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LinearR4.cpp
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/*
*
* RayTrace Software Package, release 1.0.1, May 9, 2002.
*
* Mathematics Subpackage (VrMath)
*
* Author: Samuel R. Buss
*
* Software accompanying the book
* 3D Computer Graphics: A Mathematical Introduction with OpenGL,
* by S. Buss, Cambridge University Press, 2003.
*
* Software is "as-is" and carries no warranty. It may be used without
* restriction, but if you modify it, please change the filenames to
* prevent confusion between different versions. Please acknowledge
* all use of the software in any publications or products based on it.
*
* Bug reports: Sam Buss, [email protected].
* Web page: http://math.ucsd.edu/~sbuss/MathCG
*
*/
#include "LinearR4.h"
#include <assert.h>
const VectorR4 VectorR4::Zero(0.0, 0.0, 0.0, 0.0);
const VectorR4 VectorR4::UnitX( 1.0, 0.0, 0.0, 0.0);
const VectorR4 VectorR4::UnitY( 0.0, 1.0, 0.0, 0.0);
const VectorR4 VectorR4::UnitZ( 0.0, 0.0, 1.0, 0.0);
const VectorR4 VectorR4::UnitW( 0.0, 0.0, 0.0, 1.0);
const VectorR4 VectorR4::NegUnitX(-1.0, 0.0, 0.0, 0.0);
const VectorR4 VectorR4::NegUnitY( 0.0,-1.0, 0.0, 0.0);
const VectorR4 VectorR4::NegUnitZ( 0.0, 0.0,-1.0, 0.0);
const VectorR4 VectorR4::NegUnitW( 0.0, 0.0, 0.0,-1.0);
const Matrix4x4 Matrix4x4::Identity(1.0, 0.0, 0.0, 0.0, 0.0, 1.0, 0.0, 0.0,
0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 1.0);
// ******************************************************
// * VectorR4 class - math library functions *
// * * * * * * * * * * * * * * * * * * * * * * * * * * **
double VectorR4::MaxAbs() const
{
register double m;
m = (x>0.0) ? x : -x;
if ( y>m ) m=y;
else if ( -y >m ) m = -y;
if ( z>m ) m=z;
else if ( -z>m ) m = -z;
if ( w>m ) m=w;
else if ( -w>m ) m = -w;
return m;
}
// ******************************************************
// * Matrix4x4 class - math library functions *
// * * * * * * * * * * * * * * * * * * * * * * * * * * **
void Matrix4x4::operator*= (const Matrix4x4& B) // Matrix product
{
double t1, t2, t3; // temporary values
t1 = m11*B.m11 + m12*B.m21 + m13*B.m31 + m14*B.m41;
t2 = m11*B.m12 + m12*B.m22 + m13*B.m32 + m14*B.m42;
t3 = m11*B.m13 + m12*B.m23 + m13*B.m33 + m14*B.m43;
m14 = m11*B.m14 + m12*B.m24 + m13*B.m34 + m14*B.m44;
m11 = t1;
m12 = t2;
m13 = t3;
t1 = m21*B.m11 + m22*B.m21 + m23*B.m31 + m24*B.m41;
t2 = m21*B.m12 + m22*B.m22 + m23*B.m32 + m24*B.m42;
t3 = m21*B.m13 + m22*B.m23 + m23*B.m33 + m24*B.m43;
m24 = m21*B.m14 + m22*B.m24 + m23*B.m34 + m24*B.m44;
m21 = t1;
m22 = t2;
m23 = t3;
t1 = m31*B.m11 + m32*B.m21 + m33*B.m31 + m34*B.m41;
t2 = m31*B.m12 + m32*B.m22 + m33*B.m32 + m34*B.m42;
t3 = m31*B.m13 + m32*B.m23 + m33*B.m33 + m34*B.m43;
m34 = m31*B.m14 + m32*B.m24 + m33*B.m34 + m34*B.m44;
m31 = t1;
m32 = t2;
m33 = t3;
t1 = m41*B.m11 + m42*B.m21 + m43*B.m31 + m44*B.m41;
t2 = m41*B.m12 + m42*B.m22 + m43*B.m32 + m44*B.m42;
t3 = m41*B.m13 + m42*B.m23 + m43*B.m33 + m44*B.m43;
m44 = m41*B.m14 + m42*B.m24 + m43*B.m34 + m44*B.m44;
m41 = t1;
m42 = t2;
m43 = t3;
}
inline void ReNormalizeHelper ( double &a, double &b, double &c, double &d )
{
register double scaleF = a*a+b*b+c*c+d*d; // Inner product of Vector-R4
scaleF = 1.0-0.5*(scaleF-1.0);
a *= scaleF;
b *= scaleF;
c *= scaleF;
d *= scaleF;
}
Matrix4x4& Matrix4x4::ReNormalize() {
ReNormalizeHelper( m11, m21, m31, m41 ); // Renormalize first column
ReNormalizeHelper( m12, m22, m32, m42 ); // Renormalize second column
ReNormalizeHelper( m13, m23, m33, m43 ); // Renormalize third column
ReNormalizeHelper( m14, m24, m34, m44 ); // Renormalize fourth column
double alpha = 0.5*(m11*m12 + m21*m22 + m31*m32 + m41*m42); //1st and 2nd cols
double beta = 0.5*(m11*m13 + m21*m23 + m31*m33 + m41*m43); //1st and 3rd cols
double gamma = 0.5*(m11*m14 + m21*m24 + m31*m34 + m41*m44); //1st and 4nd cols
double delta = 0.5*(m12*m13 + m22*m23 + m32*m33 + m42*m43); //2nd and 3rd cols
double eps = 0.5*(m12*m14 + m22*m24 + m32*m34 + m42*m44); //2nd and 4nd cols
double phi = 0.5*(m13*m14 + m23*m24 + m33*m34 + m43*m44); //3rd and 4nd cols
double temp1, temp2, temp3;
temp1 = m11 - alpha*m12 - beta*m13 - gamma*m14;
temp2 = m12 - alpha*m11 - delta*m13 - eps*m14;
temp3 = m13 - beta*m11 - delta*m12 - phi*m14;
m14 -= (gamma*m11 + eps*m12 + phi*m13);
m11 = temp1;
m12 = temp2;
m13 = temp3;
temp1 = m21 - alpha*m22 - beta*m23 - gamma*m24;
temp2 = m22 - alpha*m21 - delta*m23 - eps*m24;
temp3 = m23 - beta*m21 - delta*m22 - phi*m24;
m24 -= (gamma*m21 + eps*m22 + phi*m23);
m21 = temp1;
m22 = temp2;
m23 = temp3;
temp1 = m31 - alpha*m32 - beta*m33 - gamma*m34;
temp2 = m32 - alpha*m31 - delta*m33 - eps*m34;
temp3 = m33 - beta*m31 - delta*m32 - phi*m34;
m34 -= (gamma*m31 + eps*m32 + phi*m33);
m31 = temp1;
m32 = temp2;
m33 = temp3;
temp1 = m41 - alpha*m42 - beta*m43 - gamma*m44;
temp2 = m42 - alpha*m41 - delta*m43 - eps*m44;
temp3 = m43 - beta*m41 - delta*m42 - phi*m44;
m44 -= (gamma*m41 + eps*m42 + phi*m43);
m41 = temp1;
m42 = temp2;
m43 = temp3;
return *this;
}
// ******************************************************
// * LinearMapR4 class - math library functions *
// * * * * * * * * * * * * * * * * * * * * * * * * * * **
double LinearMapR4::Determinant () const // Returns the determinant
{
double Tbt34C12 = m31*m42-m32*m41; // 2x2 subdeterminants
double Tbt34C13 = m31*m43-m33*m41;
double Tbt34C14 = m31*m44-m34*m41;
double Tbt34C23 = m32*m43-m33*m42;
double Tbt34C24 = m32*m44-m34*m42;
double Tbt34C34 = m33*m44-m34*m43;
double sd11 = m22*Tbt34C34 - m23*Tbt34C24 + m24*Tbt34C23; // 3x3 subdeterminants
double sd12 = m21*Tbt34C34 - m23*Tbt34C14 + m24*Tbt34C13;
double sd13 = m21*Tbt34C24 - m22*Tbt34C14 + m24*Tbt34C12;
double sd14 = m21*Tbt34C23 - m22*Tbt34C13 + m23*Tbt34C12;
return ( m11*sd11 - m12*sd12 + m13*sd13 - m14*sd14 );
}
LinearMapR4 LinearMapR4::Inverse() const // Returns inverse
{
double Tbt34C12 = m31*m42-m32*m41; // 2x2 subdeterminants
double Tbt34C13 = m31*m43-m33*m41;
double Tbt34C14 = m31*m44-m34*m41;
double Tbt34C23 = m32*m43-m33*m42;
double Tbt34C24 = m32*m44-m34*m42;
double Tbt34C34 = m33*m44-m34*m43;
double Tbt24C12 = m21*m42-m22*m41; // 2x2 subdeterminants
double Tbt24C13 = m21*m43-m23*m41;
double Tbt24C14 = m21*m44-m24*m41;
double Tbt24C23 = m22*m43-m23*m42;
double Tbt24C24 = m22*m44-m24*m42;
double Tbt24C34 = m23*m44-m24*m43;
double Tbt23C12 = m21*m32-m22*m31; // 2x2 subdeterminants
double Tbt23C13 = m21*m33-m23*m31;
double Tbt23C14 = m21*m34-m24*m31;
double Tbt23C23 = m22*m33-m23*m32;
double Tbt23C24 = m22*m34-m24*m32;
double Tbt23C34 = m23*m34-m24*m33;
double sd11 = m22*Tbt34C34 - m23*Tbt34C24 + m24*Tbt34C23; // 3x3 subdeterminants
double sd12 = m21*Tbt34C34 - m23*Tbt34C14 + m24*Tbt34C13;
double sd13 = m21*Tbt34C24 - m22*Tbt34C14 + m24*Tbt34C12;
double sd14 = m21*Tbt34C23 - m22*Tbt34C13 + m23*Tbt34C12;
double sd21 = m12*Tbt34C34 - m13*Tbt34C24 + m14*Tbt34C23; // 3x3 subdeterminants
double sd22 = m11*Tbt34C34 - m13*Tbt34C14 + m14*Tbt34C13;
double sd23 = m11*Tbt34C24 - m12*Tbt34C14 + m14*Tbt34C12;
double sd24 = m11*Tbt34C23 - m12*Tbt34C13 + m13*Tbt34C12;
double sd31 = m12*Tbt24C34 - m13*Tbt24C24 + m14*Tbt24C23; // 3x3 subdeterminants
double sd32 = m11*Tbt24C34 - m13*Tbt24C14 + m14*Tbt24C13;
double sd33 = m11*Tbt24C24 - m12*Tbt24C14 + m14*Tbt24C12;
double sd34 = m11*Tbt24C23 - m12*Tbt24C13 + m13*Tbt24C12;
double sd41 = m12*Tbt23C34 - m13*Tbt23C24 + m14*Tbt23C23; // 3x3 subdeterminants
double sd42 = m11*Tbt23C34 - m13*Tbt23C14 + m14*Tbt23C13;
double sd43 = m11*Tbt23C24 - m12*Tbt23C14 + m14*Tbt23C12;
double sd44 = m11*Tbt23C23 - m12*Tbt23C13 + m13*Tbt23C12;
register double detInv = 1.0/(m11*sd11 - m12*sd12 + m13*sd13 - m14*sd14);
return( LinearMapR4( sd11*detInv, -sd12*detInv, sd13*detInv, -sd14*detInv,
-sd21*detInv, sd22*detInv, -sd23*detInv, sd24*detInv,
sd31*detInv, -sd32*detInv, sd33*detInv, -sd34*detInv,
-sd41*detInv, sd42*detInv, -sd43*detInv, sd44*detInv ) );
}
LinearMapR4& LinearMapR4::Invert() // Converts into inverse.
{
double Tbt34C12 = m31*m42-m32*m41; // 2x2 subdeterminants
double Tbt34C13 = m31*m43-m33*m41;
double Tbt34C14 = m31*m44-m34*m41;
double Tbt34C23 = m32*m43-m33*m42;
double Tbt34C24 = m32*m44-m34*m42;
double Tbt34C34 = m33*m44-m34*m43;
double Tbt24C12 = m21*m42-m22*m41; // 2x2 subdeterminants
double Tbt24C13 = m21*m43-m23*m41;
double Tbt24C14 = m21*m44-m24*m41;
double Tbt24C23 = m22*m43-m23*m42;
double Tbt24C24 = m22*m44-m24*m42;
double Tbt24C34 = m23*m44-m24*m43;
double Tbt23C12 = m21*m32-m22*m31; // 2x2 subdeterminants
double Tbt23C13 = m21*m33-m23*m31;
double Tbt23C14 = m21*m34-m24*m31;
double Tbt23C23 = m22*m33-m23*m32;
double Tbt23C24 = m22*m34-m24*m32;
double Tbt23C34 = m23*m34-m24*m33;
double sd11 = m22*Tbt34C34 - m23*Tbt34C24 + m24*Tbt34C23; // 3x3 subdeterminants
double sd12 = m21*Tbt34C34 - m23*Tbt34C14 + m24*Tbt34C13;
double sd13 = m21*Tbt34C24 - m22*Tbt34C14 + m24*Tbt34C12;
double sd14 = m21*Tbt34C23 - m22*Tbt34C13 + m23*Tbt34C12;
double sd21 = m12*Tbt34C34 - m13*Tbt34C24 + m14*Tbt34C23; // 3x3 subdeterminants
double sd22 = m11*Tbt34C34 - m13*Tbt34C14 + m14*Tbt34C13;
double sd23 = m11*Tbt34C24 - m12*Tbt34C14 + m14*Tbt34C12;
double sd24 = m11*Tbt34C23 - m12*Tbt34C13 + m13*Tbt34C12;
double sd31 = m12*Tbt24C34 - m13*Tbt24C24 + m14*Tbt24C23; // 3x3 subdeterminants
double sd32 = m11*Tbt24C34 - m13*Tbt24C14 + m14*Tbt24C13;
double sd33 = m11*Tbt24C24 - m12*Tbt24C14 + m14*Tbt24C12;
double sd34 = m11*Tbt24C23 - m12*Tbt24C13 + m13*Tbt24C12;
double sd41 = m12*Tbt23C34 - m13*Tbt23C24 + m14*Tbt23C23; // 3x3 subdeterminants
double sd42 = m11*Tbt23C34 - m13*Tbt23C14 + m14*Tbt23C13;
double sd43 = m11*Tbt23C24 - m12*Tbt23C14 + m14*Tbt23C12;
double sd44 = m11*Tbt23C23 - m12*Tbt23C13 + m13*Tbt23C12;
register double detInv = 1.0/(m11*sd11 - m12*sd12 + m13*sd13 - m14*sd14);
m11 = sd11*detInv;
m12 = -sd21*detInv;
m13 = sd31*detInv;
m14 = -sd41*detInv;
m21 = -sd12*detInv;
m22 = sd22*detInv;
m23 = -sd32*detInv;
m24 = sd42*detInv;
m31 = sd13*detInv;
m32 = -sd23*detInv;
m33 = sd33*detInv;
m34 = -sd43*detInv;
m41 = -sd14*detInv;
m42 = sd24*detInv;
m43 = -sd34*detInv;
m44 = sd44*detInv;
return ( *this );
}
VectorR4 LinearMapR4::Solve(const VectorR4& u) const // Returns solution
{
// Just uses Inverse() for now.
return ( Inverse()*u );
}
// ******************************************************
// * RotationMapR4 class - math library functions *
// * * * * * * * * * * * * * * * * * * * * * * * * * * **
// ***************************************************************
// * 4-space vector and matrix utilities *
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
// Returns u * v^T
LinearMapR4 TimesTranspose( const VectorR4& u, const VectorR4& v)
{
LinearMapR4 result;
TimesTranspose( u, v, result );
return result;
}
// The following routines are use to obtain
// a righthanded orthonormal basis to complement vectors u,v,w.
// The vectors u,v,w must be unit and orthonormal.
// The value is returned in "rotmat" with the first column(s) of
// rotmat equal to u,v,w as appropriate.
void GetOrtho( const VectorR4& u, RotationMapR4& rotmat )
{
rotmat.SetColumn1(u);
GetOrtho( 1, rotmat );
}
void GetOrtho( const VectorR4& u, const VectorR4& v, RotationMapR4& rotmat )
{
rotmat.SetColumn1(u);
rotmat.SetColumn2(v);
GetOrtho( 2, rotmat );
}
void GetOrtho( const VectorR4& u, const VectorR4& v, const VectorR4& s,
RotationMapR4& rotmat )
{
rotmat.SetColumn1(u);
rotmat.SetColumn2(v);
rotmat.SetColumn3(s);
GetOrtho( 3, rotmat );
}
// This final version of GetOrtho is mainly for internal use.
// It uses a Gram-Schmidt procedure to extend a partial orthonormal
// basis to a complete orthonormal basis.
// j = number of columns of rotmat that have already been set.
void GetOrtho( int j, RotationMapR4& rotmat)
{
if ( j==0 ) {
rotmat.SetIdentity();
return;
}
if ( j==1 ) {
rotmat.SetColumn2( -rotmat.m21, rotmat.m11, -rotmat.m41, rotmat.m31 );
j = 2;
}
assert ( rotmat.Column1().Norm()<1.0001 && 0.9999<rotmat.Column1().Norm()
&& rotmat.Column1().Norm()<1.0001 && 0.9999<rotmat.Column1().Norm()
&& (rotmat.Column1()^rotmat.Column2()) < 0.001
&& (rotmat.Column1()^rotmat.Column2()) > -0.001 );
// 2x2 subdeterminants in first 2 columns
double d12 = rotmat.m11*rotmat.m22-rotmat.m12*rotmat.m21;
double d13 = rotmat.m11*rotmat.m32-rotmat.m12*rotmat.m31;
double d14 = rotmat.m11*rotmat.m42-rotmat.m12*rotmat.m41;
double d23 = rotmat.m21*rotmat.m32-rotmat.m22*rotmat.m31;
double d24 = rotmat.m21*rotmat.m42-rotmat.m22*rotmat.m41;
double d34 = rotmat.m31*rotmat.m42-rotmat.m32*rotmat.m41;
VectorR4 vec3;
if ( j==2 ) {
if ( d12>0.4 || d12<-0.4 || d13>0.4 || d13<-0.4
|| d23>0.4 || d23<-0.4 ) {
vec3.Set( d23, -d13, d12, 0.0);
}
else if ( d24>0.4 || d24<-0.4 || d14>0.4 || d14<-0.4 ) {
vec3.Set( d24, -d14, 0.0, d12 );
}
else {
vec3.Set( d34, 0.0, -d14, d13 );
}
vec3.Normalize();
rotmat.SetColumn3(vec3);
}
// Do the final column
rotmat.SetColumn4 (
-rotmat.m23*d34 + rotmat.m33*d24 - rotmat.m43*d23,
rotmat.m13*d34 - rotmat.m33*d14 + rotmat.m43*d13,
-rotmat.m13*d24 + rotmat.m23*d14 - rotmat.m43*d12,
rotmat.m13*d23 - rotmat.m23*d13 + rotmat.m33*d12 );
assert ( 0.99 < (((LinearMapR4)rotmat)).Determinant()
&& (((LinearMapR4)rotmat)).Determinant() < 1.01 );
}
// *********************************************************************
// Rotation routines *
// *********************************************************************
// Rotate unit vector x in the direction of "dir": length of dir is rotation angle.
// x must be a unit vector. dir must be perpindicular to x.
VectorR4& VectorR4::RotateUnitInDirection ( const VectorR4& dir)
{
assert ( this->Norm()<1.0001 && this->Norm()>0.9999 &&
(dir^(*this))<0.0001 && (dir^(*this))>-0.0001 );
double theta = dir.NormSq();
if ( theta==0.0 ) {
return *this;
}
else {
theta = sqrt(theta);
double costheta = cos(theta);
double sintheta = sin(theta);
VectorR4 dirUnit = dir/theta;
*this = costheta*(*this) + sintheta*dirUnit;
// this->NormalizeFast();
return ( *this );
}
}
// RotateToMap returns a RotationMapR4 that rotates fromVec to toVec,
// leaving the orthogonal subspace fixed.
// fromVec and toVec should be unit vectors
RotationMapR4 RotateToMap( const VectorR4& fromVec, const VectorR4& toVec)
{
LinearMapR4 result;
result.SetIdentity();
LinearMapR4 temp;
VectorR4 vPerp = ProjectPerpUnitDiff( toVec, fromVec );
double sintheta = vPerp.Norm(); // theta = angle between toVec and fromVec
VectorR4 vProj = toVec-vPerp;
double costheta = vProj^fromVec;
if ( sintheta == 0.0 ) {
// The vectors either coincide (return identity) or directly oppose
if ( costheta < 0.0 ) {
result = -result; // Vectors directly oppose: return -identity.
}
}
else {
vPerp /= sintheta; // Normalize
VectorProjectMap ( fromVec, temp ); // project in fromVec direction
temp *= (costheta-1.0);
result += temp;
VectorProjectMap ( vPerp, temp ); // Project in vPerp direction
temp *= (costheta-1.0);
result += temp;
TimesTranspose ( vPerp, fromVec, temp ); // temp = (vPerp)*(fromVec^T)
temp *= sintheta;
result += temp;
temp.MakeTranspose();
result -= temp; // (-sintheta)*(fromVec)*(vPerp^T)
}
RotationMapR4 rotationResult;
rotationResult.Set(result); // Make explicitly a RotationMapR4
return rotationResult;
}
// ***************************************************************
// Stream Output Routines *
// * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
ostream& operator<< ( ostream& os, const VectorR4& u )
{
return (os << "<" << u.x << "," << u.y << "," << u.z << "," << u.w << ">");
}