/*-------------------------------------------------------------------------*
* Coord *
* *
* Compute the two coordinates (x1,x2) corresponding to a point in the *
* spherical triangle. This is the inverse of "Chart". *
* *
*-------------------------------------------------------------------------*/
Vec2 SphericalTriangle::Coord( const Vec3 &P1 ) const
{
Vec3 P = Unit( P1 );
// Compute the new C vertex, which lies on the arc defined by B-P
// and the arc defined by A-C.
Vec3 C_new = Unit( ( B() ^ P ) ^ ( C() ^ A() ) );
// Adjust the sign of C_new. Make sure it's on the arc between A and C.
if( C_new * ( A() + C() ) < 0.0 ) C_new = -C_new;
// Compute x1, the area of the sub-triangle over the original area.
float cos_beta = CosDihedralAngle( A(), B(), C_new );
float cos_gamma = CosDihedralAngle( A(), C_new , B() );
float sub_area = Alpha() + acos( cos_beta ) + acos( cos_gamma ) - Pi;
float x1 = sub_area / SolidAngle();
// Now compute the second coordinate using the new C vertex.
float z = P * B();
float x2 = ( 1.0 - z ) / ( 1.0 - C_new * B() );
if( x1 < 0.0 ) x1 = 0.0; if( x1 > 1.0 ) x1 = 1.0;
if( x2 < 0.0 ) x2 = 0.0; if( x2 > 1.0 ) x2 = 1.0;
return Vec2( x1, x2 );
}
/*-------------------------------------------------------------------------*
* Init *
* *
* Construct the spherical triange from three vertices. Assume that the *
* sphere is centered at the origin. The vectors A, B, and C need not *
* be normalized. *
* *
*-------------------------------------------------------------------------*/
void SphericalTriangle::Init(
const Vector3d &A0, const Vector3d &B0, const Vector3d &C0
)
{
// Normalize the three vectors -- these are the vertices.
A_ = Unit( A0 ) ;
B_ = Unit( B0 ) ;
C_ = Unit( C0 ) ;
// Compute and save the cosines of the edge lengths.
cos_a = B_ * C_;
cos_b = A_ * C_;
cos_c = A_ * B_;
// Compute and save the edge lengths.
a_ = ArcCos( cos_a );
b_ = ArcCos( cos_b );
c_ = ArcCos( cos_c );
// Compute the cosines of the internal (i.e. dihedral) angles.
cos_alpha = CosDihedralAngle( C_, A_, B_ );
cos_beta = CosDihedralAngle( A_, B_, C_ );
cos_gamma = CosDihedralAngle( A_, C_, B_ );
// Compute the (dihedral) angles.
alpha = ArcCos( cos_alpha );
beta = ArcCos( cos_beta );
gamma = ArcCos( cos_gamma );
// Compute the solid angle of the spherical triangle.
area = alpha + beta + gamma - M_PI ;
// Compute the orientation of the triangle.
double mix = A_ * ( B_ ^ C_ ) ;
orient = (mix > 0) ? 1 : ((mix < 0) ? -1 : 0) ;
// Initialize three variables that are used for sampling the triangle.
U = Unit( C_ / A_ ); // In plane of AC orthogonal to A.
sin_alpha = sin( alpha );
product = sin_alpha * cos_c;
}