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public/mathlib/spherical_geometry.h

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+//========= Copyright Valve Corporation, All rights reserved. ============//
+//
+// Purpose: Functions for spherical geometry.
+//
+// $NoKeywords: $
+//
+//=============================================================================//
+
+#ifndef SPHERICAL_GEOMETRY_H
+#define SPHERICAL_GEOMETRY_H
+
+#ifdef _WIN32
+#pragma once
+#endif
+
+#include <math.h>
+#include <float.h>
+
+// see http://mathworld.wolfram.com/SphericalTrigonometry.html
+
+// return the spherical distance, in radians, between 2 points on the unit sphere.
+FORCEINLINE float UnitSphereLineSegmentLength( Vector const &a, Vector const &b )
+{
+	// check unit length
+	Assert( fabs( VectorLength( a ) - 1.0 ) < 1.0e-3 );
+	Assert( fabs( VectorLength( b ) - 1.0 ) < 1.0e-3 );
+	return acos( DotProduct( a, b ) );
+}
+
+
+// given 3 points on the unit sphere, return the spherical area (in radians) of the triangle they form.
+// valid for "small" triangles.
+FORCEINLINE float UnitSphereTriangleArea( Vector const &a, Vector const &b , Vector const &c )
+{
+	float flLengthA = UnitSphereLineSegmentLength( b, c );
+	float flLengthB = UnitSphereLineSegmentLength( c, a );
+	float flLengthC = UnitSphereLineSegmentLength( a, b );
+	
+	if ( ( flLengthA == 0. ) || ( flLengthB == 0. ) || ( flLengthC == 0. ) )
+		return 0.;											// zero area triangle
+			
+	// now, find the 3 incribed angles for the triangle
+	float flHalfSumLens = 0.5 * ( flLengthA + flLengthB + flLengthC );
+	float flSinSums = sin( flHalfSumLens );
+	float flSinSMinusA= sin( flHalfSumLens - flLengthA );
+	float flSinSMinusB= sin( flHalfSumLens - flLengthB );
+	float flSinSMinusC= sin( flHalfSumLens - flLengthC );
+	
+	float flTanAOver2 = sqrt ( ( flSinSMinusB * flSinSMinusC ) / ( flSinSums * flSinSMinusA ) );
+	float flTanBOver2 = sqrt ( ( flSinSMinusA * flSinSMinusC ) / ( flSinSums * flSinSMinusB ) );
+	float flTanCOver2 = sqrt ( ( flSinSMinusA * flSinSMinusB ) / ( flSinSums * flSinSMinusC ) );
+
+	// Girards formula : area = sum of angles - pi.
+	return 2.0 * ( atan( flTanAOver2 ) + atan( flTanBOver2 ) + atan( flTanCOver2 ) ) - M_PI;
+}
+
+// spherical harmonics-related functions. Best explanation at http://www.research.scea.com/gdc2003/spherical-harmonic-lighting.pdf
+
+// Evaluate associated legendre polynomial P( l, m ) at flX, using recurrence relation
+float AssociatedLegendrePolynomial( int nL, int nM, float flX );
+
+// Evaluate order N spherical harmonic with spherical coordinates
+// nL = band, 0..N
+// nM = -nL .. nL
+// theta = 0..M_PI
+// phi = 0.. 2 * M_PHI
+float SphericalHarmonic( int nL, int nM, float flTheta, float flPhi );
+
+// evaluate spherical harmonic with normalized vector direction
+float SphericalHarmonic( int nL, int nM, Vector const &vecDirection );
+
+
+#endif // SPHERICAL_GEOMETRY_H