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/* -----------------------------------------------------------------------------
Copyright (c) 2006 Simon Brown si@sjbrown.co.uk
Permission is hereby granted, free of charge, to any person obtaining
a copy of this software and associated documentation files (the
"Software"), to deal in the Software without restriction, including
without limitation the rights to use, copy, modify, merge, publish,
distribute, sublicense, and/or sell copies of the Software, and to
permit persons to whom the Software is furnished to do so, subject to
the following conditions:
The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY
CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT,
TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE
SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
-------------------------------------------------------------------------- */
/*! @file
The symmetric eigensystem solver algorithm is from
http://www.geometrictools.com/Documentation/EigenSymmetric3x3.pdf
*/
#include "maths.h"
#include <cfloat>
namespace squish {
Sym3x3 ComputeWeightedCovariance( int n, Vec3 const* points, float const* weights )
{
// compute the centroid
float total = 0.0f;
Vec3 centroid( 0.0f );
for( int i = 0; i < n; ++i )
{
total += weights[i];
centroid += weights[i]*points[i];
}
centroid /= total;
// accumulate the covariance matrix
Sym3x3 covariance( 0.0f );
for( int i = 0; i < n; ++i )
{
Vec3 a = points[i] - centroid;
Vec3 b = weights[i]*a;
covariance[0] += a.X()*b.X();
covariance[1] += a.X()*b.Y();
covariance[2] += a.X()*b.Z();
covariance[3] += a.Y()*b.Y();
covariance[4] += a.Y()*b.Z();
covariance[5] += a.Z()*b.Z();
}
// return it
return covariance;
}
static Vec3 GetMultiplicity1Evector( Sym3x3 const& matrix, float evalue )
{
// compute M
Sym3x3 m;
m[0] = matrix[0] - evalue;
m[1] = matrix[1];
m[2] = matrix[2];
m[3] = matrix[3] - evalue;
m[4] = matrix[4];
m[5] = matrix[5] - evalue;
// compute U
Sym3x3 u;
u[0] = m[3]*m[5] - m[4]*m[4];
u[1] = m[2]*m[4] - m[1]*m[5];
u[2] = m[1]*m[4] - m[2]*m[3];
u[3] = m[0]*m[5] - m[2]*m[2];
u[4] = m[1]*m[2] - m[4]*m[0];
u[5] = m[0]*m[3] - m[1]*m[1];
// find the largest component
float mc = std::fabs( u[0] );
int mi = 0;
for( int i = 1; i < 6; ++i )
{
float c = std::fabs( u[i] );
if( c > mc )
{
mc = c;
mi = i;
}
}
// pick the column with this component
switch( mi )
{
case 0:
return Vec3( u[0], u[1], u[2] );
case 1:
case 3:
return Vec3( u[1], u[3], u[4] );
default:
return Vec3( u[2], u[4], u[5] );
}
}
static Vec3 GetMultiplicity2Evector( Sym3x3 const& matrix, float evalue )
{
// compute M
Sym3x3 m;
m[0] = matrix[0] - evalue;
m[1] = matrix[1];
m[2] = matrix[2];
m[3] = matrix[3] - evalue;
m[4] = matrix[4];
m[5] = matrix[5] - evalue;
// find the largest component
float mc = std::fabs( m[0] );
int mi = 0;
for( int i = 1; i < 6; ++i )
{
float c = std::fabs( m[i] );
if( c > mc )
{
mc = c;
mi = i;
}
}
// pick the first eigenvector based on this index
switch( mi )
{
case 0:
case 1:
return Vec3( -m[1], m[0], 0.0f );
case 2:
return Vec3( m[2], 0.0f, -m[0] );
case 3:
case 4:
return Vec3( 0.0f, -m[4], m[3] );
default:
return Vec3( 0.0f, -m[5], m[4] );
}
}
Vec3 ComputePrincipleComponent( Sym3x3 const& matrix )
{
// compute the cubic coefficients
float c0 = matrix[0]*matrix[3]*matrix[5]
+ 2.0f*matrix[1]*matrix[2]*matrix[4]
- matrix[0]*matrix[4]*matrix[4]
- matrix[3]*matrix[2]*matrix[2]
- matrix[5]*matrix[1]*matrix[1];
float c1 = matrix[0]*matrix[3] + matrix[0]*matrix[5] + matrix[3]*matrix[5]
- matrix[1]*matrix[1] - matrix[2]*matrix[2] - matrix[4]*matrix[4];
float c2 = matrix[0] + matrix[3] + matrix[5];
// compute the quadratic coefficients
float a = c1 - ( 1.0f/3.0f )*c2*c2;
float b = ( -2.0f/27.0f )*c2*c2*c2 + ( 1.0f/3.0f )*c1*c2 - c0;
// compute the root count check
float Q = 0.25f*b*b + ( 1.0f/27.0f )*a*a*a;
// test the multiplicity
if( FLT_EPSILON < Q )
{
// only one root, which implies we have a multiple of the identity
return Vec3( 1.0f );
}
else if( Q < -FLT_EPSILON )
{
// three distinct roots
float theta = std::atan2( std::sqrt( -Q ), -0.5f*b );
float rho = std::sqrt( 0.25f*b*b - Q );
float rt = std::pow( rho, 1.0f/3.0f );
float ct = std::cos( theta/3.0f );
float st = std::sin( theta/3.0f );
float l1 = ( 1.0f/3.0f )*c2 + 2.0f*rt*ct;
float l2 = ( 1.0f/3.0f )*c2 - rt*( ct + ( float )sqrt( 3.0f )*st );
float l3 = ( 1.0f/3.0f )*c2 - rt*( ct - ( float )sqrt( 3.0f )*st );
// pick the larger
if( std::fabs( l2 ) > std::fabs( l1 ) )
l1 = l2;
if( std::fabs( l3 ) > std::fabs( l1 ) )
l1 = l3;
// get the eigenvector
return GetMultiplicity1Evector( matrix, l1 );
}
else // if( -FLT_EPSILON <= Q && Q <= FLT_EPSILON )
{
// two roots
float rt;
if( b < 0.0f )
rt = -std::pow( -0.5f*b, 1.0f/3.0f );
else
rt = std::pow( 0.5f*b, 1.0f/3.0f );
float l1 = ( 1.0f/3.0f )*c2 + rt; // repeated
float l2 = ( 1.0f/3.0f )*c2 - 2.0f*rt;
// get the eigenvector
if( std::fabs( l1 ) > std::fabs( l2 ) )
return GetMultiplicity2Evector( matrix, l1 );
else
return GetMultiplicity1Evector( matrix, l2 );
}
}
} // namespace squish
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