File: maths.cpp

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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 "simd.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];
    }
    if( total > FLT_EPSILON )
        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;
}

#if 0

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 );
    }
}

#else

#define POWER_ITERATION_COUNT    8

Vec3 ComputePrincipleComponent( Sym3x3 const& matrix )
{
    Vec4 const row0( matrix[0], matrix[1], matrix[2], 0.0f );
    Vec4 const row1( matrix[1], matrix[3], matrix[4], 0.0f );
    Vec4 const row2( matrix[2], matrix[4], matrix[5], 0.0f );
    Vec4 v = VEC4_CONST( 1.0f );
    for( int i = 0; i < POWER_ITERATION_COUNT; ++i )
    {
        // matrix multiply
        Vec4 w = row0*v.SplatX();
        w = MultiplyAdd(row1, v.SplatY(), w);
        w = MultiplyAdd(row2, v.SplatZ(), w);

        // get max component from xyz in all channels
        Vec4 a = Max(w.SplatX(), Max(w.SplatY(), w.SplatZ()));

        // divide through and advance
        v = w*Reciprocal(a);
    }
    return v.GetVec3();
}

#endif

} // namespace squish