polynomial in n of order 1
-1, 3, 3, 64, // C3[1], coeff of eps^3, polynomial in n of order 2
-1, 0, 1, 8, // C3[1], coeff of eps^2, polynomial in n of order 2
-1, 1, 4, // C3[1], coeff of eps^1, polynomial in n of order 1
5, 256, // C3[2], coeff of eps^5, polynomial in n of order 0
1, 3,128, // C3[2], coeff of eps^4, polynomial in n of order 1
-3, -2, 3, 64, // C3[2], coeff of eps^3, polynomial in n of order 2
1, -3, 2, 32, // C3[2], coeff of eps^2, polynomial in n of order 2
7, 512, // C3[3], coeff of eps^5, polynomial in n of order 0
-10, 9,384, // C3[3], coeff of eps^4, polynomial in n of order 1
5, -9, 5, 192, // C3[3], coeff of eps^3, polynomial in n of order 2
7, 512, // C3[4], coeff of eps^5, polynomial in n of order 0
-14, 7,512, // C3[4], coeff of eps^4, polynomial in n of order 1
21, 2560 // C3[5], coeff of eps^5, polynomial in n of order 0
};
var C3x = new double[nC3x_];
int o = 0, k = 0;
for (int l = 1; l < nC3_; ++l) // l is index of C3[l]
{
for (int j = nC3_ - 1; j >= l; --j) // coeff of eps^j
{
int m = Math.Min(nC3_ - j - 1, j); // order of polynomial in n
C3x[k++] = GeoMath.PolyVal(m, coeff, o, n) / coeff[o + m + 1];
o += m + 2;
}
}
return(C3x);
}
//TODO: Can make this more efficient by cloning the array once it has been created
public static double[] GenerateA3(double n)
{
double[] coeff =
{
-3,128, // coeff of eps^5, polynomial in n of order 0
-2, -3, 64, // coeff of eps^4, polynomial in n of order 1
-1, -3, -1, 16, // coeff of eps^3, polynomial in n of order 2
3, -1, -2, 8, // coeff of eps^2, polynomial in n of order 2
1, -1, 2, // coeff of eps^1, polynomial in n of order 1
1, 1, // coeff of eps^0, polynomial in n of order 0
};
var A3x = new double[nA3x_];
int o = 0, k = 0;
for (int j = nA3_ - 1; j >= 0; --j) // coeff of eps^j
{
int m = Math.Min(nA3_ - j - 1, j); // order of polynomial in n
A3x[k++] = GeoMath.PolyVal(m, coeff, o, n) / coeff[o + m + 1];
o += m + 2;
}
return(A3x);
}
/**
* Constructor for a ellipsoid with
*
* @param[in] a equatorial radius (meters).
* @param[in] f flattening of ellipsoid. Setting \e f = 0 gives a sphere.
* Negative \e f gives a prolate ellipsoid.
* @param[in] k0 central scale factor.
* @exception GeographicErr if \e a, (1 − \e f) \e a, or \e k0 is
* not positive.
**********************************************************************/
public TransverseMercator(double a, double f, double k0)
{
_a = a;
_f = f;
_k0 = k0;
_e2 = _f * (2 - _f);
_es = (f < 0 ? -1 : 1) * Math.Sqrt(Math.Abs(_e2));
_e2m = 1 - _e2;
// _c = Math.Sqrt( pow(1 + _e, 1 + _e) * pow(1 - _e, 1 - _e) ) )
// See, for example, Lee (1976), p 100.
_c = Math.Sqrt(_e2m) * Math.Exp(GeoMath.Eatanhe(1, _es));
_n = _f / (2 - _f);
if (!(GeoMath.IsFinite(_a) && _a > 0))
{
throw new GeographicException("Equatorial radius is not positive");
}
if (!(GeoMath.IsFinite(_f) && _f < 1))
{
throw new GeographicException("Polar semi-axis is not positive");
}
if (!(GeoMath.IsFinite(_k0) && _k0 > 0))
{
throw new GeographicException("Scale is not positive");
}
double[] b1coeff =
{
// b1*(n+1), polynomial in n2 of order 3
1, 4, 64, 256, 256,
};
double[] alpcoeff =
{
// alp[1]/n^1, polynomial in n of order 5
31564, -66675, 34440, 47250, -100800, 75600, 151200,
// alp[2]/n^2, polynomial in n of order 4
-1983433, 863232, 748608, -1161216, 524160, 1935360,
// alp[3]/n^3, polynomial in n of order 3
670412, 406647, -533952, 184464, 725760,
// alp[4]/n^4, polynomial in n of order 2
6601661, -7732800, 2230245, 7257600,
// alp[5]/n^5, polynomial in n of order 1
-13675556, 3438171, 7983360,
// alp[6]/n^6, polynomial in n of order 0
212378941, 319334400,
}; // count = 27
double[] betcoeff =
{
// bet[1]/n^1, polynomial in n of order 5
384796, -382725, -6720, 932400, -1612800, 1209600, 2419200,
// bet[2]/n^2, polynomial in n of order 4
-1118711, 1695744, -1174656, 258048, 80640, 3870720,
// bet[3]/n^3, polynomial in n of order 3
22276, -16929, -15984, 12852, 362880,
// bet[4]/n^4, polynomial in n of order 2
-830251, -158400, 197865, 7257600,
// bet[5]/n^5, polynomial in n of order 1
-435388, 453717, 15966720,
// bet[6]/n^6, polynomial in n of order 0
20648693, 638668800,
}; // count = 27
int m = maxpow_ / 2;
_b1 = GeoMath.PolyVal(m, b1coeff, 0, GeoMath.Square(_n)) / (b1coeff[m + 1] * (1 + _n));
// _a1 is the equivalent radius for computing the circumference of
// ellipse.
_a1 = _b1 * _a;
int o = 0;
double d = _n;
for (int l = 1; l <= maxpow_; ++l)
{
m = maxpow_ - l;
_alp[l] = d * GeoMath.PolyVal(m, alpcoeff, o, _n) / alpcoeff[o + m + 1];
_bet[l] = d * GeoMath.PolyVal(m, betcoeff, o, _n) / betcoeff[o + m + 1];
o += m + 2;
d *= _n;
}
// Post condition: o == sizeof(alpcoeff) / sizeof(double) &&
// o == sizeof(betcoeff) / sizeof(double)
}