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Improve short-distance accuracy of Andoyer inverse #1461

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42 changes: 23 additions & 19 deletions include/boost/geometry/formulas/andoyer_inverse.hpp
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Original file line number Diff line number Diff line change
Expand Up @@ -73,28 +73,35 @@ class andoyer_inverse

CT const c0 = CT(0);
CT const c1 = CT(1);
CT const c2 = CT(2);
CT const pi = math::pi<CT>();
CT const f = formula::flattening<CT>(spheroid);

CT const dlon = lon2 - lon1;
CT const dlat = lat2 - lat1;
CT const sin_dlon = sin(dlon);
CT const cos_dlon = cos(dlon);
CT const hav_dlon = math::hav(dlon);
CT const sin_lat1 = sin(lat1);
CT const cos_lat1 = cos(lat1);
CT const sin_lat2 = sin(lat2);
CT const cos_lat2 = cos(lat2);
CT const hav_dlat = math::hav(dlat);

// H,G,T = infinity if cos_d = 1 or cos_d = -1
// using the haversine formula instead of the cosine law for better accuracy around short distances
// H,G,T = infinity if hav_d = 0 or hav_d = 1
// lat1 == +-90 && lat2 == +-90
// lat1 == lat2 && lon1 == lon2
CT cos_d = sin_lat1*sin_lat2 + cos_lat1*cos_lat2*cos_dlon;
// on some platforms cos_d may be outside valid range
if (cos_d < -c1)
cos_d = -c1;
else if (cos_d > c1)
cos_d = c1;

CT const d = acos(cos_d); // [0, pi]
CT hav_d = hav_dlat + cos_lat1*cos_lat2*hav_dlon;
// on some platforms hav_d may be outside valid range
if (hav_d < c0)
hav_d = c0;
else if (hav_d > c1)
hav_d = c1;

CT const sin_d_half = sqrt(hav_d);
CT const d_half = asin(sin_d_half);
CT const d = c2 * d_half; // [0, pi]
CT const sin_d = sin(d); // [-1, 1]

if BOOST_GEOMETRY_CONSTEXPR (EnableDistance)
Expand All @@ -103,8 +110,8 @@ class andoyer_inverse
CT const L = math::sqr(sin_lat1+sin_lat2);
CT const three_sin_d = CT(3) * sin_d;

CT const one_minus_cos_d = c1 - cos_d;
CT const one_plus_cos_d = c1 + cos_d;
CT const one_minus_cos_d = c2 * hav_d;
CT const one_plus_cos_d = c2 - one_minus_cos_d;
// cos_d = 1 means that the points are very close
// cos_d = -1 means that the points are antipodal

Expand Down Expand Up @@ -141,7 +148,7 @@ class andoyer_inverse
// correctly and consistently across all formulas.

// points very close
if (cos_d >= c0)
if (hav_d <= CT(0.5))
{
result.azimuth = c0;
result.reverse_azimuth = c0;
Expand All @@ -164,7 +171,8 @@ class andoyer_inverse
}
else
{
CT const c2 = CT(2);
CT const M = cos_lat1 * sin_lat2;
CT const N = sin_lat1 * cos_lat2;

CT A = c0;
CT U = c0;
Expand All @@ -177,9 +185,7 @@ class andoyer_inverse
}
else
{
CT const tan_lat2 = sin_lat2/cos_lat2;
CT const M = cos_lat1*tan_lat2-sin_lat1*cos_dlon;
A = atan2(sin_dlon, M);
A = atan2(sin_dlon*cos_lat2, M - N*cos_dlon);
CT const sin_2A = sin(c2*A);
U = (f/ c2)*math::sqr(cos_lat1)*sin_2A;
}
Expand All @@ -195,9 +201,7 @@ class andoyer_inverse
}
else
{
CT const tan_lat1 = sin_lat1/cos_lat1;
CT const N = cos_lat2*tan_lat1-sin_lat2*cos_dlon;
B = atan2(sin_dlon, N);
B = atan2(sin_dlon*cos_lat1, N - M*cos_dlon);
CT const sin_2B = sin(c2*B);
V = (f/ c2)*math::sqr(cos_lat2)*sin_2B;
}
Expand Down