Matrix3D
HorizontalToEquatorialMatrix( Angle localSiderealTime,
Angle geographicLatitude )
{
// This is equivalent to
// Matrix3D( 2, -lst ) * Matrix3D( 1,0,0, 0,-1,0, 0,0,1 )
// * Matrix3D( 1, lat - pi/2 ) * Matrix3D( 2, pi ).
double sinLST = localSiderealTime.Sin( );
double cosLST = localSiderealTime.Cos( );
double sinLat = geographicLatitude.Sin( );
double cosLat = geographicLatitude.Cos( );
return Matrix3D( - cosLST * sinLat, - sinLST, cosLST * cosLat,
- sinLST * sinLat, cosLST, sinLST * cosLat,
cosLat, 0, sinLat );
}
Point3D
Geodetic::Rectangular( ) const
{
double sinLat = m_latitude.Sin( );
double cosLat = m_latitude.Cos( );
double sinLong = m_longitude.Sin( );
double cosLong = m_longitude.Cos( );
double eccentricity = std::sqrt( 2 * flattening
- flattening * flattening );
double eccentSqr = eccentricity * eccentricity;
double radCurv = radius
/ sqrt( 1. - eccentSqr * sinLat * sinLat );
return Point3D( (radCurv + m_height) * cosLat * cosLong,
(radCurv + m_height) * cosLat * sinLong,
((1 + eccentSqr) * radCurv + m_height) * sinLat );
}
Horizontal
EquatorialToHorizontal( const Equatorial & equatorial,
Angle localSiderealTime, Angle geographicLatitude )
{
Angle hourAngle = localSiderealTime - equatorial.RightAscension();
double sinHA = hourAngle.Sin( );
double cosHA = hourAngle.Cos( );
double sinDec = equatorial.Declination().Sin( );
double cosDec = equatorial.Declination().Cos( );
double tanDec = (cosDec == 0.) ? infinity : sinDec / cosDec;
double sinLat = geographicLatitude.Sin( );
double cosLat = geographicLatitude.Cos( );
Angle az = ArcTan( sinHA, cosHA * sinLat - tanDec * cosLat );
az += Angle( M_PI );
az.NormalizePositive( );
Angle alt = ArcSin( sinDec * sinLat + cosHA * cosDec * cosLat );
return Horizontal( az, alt, equatorial.Distance() );
}
Equatorial
HorizontalToEquatorial( const Horizontal & horizontal,
Angle localSiderealTime, Angle geographicLatitude )
{
Angle az = horizontal.Azimuth() - Angle( M_PI );
double sinAz = az.Sin( );
double cosAz = az.Cos( );
double sinAlt = horizontal.Altitude().Sin( );
double cosAlt = horizontal.Altitude().Cos( );
double tanAlt = (cosAlt == 0.) ? infinity : sinAlt / cosAlt;
double sinLat = geographicLatitude.Sin( );
double cosLat = geographicLatitude.Cos( );
// (Meeus has an error here, which I've corrected.)
Angle hourAngle = ArcTan( sinAz, cosAz * sinLat + tanAlt * cosLat );
Angle ra = localSiderealTime - hourAngle;
ra.NormalizePositive( );
Angle dec = ArcSin( sinAlt * sinLat - cosAz * cosAlt * cosLat );
return Equatorial( ra, dec, horizontal.Distance() );
}
Angle
EclipticalLongitude( const Point3D & equatorialRect, Angle obliquity )
{
double x = equatorialRect.X();
double y = obliquity.Cos() * equatorialRect.Y()
+ obliquity.Sin() * equatorialRect.Z();
if ( (x == 0.) && (y == 0.) )
return Angle( 0. );
else
return ArcTan( y, x );
}
void
Geodetic::Set( const Point3D & rectangular )
{
//Explanatory Supplement (4.22-11 to 4.22-24)
double X = rectangular.X();
double Y = rectangular.Y();
double Z = rectangular.Z();
double a = radius;
double b = a - flattening * a;
if ( Z < 0.)
b = -b;
if ( (X == 0.) && (Y == 0.) )
{
m_longitude.Set( 0. );
m_latitude.Set( 0. );
m_height = Z - b;
if ( m_height < 0. )
m_height = - m_height;
return;
}
else
m_longitude = ArcTan( Y, X );
if ( Z == 0.)
{
m_latitude.Set( 0. );
m_height.Set( r - radius );
return;
}
double r = std::sqrt( X * X + Y * Y );
double aSqr = a * a;
double bSqr = b * b;
double E = (b * Z - (aSqr - bSqr)) / (a * r);
double ESqr = E * E;
double F = (b * Z + (aSqr + bSqr)) / (a * r);
double P = (4./3.) * (E * F + 1.);
double Q = 2. * (ESqr - F * F);
double D = P * P * P + Q * Q;
double sqrtD = std::sqrt( D );
double v = std::pow( (sqrtD - Q), (1./3.) )
- std::pow( (sqrtD + Q), (1./3.) );
const double epsilon = 1.0;
if ( (std::fabs( Z ) < epsilon ) || (std::fabs( r ) < epsilon) )
v = - (v * v * v + 2. * Q) / (3. * P);
double G = 0.5 * (std::sqrt( ESqr + v ) + E);
double t = std::sqrt( G * G + (F - v * G) / (G + G - E) ) - G;
m_latitude = ArcTan( (a * (1. - t * t)), (2. * b * t) );
m_height = (r - a * t) * m_latitude.Cos( )
+ (Z - b) * m_latitude.Sin( );
}
Ecliptical
EquatorialToEcliptical( const Equatorial & equatorial, Angle obliquity )
{
double sinRA = equatorial.RightAscension().Sin( );
double cosRA = equatorial.RightAscension().Cos( );
double sinDec = equatorial.Declination().Sin( );
double cosDec = equatorial.Declination().Cos( );
double tanDec = (cosDec == 0.) ? infinity : sinDec / cosDec;
double sinObl = obliquity.Sin( );
double cosObl = obliquity.Cos( );
Angle lng = ArcTan( sinRA * cosObl + tanDec * sinObl, cosRA );
lng.NormalizePositive( );
Angle lat = ArcSin( sinDec * cosObl - sinRA * cosDec * sinObl );
return Ecliptical( lng, lat, equatorial.Distance() );
}
Equatorial
EclipticalToEquatorial( const Ecliptical & ecliptical, Angle obliquity )
{
double sinLong = ecliptical.Longitude().Sin( );
double cosLong = ecliptical.Longitude().Cos( );
double sinLat = ecliptical.Latitude().Sin( );
double cosLat = ecliptical.Latitude().Cos( );
double tanLat = (cosLat == 0.) ? infinity : sinLat / cosLat;
double sinObl = obliquity.Sin( );
double cosObl = obliquity.Cos( );
Angle ra = ArcTan( sinLong * cosObl - tanLat * sinObl, cosLong );
ra.NormalizePositive( );
Angle dec = ArcSin( sinLat * cosObl + sinLong * cosLat * sinObl );
return Equatorial( ra, dec, ecliptical.Distance() );
}
void
Matrix3<T>::Set( int axis, const Angle & angle )
{
if ( (axis < 0) || (axis >= 3) )
throw std::out_of_range( "Matrix3: out_of_range error" );
static const int indices[3][3]
= { { 0, 1, 2 }, { 1, 2, 0 }, { 2, 0, 1 } };
const int i0 = indices[ axis ][ 0 ];
const int i1 = indices[ axis ][ 1 ];
const int i2 = indices[ axis ][ 2 ];
const T cos = static_cast< T >( angle.Cos( ) );
const T sin = static_cast< T >( angle.Sin( ) );
m_elements[ i0 ][ i0 ] = 1.;
m_elements[ i0 ][ i1 ] = m_elements[ i0 ][ i2 ] = m_elements[ i1 ][ i0 ]
= m_elements[ i2 ][ i0 ] = 0.;
m_elements[ i1 ][ i1 ] = m_elements[ i2 ][ i2 ] = cos;
m_elements[ i2 ][ i1 ] = - sin;
m_elements[ i1 ][ i2 ] = sin;
}
Equatorial
GalacticToEquatorial( const Galactic & galactic )
{
Angle galNorthRAAdj( 192.25 - 180., Angle::Degree );
Angle galNorthDec( 27.4, Angle::Degree );
Angle longOffset( 33. + 90., Angle::Degree );
Angle adjLong = galactic.Longitude() - longOffset;
double sinAdjLong = adjLong.Sin( );
double cosAdjLong = adjLong.Cos( );
double sinLat = galactic.Latitude().Sin( );
double cosLat = galactic.Latitude().Cos( );
double tanLat = (cosLat == 0.) ? infinity : sinLat / cosLat;
double sinNDec = galNorthDec.Sin( );
double cosNDec = galNorthDec.Cos( );
Angle y = ArcTan( sinAdjLong, cosAdjLong * sinNDec - tanLat * cosNDec );
Angle ra = y + galNorthRAAdj;
ra.NormalizePositive( );
Angle dec = ArcSin( sinLat * sinNDec + cosAdjLong * cosLat * cosNDec );
return Equatorial( ra, dec, galactic.Distance() );
}
Galactic
EquatorialToGalactic( const Equatorial & equatorial )
{
Angle galNorthRA( 192.25, Angle::Degree );
Angle galNorthDec( 27.4, Angle::Degree );
Angle longOffset( 33. + 270., Angle::Degree );
Angle adjRA = galNorthRA - equatorial.RightAscension();
double sinAdjRA = adjRA.Sin( );
double cosAdjRA = adjRA.Cos( );
double sinDec = equatorial.Declination().Sin( );
double cosDec = equatorial.Declination().Cos( );
double tanDec = (cosDec == 0.) ? infinity : sinDec / cosDec;
double sinNDec = galNorthDec.Sin( );
double cosNDec = galNorthDec.Cos( );
Angle x = ArcTan( sinAdjRA, cosAdjRA * sinNDec - tanDec * cosNDec );
Angle lng = longOffset - x;
lng.NormalizePositive( );
Angle lat = ArcSin( sinDec * sinNDec + cosAdjRA * cosDec * cosNDec );
return Galactic( lng, lat, equatorial.Distance() );
}
void
Matrix3<T>::Set( const AxisAngle<T> & axisAngle )
{
Angle angle = axisAngle.GetAngle( );
const Vector3<T> & axis = axisAngle.Axis( );
const T cc = static_cast<T>( 1. - angle.Cos( ) );
const T s = static_cast<T>( angle.Sin( ) );
const T a00 = axis[0] * axis[0];
const T a01 = axis[0] * axis[1];
const T a02 = axis[0] * axis[2];
const T a11 = axis[1] * axis[1];
const T a12 = axis[1] * axis[2];
const T a22 = axis[2] * axis[2];
m_elements[0][0] = static_cast<T>( 1. - cc * (a11 + a22) );
m_elements[1][0] = static_cast<T>( -s * axis[2] + cc * a01 );
m_elements[2][0] = static_cast<T>( s * axis[1] + cc * a02 );
m_elements[0][1] = static_cast<T>( s * axis[2] + cc * a01 );
m_elements[1][1] = static_cast<T>( 1. - cc * (a00 + a22) );
m_elements[2][1] = static_cast<T>( -s * axis[0] + cc * a12 );
m_elements[0][2] = static_cast<T>( -s * axis[1] + cc * a02 );
m_elements[1][2] = static_cast<T>( s * axis[0] + cc * a12 );
m_elements[2][2] = static_cast<T>( 1. - cc * (a00 + a11) );
}