/// <summary>
/// The great circle distance in radians between two spherical coordinates.
/// This function uses the Haversine formula.
/// For math details, see:
/// https://en.wikipedia.org/wiki/Haversine_formula
/// https://www.movable-type.co.uk/scripts/latlong.html
/// </summary>
/// <param name="a">the first lat/lng pair (in radians)</param>
/// <param name="b">the second lat/lng pair (in radians)</param>
/// <returns>
/// the great circle distance in radians between a and b
/// </returns>
/// <!--
/// geoCoord.c
/// double H3_EXPORT(pointDistRads)
/// -->
public static decimal DistanceToRadians(this GeoCoord a, GeoCoord b)
{
decimal sinLat = DecimalEx.Sin((b.Latitude - a.Latitude) / 2.0m);
decimal sinLng = DecimalEx.Sin((b.Longitude - a.Longitude) / 2.0m);
decimal p = sinLat * sinLat + DecimalEx.Cos(a.Latitude) *
DecimalEx.Cos(b.Latitude) * sinLng * sinLng;
return(2 * DecimalEx.ATan2(DecimalEx.Sqrt(p), DecimalEx.Sqrt(1 - p)));
}
/// <summary>
/// Determines the azimuth to p2 from p1 in radians
/// </summary>
/// <param name="p1">The first spherical coordinates</param>
/// <param name="p2">The second spherical coordinates</param>
/// <returns>The azimuth in radians from p1 to p2</returns>
/// <!--
/// geoCoord.c
/// double _geoAzimuthRads
/// -->
internal static decimal AzimuthRadiansTo(this GeoCoord p1, GeoCoord p2)
{
return
(DecimalEx.ATan2
(
DecimalEx.Cos(p2.Latitude) * DecimalEx.Sin(p2.Longitude - p1.Longitude),
DecimalEx.Cos(p1.Latitude) * DecimalEx.Sin(p2.Latitude) -
DecimalEx.Sin(p1.Latitude) * DecimalEx.Cos(p2.Latitude) *
DecimalEx.Cos(p2.Longitude - p1.Longitude)
));
}
/// <summary>
/// Determines the center point in spherical coordinates of a cell given by 2D
/// hex coordinates on a particular icosahedral face.
/// </summary>
/// <param name="v">The 2D hex coordinates of the cell</param>
/// <param name="face">The icosahedral face upon which the 2D hex coordinate system is centered</param>
/// <param name="res">The H3 resolution of the cell</param>
/// <param name="substrate">
/// Indicates whether or not this grid is actually a substrate
/// grid relative to the specified resolution.
/// </param>
/// <returns>The spherical coordinates of the cell center point</returns>
/// <!--
/// faceIjk.c
/// void _hex2dToGeo
/// -->
public static GeoCoord ToGeoCoord(this Vec2d v, int face, int res, int substrate)
{
// calculate (r, theta) in hex2d
decimal r = v.Magnitude;
bool bSubstrate = substrate != 0;
if (r < Constants.H3.EPSILON)
{
return(Constants.FaceIjk.FaceCenterGeo[face]);
}
decimal theta = DecimalEx.ATan2(v.Y, v.X);
// scale for current resolution length u
for (var i = 0; i < res; i++)
{
r /= Constants.FaceIjk.MSqrt7;
}
// scale accordingly if this is a substrate grid
if (substrate != 0)
{
r /= 3.0m;
if (res.IsResClassIii())
{
r /= Constants.FaceIjk.MSqrt7;
}
}
r *= Constants.H3.RES0_U_GNOMONIC;
// perform inverse gnomonic scaling of r
r = DecimalEx.ATan(r);
// adjust theta for Class III
// if a substrate grid, then it's already been adjusted for Class III
if (!bSubstrate && res.IsResClassIii())
{
theta = (theta + Constants.H3.M_AP7_ROT_RADS).NormalizeRadians();
}
// find theta as an azimuth
theta = (Constants.FaceIjk.FaceAxesAzRadsCii[face, 0] - theta).NormalizeRadians();
// now find the point at (r,theta) from the face center
return(Constants.FaceIjk.FaceCenterGeo[face]
.GetAzimuthDistancePoint(theta, r));
}
public void Test(decimal y, decimal x, decimal expected, decimal tolerance)
{
tolerance = Helper.GetScaledTolerance(expected, (int)tolerance, true);
Assert.That(DecimalEx.ATan2(y, x), Is.EqualTo(expected).Within(tolerance));
}
/// <summary>
/// Computes the point on the sphere a specified azimuth and distance from
/// another point.
/// </summary>
/// <param name="p1">The first spherical coordinates.</param>
/// <param name="azimuth">The desired azimuth from p1.</param>
/// <param name="distance">The desired distance from p1, must be non-negative.</param>
/// <returns>The spherical coordinates at the desired azimuth and distance from p1.</returns>
/// <!--
/// geoCoord.c
/// void _geoAzDistanceRads
/// -->
internal static GeoCoord GetAzimuthDistancePoint(this GeoCoord p1, decimal azimuth, decimal distance)
{
if (distance < Constants.H3.EPSILON)
{
return(p1);
}
azimuth = azimuth.NormalizeRadians().ConstrainToPiAccuracy();
var p2 = new GeoCoord();
// check for due north/south azimuth
if (azimuth < Constants.H3.EPSILON ||
Math.Abs(azimuth - Constants.H3.M_PI) < Constants.H3.EPSILON)
{
if (azimuth < Constants.H3.EPSILON) // due north
{
p2 = p2.SetLatitude(p1.Latitude + distance);
}
else // due south
{
p2 = p2.SetLatitude(p1.Latitude - distance);
}
if (Math.Abs(p2.Latitude - Constants.H3.M_PI_2) < Constants.H3.EPSILON) // north pole
{
p2 = new GeoCoord(Constants.H3.M_PI_2, 0.0m);
}
else if (Math.Abs(p2.Latitude + Constants.H3.M_PI_2) < Constants.H3.EPSILON) // south pole
{
p2 = new GeoCoord(-Constants.H3.M_PI_2, 0.0m);
}
else
{
p2 = p2.SetLongitude(p1.Longitude.ConstrainLongitude());
}
}
else // Not due north or south
{
decimal sinLatitude = DecimalEx.Sin(p1.Latitude) * DecimalEx.Cos(distance) +
DecimalEx.Cos(p1.Latitude) * DecimalEx.Sin(distance) * DecimalEx.Cos(azimuth);
sinLatitude = sinLatitude.ConstrainToPiAccuracy();
if (sinLatitude > 1.0m)
{
sinLatitude = 1.0m;
}
if (sinLatitude < -1.0m)
{
sinLatitude = 1.0m;
}
p2 = p2.SetLatitude(DecimalEx.ASin(sinLatitude).ConstrainToPiAccuracy());
if (Math.Abs(p2.Latitude - Constants.H3.M_PI_2) < Constants.H3.EPSILON) // north pole
{
p2 = new GeoCoord(Constants.H3.M_PI_2, 0.0m);
}
else if (Math.Abs(p2.Latitude + Constants.H3.M_PI_2) < Constants.H3.EPSILON) // south pole
{
p2 = new GeoCoord(-Constants.H3.M_PI_2, 0.0m);
}
else
{
decimal sinLongitude = DecimalEx.Sin(azimuth) * DecimalEx.Sin(distance) / DecimalEx.Cos(p2.Latitude);
decimal cosLongitude = (DecimalEx.Cos(distance) - DecimalEx.Sin(p1.Latitude) * DecimalEx.Sin(p2.Latitude)) /
DecimalEx.Cos(p1.Latitude) / DecimalEx.Cos(p2.Latitude);
if (sinLongitude > 1.0m)
{
sinLongitude = 1.0m;
}
if (sinLongitude < -1.0m)
{
sinLongitude = -1.0m;
}
if (cosLongitude > 1.0m)
{
cosLongitude = 1.0m;
}
if (cosLongitude < -1.0m)
{
cosLongitude = -1.0m;
}
p2 = p2.SetLongitude
(
(p1.Longitude + DecimalEx.ATan2(sinLongitude, cosLongitude))
.ConstrainLongitude().ConstrainToPiAccuracy()
);
}
}
return(p2);
}