[7908] | 1 | /* |
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| 2 | ----------------------------------------------------------------------------- |
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| 3 | This source file is part of OGRE |
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| 4 | (Object-oriented Graphics Rendering Engine) |
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| 5 | For the latest info, see http://www.ogre3d.org/ |
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| 6 | |
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| 7 | Copyright (c) 2000-2006 Torus Knot Software Ltd |
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| 8 | Also see acknowledgements in Readme.html |
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| 9 | |
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| 10 | This program is free software; you can redistribute it and/or modify it under |
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| 11 | the terms of the GNU Lesser General Public License as published by the Free Software |
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| 12 | Foundation; either version 2 of the License, or (at your option) any later |
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| 13 | version. |
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| 14 | |
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| 15 | This program is distributed in the hope that it will be useful, but WITHOUT |
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| 16 | ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS |
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| 17 | FOR A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more details. |
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| 18 | |
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| 19 | You should have received a copy of the GNU Lesser General Public License along with |
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| 20 | this program; if not, write to the Free Software Foundation, Inc., 59 Temple |
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| 21 | Place - Suite 330, Boston, MA 02111-1307, USA, or go to |
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| 22 | http://www.gnu.org/copyleft/lesser.txt. |
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| 23 | |
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| 24 | You may alternatively use this source under the terms of a specific version of |
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| 25 | the OGRE Unrestricted License provided you have obtained such a license from |
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| 26 | Torus Knot Software Ltd. |
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| 27 | ----------------------------------------------------------------------------- |
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| 28 | */ |
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| 29 | #include "OgreMatrix3.h" |
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| 30 | |
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| 31 | #include "OgreMath.h" |
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| 32 | |
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| 33 | // Adapted from Matrix math by Wild Magic http://www.geometrictools.com/ |
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| 34 | |
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| 35 | namespace Ogre |
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| 36 | { |
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| 37 | const Real Matrix3::EPSILON = 1e-06; |
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| 38 | const Matrix3 Matrix3::ZERO(0,0,0,0,0,0,0,0,0); |
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| 39 | const Matrix3 Matrix3::IDENTITY(1,0,0,0,1,0,0,0,1); |
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| 40 | const Real Matrix3::ms_fSvdEpsilon = 1e-04; |
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| 41 | const unsigned int Matrix3::ms_iSvdMaxIterations = 32; |
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| 42 | |
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| 43 | //----------------------------------------------------------------------- |
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| 44 | Vector3 Matrix3::GetColumn (size_t iCol) const |
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| 45 | { |
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| 46 | assert( 0 <= iCol && iCol < 3 ); |
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| 47 | return Vector3(m[0][iCol],m[1][iCol], |
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| 48 | m[2][iCol]); |
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| 49 | } |
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| 50 | //----------------------------------------------------------------------- |
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| 51 | void Matrix3::SetColumn(size_t iCol, const Vector3& vec) |
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| 52 | { |
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| 53 | assert( 0 <= iCol && iCol < 3 ); |
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| 54 | m[0][iCol] = vec.x; |
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| 55 | m[1][iCol] = vec.y; |
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| 56 | m[2][iCol] = vec.z; |
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| 57 | |
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| 58 | } |
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| 59 | //----------------------------------------------------------------------- |
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| 60 | void Matrix3::FromAxes(const Vector3& xAxis, const Vector3& yAxis, const Vector3& zAxis) |
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| 61 | { |
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| 62 | SetColumn(0,xAxis); |
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| 63 | SetColumn(1,yAxis); |
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| 64 | SetColumn(2,zAxis); |
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| 65 | |
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| 66 | } |
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| 67 | |
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| 68 | //----------------------------------------------------------------------- |
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| 69 | bool Matrix3::operator== (const Matrix3& rkMatrix) const |
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| 70 | { |
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| 71 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 72 | { |
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| 73 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 74 | { |
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| 75 | if ( m[iRow][iCol] != rkMatrix.m[iRow][iCol] ) |
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| 76 | return false; |
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| 77 | } |
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| 78 | } |
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| 79 | |
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| 80 | return true; |
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| 81 | } |
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| 82 | //----------------------------------------------------------------------- |
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| 83 | Matrix3 Matrix3::operator+ (const Matrix3& rkMatrix) const |
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| 84 | { |
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| 85 | Matrix3 kSum; |
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| 86 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 87 | { |
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| 88 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 89 | { |
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| 90 | kSum.m[iRow][iCol] = m[iRow][iCol] + |
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| 91 | rkMatrix.m[iRow][iCol]; |
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| 92 | } |
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| 93 | } |
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| 94 | return kSum; |
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| 95 | } |
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| 96 | //----------------------------------------------------------------------- |
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| 97 | Matrix3 Matrix3::operator- (const Matrix3& rkMatrix) const |
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| 98 | { |
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| 99 | Matrix3 kDiff; |
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| 100 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 101 | { |
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| 102 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 103 | { |
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| 104 | kDiff.m[iRow][iCol] = m[iRow][iCol] - |
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| 105 | rkMatrix.m[iRow][iCol]; |
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| 106 | } |
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| 107 | } |
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| 108 | return kDiff; |
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| 109 | } |
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| 110 | //----------------------------------------------------------------------- |
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| 111 | Matrix3 Matrix3::operator* (const Matrix3& rkMatrix) const |
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| 112 | { |
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| 113 | Matrix3 kProd; |
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| 114 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 115 | { |
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| 116 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 117 | { |
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| 118 | kProd.m[iRow][iCol] = |
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| 119 | m[iRow][0]*rkMatrix.m[0][iCol] + |
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| 120 | m[iRow][1]*rkMatrix.m[1][iCol] + |
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| 121 | m[iRow][2]*rkMatrix.m[2][iCol]; |
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| 122 | } |
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| 123 | } |
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| 124 | return kProd; |
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| 125 | } |
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| 126 | //----------------------------------------------------------------------- |
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| 127 | Vector3 Matrix3::operator* (const Vector3& rkPoint) const |
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| 128 | { |
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| 129 | Vector3 kProd; |
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| 130 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 131 | { |
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| 132 | kProd[iRow] = |
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| 133 | m[iRow][0]*rkPoint[0] + |
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| 134 | m[iRow][1]*rkPoint[1] + |
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| 135 | m[iRow][2]*rkPoint[2]; |
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| 136 | } |
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| 137 | return kProd; |
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| 138 | } |
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| 139 | //----------------------------------------------------------------------- |
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| 140 | Vector3 operator* (const Vector3& rkPoint, const Matrix3& rkMatrix) |
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| 141 | { |
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| 142 | Vector3 kProd; |
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| 143 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 144 | { |
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| 145 | kProd[iRow] = |
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| 146 | rkPoint[0]*rkMatrix.m[0][iRow] + |
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| 147 | rkPoint[1]*rkMatrix.m[1][iRow] + |
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| 148 | rkPoint[2]*rkMatrix.m[2][iRow]; |
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| 149 | } |
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| 150 | return kProd; |
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| 151 | } |
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| 152 | //----------------------------------------------------------------------- |
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| 153 | Matrix3 Matrix3::operator- () const |
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| 154 | { |
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| 155 | Matrix3 kNeg; |
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| 156 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 157 | { |
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| 158 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 159 | kNeg[iRow][iCol] = -m[iRow][iCol]; |
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| 160 | } |
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| 161 | return kNeg; |
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| 162 | } |
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| 163 | //----------------------------------------------------------------------- |
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| 164 | Matrix3 Matrix3::operator* (Real fScalar) const |
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| 165 | { |
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| 166 | Matrix3 kProd; |
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| 167 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 168 | { |
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| 169 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 170 | kProd[iRow][iCol] = fScalar*m[iRow][iCol]; |
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| 171 | } |
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| 172 | return kProd; |
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| 173 | } |
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| 174 | //----------------------------------------------------------------------- |
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| 175 | Matrix3 operator* (Real fScalar, const Matrix3& rkMatrix) |
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| 176 | { |
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| 177 | Matrix3 kProd; |
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| 178 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 179 | { |
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| 180 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 181 | kProd[iRow][iCol] = fScalar*rkMatrix.m[iRow][iCol]; |
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| 182 | } |
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| 183 | return kProd; |
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| 184 | } |
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| 185 | //----------------------------------------------------------------------- |
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| 186 | Matrix3 Matrix3::Transpose () const |
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| 187 | { |
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| 188 | Matrix3 kTranspose; |
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| 189 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 190 | { |
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| 191 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 192 | kTranspose[iRow][iCol] = m[iCol][iRow]; |
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| 193 | } |
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| 194 | return kTranspose; |
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| 195 | } |
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| 196 | //----------------------------------------------------------------------- |
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| 197 | bool Matrix3::Inverse (Matrix3& rkInverse, Real fTolerance) const |
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| 198 | { |
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| 199 | // Invert a 3x3 using cofactors. This is about 8 times faster than |
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| 200 | // the Numerical Recipes code which uses Gaussian elimination. |
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| 201 | |
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| 202 | rkInverse[0][0] = m[1][1]*m[2][2] - |
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| 203 | m[1][2]*m[2][1]; |
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| 204 | rkInverse[0][1] = m[0][2]*m[2][1] - |
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| 205 | m[0][1]*m[2][2]; |
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| 206 | rkInverse[0][2] = m[0][1]*m[1][2] - |
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| 207 | m[0][2]*m[1][1]; |
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| 208 | rkInverse[1][0] = m[1][2]*m[2][0] - |
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| 209 | m[1][0]*m[2][2]; |
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| 210 | rkInverse[1][1] = m[0][0]*m[2][2] - |
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| 211 | m[0][2]*m[2][0]; |
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| 212 | rkInverse[1][2] = m[0][2]*m[1][0] - |
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| 213 | m[0][0]*m[1][2]; |
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| 214 | rkInverse[2][0] = m[1][0]*m[2][1] - |
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| 215 | m[1][1]*m[2][0]; |
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| 216 | rkInverse[2][1] = m[0][1]*m[2][0] - |
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| 217 | m[0][0]*m[2][1]; |
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| 218 | rkInverse[2][2] = m[0][0]*m[1][1] - |
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| 219 | m[0][1]*m[1][0]; |
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| 220 | |
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| 221 | Real fDet = |
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| 222 | m[0][0]*rkInverse[0][0] + |
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| 223 | m[0][1]*rkInverse[1][0]+ |
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| 224 | m[0][2]*rkInverse[2][0]; |
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| 225 | |
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| 226 | if ( Math::Abs(fDet) <= fTolerance ) |
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| 227 | return false; |
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| 228 | |
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| 229 | Real fInvDet = 1.0/fDet; |
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| 230 | for (size_t iRow = 0; iRow < 3; iRow++) |
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| 231 | { |
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| 232 | for (size_t iCol = 0; iCol < 3; iCol++) |
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| 233 | rkInverse[iRow][iCol] *= fInvDet; |
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| 234 | } |
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| 235 | |
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| 236 | return true; |
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| 237 | } |
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| 238 | //----------------------------------------------------------------------- |
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| 239 | Matrix3 Matrix3::Inverse (Real fTolerance) const |
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| 240 | { |
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| 241 | Matrix3 kInverse = Matrix3::ZERO; |
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| 242 | Inverse(kInverse,fTolerance); |
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| 243 | return kInverse; |
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| 244 | } |
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| 245 | //----------------------------------------------------------------------- |
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| 246 | Real Matrix3::Determinant () const |
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| 247 | { |
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| 248 | Real fCofactor00 = m[1][1]*m[2][2] - |
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| 249 | m[1][2]*m[2][1]; |
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| 250 | Real fCofactor10 = m[1][2]*m[2][0] - |
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| 251 | m[1][0]*m[2][2]; |
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| 252 | Real fCofactor20 = m[1][0]*m[2][1] - |
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| 253 | m[1][1]*m[2][0]; |
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| 254 | |
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| 255 | Real fDet = |
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| 256 | m[0][0]*fCofactor00 + |
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| 257 | m[0][1]*fCofactor10 + |
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| 258 | m[0][2]*fCofactor20; |
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| 259 | |
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| 260 | return fDet; |
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| 261 | } |
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| 262 | //----------------------------------------------------------------------- |
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| 263 | void Matrix3::Bidiagonalize (Matrix3& kA, Matrix3& kL, |
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| 264 | Matrix3& kR) |
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| 265 | { |
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| 266 | Real afV[3], afW[3]; |
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| 267 | Real fLength, fSign, fT1, fInvT1, fT2; |
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| 268 | bool bIdentity; |
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| 269 | |
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| 270 | // map first column to (*,0,0) |
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| 271 | fLength = Math::Sqrt(kA[0][0]*kA[0][0] + kA[1][0]*kA[1][0] + |
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| 272 | kA[2][0]*kA[2][0]); |
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| 273 | if ( fLength > 0.0 ) |
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| 274 | { |
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| 275 | fSign = (kA[0][0] > 0.0 ? 1.0 : -1.0); |
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| 276 | fT1 = kA[0][0] + fSign*fLength; |
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| 277 | fInvT1 = 1.0/fT1; |
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| 278 | afV[1] = kA[1][0]*fInvT1; |
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| 279 | afV[2] = kA[2][0]*fInvT1; |
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| 280 | |
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| 281 | fT2 = -2.0/(1.0+afV[1]*afV[1]+afV[2]*afV[2]); |
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| 282 | afW[0] = fT2*(kA[0][0]+kA[1][0]*afV[1]+kA[2][0]*afV[2]); |
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| 283 | afW[1] = fT2*(kA[0][1]+kA[1][1]*afV[1]+kA[2][1]*afV[2]); |
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| 284 | afW[2] = fT2*(kA[0][2]+kA[1][2]*afV[1]+kA[2][2]*afV[2]); |
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| 285 | kA[0][0] += afW[0]; |
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| 286 | kA[0][1] += afW[1]; |
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| 287 | kA[0][2] += afW[2]; |
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| 288 | kA[1][1] += afV[1]*afW[1]; |
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| 289 | kA[1][2] += afV[1]*afW[2]; |
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| 290 | kA[2][1] += afV[2]*afW[1]; |
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| 291 | kA[2][2] += afV[2]*afW[2]; |
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| 292 | |
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| 293 | kL[0][0] = 1.0+fT2; |
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| 294 | kL[0][1] = kL[1][0] = fT2*afV[1]; |
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| 295 | kL[0][2] = kL[2][0] = fT2*afV[2]; |
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| 296 | kL[1][1] = 1.0+fT2*afV[1]*afV[1]; |
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| 297 | kL[1][2] = kL[2][1] = fT2*afV[1]*afV[2]; |
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| 298 | kL[2][2] = 1.0+fT2*afV[2]*afV[2]; |
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| 299 | bIdentity = false; |
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| 300 | } |
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| 301 | else |
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| 302 | { |
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| 303 | kL = Matrix3::IDENTITY; |
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| 304 | bIdentity = true; |
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| 305 | } |
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| 306 | |
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| 307 | // map first row to (*,*,0) |
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| 308 | fLength = Math::Sqrt(kA[0][1]*kA[0][1]+kA[0][2]*kA[0][2]); |
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| 309 | if ( fLength > 0.0 ) |
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| 310 | { |
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| 311 | fSign = (kA[0][1] > 0.0 ? 1.0 : -1.0); |
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| 312 | fT1 = kA[0][1] + fSign*fLength; |
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| 313 | afV[2] = kA[0][2]/fT1; |
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| 314 | |
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| 315 | fT2 = -2.0/(1.0+afV[2]*afV[2]); |
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| 316 | afW[0] = fT2*(kA[0][1]+kA[0][2]*afV[2]); |
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| 317 | afW[1] = fT2*(kA[1][1]+kA[1][2]*afV[2]); |
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| 318 | afW[2] = fT2*(kA[2][1]+kA[2][2]*afV[2]); |
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| 319 | kA[0][1] += afW[0]; |
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| 320 | kA[1][1] += afW[1]; |
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| 321 | kA[1][2] += afW[1]*afV[2]; |
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| 322 | kA[2][1] += afW[2]; |
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| 323 | kA[2][2] += afW[2]*afV[2]; |
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| 324 | |
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| 325 | kR[0][0] = 1.0; |
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| 326 | kR[0][1] = kR[1][0] = 0.0; |
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| 327 | kR[0][2] = kR[2][0] = 0.0; |
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| 328 | kR[1][1] = 1.0+fT2; |
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| 329 | kR[1][2] = kR[2][1] = fT2*afV[2]; |
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| 330 | kR[2][2] = 1.0+fT2*afV[2]*afV[2]; |
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| 331 | } |
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| 332 | else |
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| 333 | { |
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| 334 | kR = Matrix3::IDENTITY; |
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| 335 | } |
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| 336 | |
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| 337 | // map second column to (*,*,0) |
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| 338 | fLength = Math::Sqrt(kA[1][1]*kA[1][1]+kA[2][1]*kA[2][1]); |
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| 339 | if ( fLength > 0.0 ) |
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| 340 | { |
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| 341 | fSign = (kA[1][1] > 0.0 ? 1.0 : -1.0); |
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| 342 | fT1 = kA[1][1] + fSign*fLength; |
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| 343 | afV[2] = kA[2][1]/fT1; |
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| 344 | |
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| 345 | fT2 = -2.0/(1.0+afV[2]*afV[2]); |
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| 346 | afW[1] = fT2*(kA[1][1]+kA[2][1]*afV[2]); |
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| 347 | afW[2] = fT2*(kA[1][2]+kA[2][2]*afV[2]); |
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| 348 | kA[1][1] += afW[1]; |
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| 349 | kA[1][2] += afW[2]; |
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| 350 | kA[2][2] += afV[2]*afW[2]; |
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| 351 | |
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| 352 | Real fA = 1.0+fT2; |
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| 353 | Real fB = fT2*afV[2]; |
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| 354 | Real fC = 1.0+fB*afV[2]; |
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| 355 | |
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| 356 | if ( bIdentity ) |
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| 357 | { |
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| 358 | kL[0][0] = 1.0; |
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| 359 | kL[0][1] = kL[1][0] = 0.0; |
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| 360 | kL[0][2] = kL[2][0] = 0.0; |
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| 361 | kL[1][1] = fA; |
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| 362 | kL[1][2] = kL[2][1] = fB; |
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| 363 | kL[2][2] = fC; |
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| 364 | } |
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| 365 | else |
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| 366 | { |
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| 367 | for (int iRow = 0; iRow < 3; iRow++) |
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| 368 | { |
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| 369 | Real fTmp0 = kL[iRow][1]; |
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| 370 | Real fTmp1 = kL[iRow][2]; |
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| 371 | kL[iRow][1] = fA*fTmp0+fB*fTmp1; |
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| 372 | kL[iRow][2] = fB*fTmp0+fC*fTmp1; |
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| 373 | } |
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| 374 | } |
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| 375 | } |
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| 376 | } |
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| 377 | //----------------------------------------------------------------------- |
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| 378 | void Matrix3::GolubKahanStep (Matrix3& kA, Matrix3& kL, |
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| 379 | Matrix3& kR) |
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| 380 | { |
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| 381 | Real fT11 = kA[0][1]*kA[0][1]+kA[1][1]*kA[1][1]; |
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| 382 | Real fT22 = kA[1][2]*kA[1][2]+kA[2][2]*kA[2][2]; |
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| 383 | Real fT12 = kA[1][1]*kA[1][2]; |
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| 384 | Real fTrace = fT11+fT22; |
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| 385 | Real fDiff = fT11-fT22; |
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| 386 | Real fDiscr = Math::Sqrt(fDiff*fDiff+4.0*fT12*fT12); |
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| 387 | Real fRoot1 = 0.5*(fTrace+fDiscr); |
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| 388 | Real fRoot2 = 0.5*(fTrace-fDiscr); |
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| 389 | |
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| 390 | // adjust right |
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| 391 | Real fY = kA[0][0] - (Math::Abs(fRoot1-fT22) <= |
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| 392 | Math::Abs(fRoot2-fT22) ? fRoot1 : fRoot2); |
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| 393 | Real fZ = kA[0][1]; |
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| 394 | Real fInvLength = Math::InvSqrt(fY*fY+fZ*fZ); |
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| 395 | Real fSin = fZ*fInvLength; |
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| 396 | Real fCos = -fY*fInvLength; |
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| 397 | |
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| 398 | Real fTmp0 = kA[0][0]; |
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| 399 | Real fTmp1 = kA[0][1]; |
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| 400 | kA[0][0] = fCos*fTmp0-fSin*fTmp1; |
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| 401 | kA[0][1] = fSin*fTmp0+fCos*fTmp1; |
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| 402 | kA[1][0] = -fSin*kA[1][1]; |
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| 403 | kA[1][1] *= fCos; |
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| 404 | |
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| 405 | size_t iRow; |
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| 406 | for (iRow = 0; iRow < 3; iRow++) |
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| 407 | { |
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| 408 | fTmp0 = kR[0][iRow]; |
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| 409 | fTmp1 = kR[1][iRow]; |
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| 410 | kR[0][iRow] = fCos*fTmp0-fSin*fTmp1; |
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| 411 | kR[1][iRow] = fSin*fTmp0+fCos*fTmp1; |
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| 412 | } |
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| 413 | |
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| 414 | // adjust left |
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| 415 | fY = kA[0][0]; |
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| 416 | fZ = kA[1][0]; |
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| 417 | fInvLength = Math::InvSqrt(fY*fY+fZ*fZ); |
---|
| 418 | fSin = fZ*fInvLength; |
---|
| 419 | fCos = -fY*fInvLength; |
---|
| 420 | |
---|
| 421 | kA[0][0] = fCos*kA[0][0]-fSin*kA[1][0]; |
---|
| 422 | fTmp0 = kA[0][1]; |
---|
| 423 | fTmp1 = kA[1][1]; |
---|
| 424 | kA[0][1] = fCos*fTmp0-fSin*fTmp1; |
---|
| 425 | kA[1][1] = fSin*fTmp0+fCos*fTmp1; |
---|
| 426 | kA[0][2] = -fSin*kA[1][2]; |
---|
| 427 | kA[1][2] *= fCos; |
---|
| 428 | |
---|
| 429 | size_t iCol; |
---|
| 430 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 431 | { |
---|
| 432 | fTmp0 = kL[iCol][0]; |
---|
| 433 | fTmp1 = kL[iCol][1]; |
---|
| 434 | kL[iCol][0] = fCos*fTmp0-fSin*fTmp1; |
---|
| 435 | kL[iCol][1] = fSin*fTmp0+fCos*fTmp1; |
---|
| 436 | } |
---|
| 437 | |
---|
| 438 | // adjust right |
---|
| 439 | fY = kA[0][1]; |
---|
| 440 | fZ = kA[0][2]; |
---|
| 441 | fInvLength = Math::InvSqrt(fY*fY+fZ*fZ); |
---|
| 442 | fSin = fZ*fInvLength; |
---|
| 443 | fCos = -fY*fInvLength; |
---|
| 444 | |
---|
| 445 | kA[0][1] = fCos*kA[0][1]-fSin*kA[0][2]; |
---|
| 446 | fTmp0 = kA[1][1]; |
---|
| 447 | fTmp1 = kA[1][2]; |
---|
| 448 | kA[1][1] = fCos*fTmp0-fSin*fTmp1; |
---|
| 449 | kA[1][2] = fSin*fTmp0+fCos*fTmp1; |
---|
| 450 | kA[2][1] = -fSin*kA[2][2]; |
---|
| 451 | kA[2][2] *= fCos; |
---|
| 452 | |
---|
| 453 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 454 | { |
---|
| 455 | fTmp0 = kR[1][iRow]; |
---|
| 456 | fTmp1 = kR[2][iRow]; |
---|
| 457 | kR[1][iRow] = fCos*fTmp0-fSin*fTmp1; |
---|
| 458 | kR[2][iRow] = fSin*fTmp0+fCos*fTmp1; |
---|
| 459 | } |
---|
| 460 | |
---|
| 461 | // adjust left |
---|
| 462 | fY = kA[1][1]; |
---|
| 463 | fZ = kA[2][1]; |
---|
| 464 | fInvLength = Math::InvSqrt(fY*fY+fZ*fZ); |
---|
| 465 | fSin = fZ*fInvLength; |
---|
| 466 | fCos = -fY*fInvLength; |
---|
| 467 | |
---|
| 468 | kA[1][1] = fCos*kA[1][1]-fSin*kA[2][1]; |
---|
| 469 | fTmp0 = kA[1][2]; |
---|
| 470 | fTmp1 = kA[2][2]; |
---|
| 471 | kA[1][2] = fCos*fTmp0-fSin*fTmp1; |
---|
| 472 | kA[2][2] = fSin*fTmp0+fCos*fTmp1; |
---|
| 473 | |
---|
| 474 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 475 | { |
---|
| 476 | fTmp0 = kL[iCol][1]; |
---|
| 477 | fTmp1 = kL[iCol][2]; |
---|
| 478 | kL[iCol][1] = fCos*fTmp0-fSin*fTmp1; |
---|
| 479 | kL[iCol][2] = fSin*fTmp0+fCos*fTmp1; |
---|
| 480 | } |
---|
| 481 | } |
---|
| 482 | //----------------------------------------------------------------------- |
---|
| 483 | void Matrix3::SingularValueDecomposition (Matrix3& kL, Vector3& kS, |
---|
| 484 | Matrix3& kR) const |
---|
| 485 | { |
---|
| 486 | // temas: currently unused |
---|
| 487 | //const int iMax = 16; |
---|
| 488 | size_t iRow, iCol; |
---|
| 489 | |
---|
| 490 | Matrix3 kA = *this; |
---|
| 491 | Bidiagonalize(kA,kL,kR); |
---|
| 492 | |
---|
| 493 | for (unsigned int i = 0; i < ms_iSvdMaxIterations; i++) |
---|
| 494 | { |
---|
| 495 | Real fTmp, fTmp0, fTmp1; |
---|
| 496 | Real fSin0, fCos0, fTan0; |
---|
| 497 | Real fSin1, fCos1, fTan1; |
---|
| 498 | |
---|
| 499 | bool bTest1 = (Math::Abs(kA[0][1]) <= |
---|
| 500 | ms_fSvdEpsilon*(Math::Abs(kA[0][0])+Math::Abs(kA[1][1]))); |
---|
| 501 | bool bTest2 = (Math::Abs(kA[1][2]) <= |
---|
| 502 | ms_fSvdEpsilon*(Math::Abs(kA[1][1])+Math::Abs(kA[2][2]))); |
---|
| 503 | if ( bTest1 ) |
---|
| 504 | { |
---|
| 505 | if ( bTest2 ) |
---|
| 506 | { |
---|
| 507 | kS[0] = kA[0][0]; |
---|
| 508 | kS[1] = kA[1][1]; |
---|
| 509 | kS[2] = kA[2][2]; |
---|
| 510 | break; |
---|
| 511 | } |
---|
| 512 | else |
---|
| 513 | { |
---|
| 514 | // 2x2 closed form factorization |
---|
| 515 | fTmp = (kA[1][1]*kA[1][1] - kA[2][2]*kA[2][2] + |
---|
| 516 | kA[1][2]*kA[1][2])/(kA[1][2]*kA[2][2]); |
---|
| 517 | fTan0 = 0.5*(fTmp+Math::Sqrt(fTmp*fTmp + 4.0)); |
---|
| 518 | fCos0 = Math::InvSqrt(1.0+fTan0*fTan0); |
---|
| 519 | fSin0 = fTan0*fCos0; |
---|
| 520 | |
---|
| 521 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 522 | { |
---|
| 523 | fTmp0 = kL[iCol][1]; |
---|
| 524 | fTmp1 = kL[iCol][2]; |
---|
| 525 | kL[iCol][1] = fCos0*fTmp0-fSin0*fTmp1; |
---|
| 526 | kL[iCol][2] = fSin0*fTmp0+fCos0*fTmp1; |
---|
| 527 | } |
---|
| 528 | |
---|
| 529 | fTan1 = (kA[1][2]-kA[2][2]*fTan0)/kA[1][1]; |
---|
| 530 | fCos1 = Math::InvSqrt(1.0+fTan1*fTan1); |
---|
| 531 | fSin1 = -fTan1*fCos1; |
---|
| 532 | |
---|
| 533 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 534 | { |
---|
| 535 | fTmp0 = kR[1][iRow]; |
---|
| 536 | fTmp1 = kR[2][iRow]; |
---|
| 537 | kR[1][iRow] = fCos1*fTmp0-fSin1*fTmp1; |
---|
| 538 | kR[2][iRow] = fSin1*fTmp0+fCos1*fTmp1; |
---|
| 539 | } |
---|
| 540 | |
---|
| 541 | kS[0] = kA[0][0]; |
---|
| 542 | kS[1] = fCos0*fCos1*kA[1][1] - |
---|
| 543 | fSin1*(fCos0*kA[1][2]-fSin0*kA[2][2]); |
---|
| 544 | kS[2] = fSin0*fSin1*kA[1][1] + |
---|
| 545 | fCos1*(fSin0*kA[1][2]+fCos0*kA[2][2]); |
---|
| 546 | break; |
---|
| 547 | } |
---|
| 548 | } |
---|
| 549 | else |
---|
| 550 | { |
---|
| 551 | if ( bTest2 ) |
---|
| 552 | { |
---|
| 553 | // 2x2 closed form factorization |
---|
| 554 | fTmp = (kA[0][0]*kA[0][0] + kA[1][1]*kA[1][1] - |
---|
| 555 | kA[0][1]*kA[0][1])/(kA[0][1]*kA[1][1]); |
---|
| 556 | fTan0 = 0.5*(-fTmp+Math::Sqrt(fTmp*fTmp + 4.0)); |
---|
| 557 | fCos0 = Math::InvSqrt(1.0+fTan0*fTan0); |
---|
| 558 | fSin0 = fTan0*fCos0; |
---|
| 559 | |
---|
| 560 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 561 | { |
---|
| 562 | fTmp0 = kL[iCol][0]; |
---|
| 563 | fTmp1 = kL[iCol][1]; |
---|
| 564 | kL[iCol][0] = fCos0*fTmp0-fSin0*fTmp1; |
---|
| 565 | kL[iCol][1] = fSin0*fTmp0+fCos0*fTmp1; |
---|
| 566 | } |
---|
| 567 | |
---|
| 568 | fTan1 = (kA[0][1]-kA[1][1]*fTan0)/kA[0][0]; |
---|
| 569 | fCos1 = Math::InvSqrt(1.0+fTan1*fTan1); |
---|
| 570 | fSin1 = -fTan1*fCos1; |
---|
| 571 | |
---|
| 572 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 573 | { |
---|
| 574 | fTmp0 = kR[0][iRow]; |
---|
| 575 | fTmp1 = kR[1][iRow]; |
---|
| 576 | kR[0][iRow] = fCos1*fTmp0-fSin1*fTmp1; |
---|
| 577 | kR[1][iRow] = fSin1*fTmp0+fCos1*fTmp1; |
---|
| 578 | } |
---|
| 579 | |
---|
| 580 | kS[0] = fCos0*fCos1*kA[0][0] - |
---|
| 581 | fSin1*(fCos0*kA[0][1]-fSin0*kA[1][1]); |
---|
| 582 | kS[1] = fSin0*fSin1*kA[0][0] + |
---|
| 583 | fCos1*(fSin0*kA[0][1]+fCos0*kA[1][1]); |
---|
| 584 | kS[2] = kA[2][2]; |
---|
| 585 | break; |
---|
| 586 | } |
---|
| 587 | else |
---|
| 588 | { |
---|
| 589 | GolubKahanStep(kA,kL,kR); |
---|
| 590 | } |
---|
| 591 | } |
---|
| 592 | } |
---|
| 593 | |
---|
| 594 | // positize diagonal |
---|
| 595 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 596 | { |
---|
| 597 | if ( kS[iRow] < 0.0 ) |
---|
| 598 | { |
---|
| 599 | kS[iRow] = -kS[iRow]; |
---|
| 600 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 601 | kR[iRow][iCol] = -kR[iRow][iCol]; |
---|
| 602 | } |
---|
| 603 | } |
---|
| 604 | } |
---|
| 605 | //----------------------------------------------------------------------- |
---|
| 606 | void Matrix3::SingularValueComposition (const Matrix3& kL, |
---|
| 607 | const Vector3& kS, const Matrix3& kR) |
---|
| 608 | { |
---|
| 609 | size_t iRow, iCol; |
---|
| 610 | Matrix3 kTmp; |
---|
| 611 | |
---|
| 612 | // product S*R |
---|
| 613 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 614 | { |
---|
| 615 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 616 | kTmp[iRow][iCol] = kS[iRow]*kR[iRow][iCol]; |
---|
| 617 | } |
---|
| 618 | |
---|
| 619 | // product L*S*R |
---|
| 620 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 621 | { |
---|
| 622 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 623 | { |
---|
| 624 | m[iRow][iCol] = 0.0; |
---|
| 625 | for (int iMid = 0; iMid < 3; iMid++) |
---|
| 626 | m[iRow][iCol] += kL[iRow][iMid]*kTmp[iMid][iCol]; |
---|
| 627 | } |
---|
| 628 | } |
---|
| 629 | } |
---|
| 630 | //----------------------------------------------------------------------- |
---|
| 631 | void Matrix3::Orthonormalize () |
---|
| 632 | { |
---|
| 633 | // Algorithm uses Gram-Schmidt orthogonalization. If 'this' matrix is |
---|
| 634 | // M = [m0|m1|m2], then orthonormal output matrix is Q = [q0|q1|q2], |
---|
| 635 | // |
---|
| 636 | // q0 = m0/|m0| |
---|
| 637 | // q1 = (m1-(q0*m1)q0)/|m1-(q0*m1)q0| |
---|
| 638 | // q2 = (m2-(q0*m2)q0-(q1*m2)q1)/|m2-(q0*m2)q0-(q1*m2)q1| |
---|
| 639 | // |
---|
| 640 | // where |V| indicates length of vector V and A*B indicates dot |
---|
| 641 | // product of vectors A and B. |
---|
| 642 | |
---|
| 643 | // compute q0 |
---|
| 644 | Real fInvLength = Math::InvSqrt(m[0][0]*m[0][0] |
---|
| 645 | + m[1][0]*m[1][0] + |
---|
| 646 | m[2][0]*m[2][0]); |
---|
| 647 | |
---|
| 648 | m[0][0] *= fInvLength; |
---|
| 649 | m[1][0] *= fInvLength; |
---|
| 650 | m[2][0] *= fInvLength; |
---|
| 651 | |
---|
| 652 | // compute q1 |
---|
| 653 | Real fDot0 = |
---|
| 654 | m[0][0]*m[0][1] + |
---|
| 655 | m[1][0]*m[1][1] + |
---|
| 656 | m[2][0]*m[2][1]; |
---|
| 657 | |
---|
| 658 | m[0][1] -= fDot0*m[0][0]; |
---|
| 659 | m[1][1] -= fDot0*m[1][0]; |
---|
| 660 | m[2][1] -= fDot0*m[2][0]; |
---|
| 661 | |
---|
| 662 | fInvLength = Math::InvSqrt(m[0][1]*m[0][1] + |
---|
| 663 | m[1][1]*m[1][1] + |
---|
| 664 | m[2][1]*m[2][1]); |
---|
| 665 | |
---|
| 666 | m[0][1] *= fInvLength; |
---|
| 667 | m[1][1] *= fInvLength; |
---|
| 668 | m[2][1] *= fInvLength; |
---|
| 669 | |
---|
| 670 | // compute q2 |
---|
| 671 | Real fDot1 = |
---|
| 672 | m[0][1]*m[0][2] + |
---|
| 673 | m[1][1]*m[1][2] + |
---|
| 674 | m[2][1]*m[2][2]; |
---|
| 675 | |
---|
| 676 | fDot0 = |
---|
| 677 | m[0][0]*m[0][2] + |
---|
| 678 | m[1][0]*m[1][2] + |
---|
| 679 | m[2][0]*m[2][2]; |
---|
| 680 | |
---|
| 681 | m[0][2] -= fDot0*m[0][0] + fDot1*m[0][1]; |
---|
| 682 | m[1][2] -= fDot0*m[1][0] + fDot1*m[1][1]; |
---|
| 683 | m[2][2] -= fDot0*m[2][0] + fDot1*m[2][1]; |
---|
| 684 | |
---|
| 685 | fInvLength = Math::InvSqrt(m[0][2]*m[0][2] + |
---|
| 686 | m[1][2]*m[1][2] + |
---|
| 687 | m[2][2]*m[2][2]); |
---|
| 688 | |
---|
| 689 | m[0][2] *= fInvLength; |
---|
| 690 | m[1][2] *= fInvLength; |
---|
| 691 | m[2][2] *= fInvLength; |
---|
| 692 | } |
---|
| 693 | //----------------------------------------------------------------------- |
---|
| 694 | void Matrix3::QDUDecomposition (Matrix3& kQ, |
---|
| 695 | Vector3& kD, Vector3& kU) const |
---|
| 696 | { |
---|
| 697 | // Factor M = QR = QDU where Q is orthogonal, D is diagonal, |
---|
| 698 | // and U is upper triangular with ones on its diagonal. Algorithm uses |
---|
| 699 | // Gram-Schmidt orthogonalization (the QR algorithm). |
---|
| 700 | // |
---|
| 701 | // If M = [ m0 | m1 | m2 ] and Q = [ q0 | q1 | q2 ], then |
---|
| 702 | // |
---|
| 703 | // q0 = m0/|m0| |
---|
| 704 | // q1 = (m1-(q0*m1)q0)/|m1-(q0*m1)q0| |
---|
| 705 | // q2 = (m2-(q0*m2)q0-(q1*m2)q1)/|m2-(q0*m2)q0-(q1*m2)q1| |
---|
| 706 | // |
---|
| 707 | // where |V| indicates length of vector V and A*B indicates dot |
---|
| 708 | // product of vectors A and B. The matrix R has entries |
---|
| 709 | // |
---|
| 710 | // r00 = q0*m0 r01 = q0*m1 r02 = q0*m2 |
---|
| 711 | // r10 = 0 r11 = q1*m1 r12 = q1*m2 |
---|
| 712 | // r20 = 0 r21 = 0 r22 = q2*m2 |
---|
| 713 | // |
---|
| 714 | // so D = diag(r00,r11,r22) and U has entries u01 = r01/r00, |
---|
| 715 | // u02 = r02/r00, and u12 = r12/r11. |
---|
| 716 | |
---|
| 717 | // Q = rotation |
---|
| 718 | // D = scaling |
---|
| 719 | // U = shear |
---|
| 720 | |
---|
| 721 | // D stores the three diagonal entries r00, r11, r22 |
---|
| 722 | // U stores the entries U[0] = u01, U[1] = u02, U[2] = u12 |
---|
| 723 | |
---|
| 724 | // build orthogonal matrix Q |
---|
| 725 | Real fInvLength = Math::InvSqrt(m[0][0]*m[0][0] |
---|
| 726 | + m[1][0]*m[1][0] + |
---|
| 727 | m[2][0]*m[2][0]); |
---|
| 728 | kQ[0][0] = m[0][0]*fInvLength; |
---|
| 729 | kQ[1][0] = m[1][0]*fInvLength; |
---|
| 730 | kQ[2][0] = m[2][0]*fInvLength; |
---|
| 731 | |
---|
| 732 | Real fDot = kQ[0][0]*m[0][1] + kQ[1][0]*m[1][1] + |
---|
| 733 | kQ[2][0]*m[2][1]; |
---|
| 734 | kQ[0][1] = m[0][1]-fDot*kQ[0][0]; |
---|
| 735 | kQ[1][1] = m[1][1]-fDot*kQ[1][0]; |
---|
| 736 | kQ[2][1] = m[2][1]-fDot*kQ[2][0]; |
---|
| 737 | fInvLength = Math::InvSqrt(kQ[0][1]*kQ[0][1] + kQ[1][1]*kQ[1][1] + |
---|
| 738 | kQ[2][1]*kQ[2][1]); |
---|
| 739 | kQ[0][1] *= fInvLength; |
---|
| 740 | kQ[1][1] *= fInvLength; |
---|
| 741 | kQ[2][1] *= fInvLength; |
---|
| 742 | |
---|
| 743 | fDot = kQ[0][0]*m[0][2] + kQ[1][0]*m[1][2] + |
---|
| 744 | kQ[2][0]*m[2][2]; |
---|
| 745 | kQ[0][2] = m[0][2]-fDot*kQ[0][0]; |
---|
| 746 | kQ[1][2] = m[1][2]-fDot*kQ[1][0]; |
---|
| 747 | kQ[2][2] = m[2][2]-fDot*kQ[2][0]; |
---|
| 748 | fDot = kQ[0][1]*m[0][2] + kQ[1][1]*m[1][2] + |
---|
| 749 | kQ[2][1]*m[2][2]; |
---|
| 750 | kQ[0][2] -= fDot*kQ[0][1]; |
---|
| 751 | kQ[1][2] -= fDot*kQ[1][1]; |
---|
| 752 | kQ[2][2] -= fDot*kQ[2][1]; |
---|
| 753 | fInvLength = Math::InvSqrt(kQ[0][2]*kQ[0][2] + kQ[1][2]*kQ[1][2] + |
---|
| 754 | kQ[2][2]*kQ[2][2]); |
---|
| 755 | kQ[0][2] *= fInvLength; |
---|
| 756 | kQ[1][2] *= fInvLength; |
---|
| 757 | kQ[2][2] *= fInvLength; |
---|
| 758 | |
---|
| 759 | // guarantee that orthogonal matrix has determinant 1 (no reflections) |
---|
| 760 | Real fDet = kQ[0][0]*kQ[1][1]*kQ[2][2] + kQ[0][1]*kQ[1][2]*kQ[2][0] + |
---|
| 761 | kQ[0][2]*kQ[1][0]*kQ[2][1] - kQ[0][2]*kQ[1][1]*kQ[2][0] - |
---|
| 762 | kQ[0][1]*kQ[1][0]*kQ[2][2] - kQ[0][0]*kQ[1][2]*kQ[2][1]; |
---|
| 763 | |
---|
| 764 | if ( fDet < 0.0 ) |
---|
| 765 | { |
---|
| 766 | for (size_t iRow = 0; iRow < 3; iRow++) |
---|
| 767 | for (size_t iCol = 0; iCol < 3; iCol++) |
---|
| 768 | kQ[iRow][iCol] = -kQ[iRow][iCol]; |
---|
| 769 | } |
---|
| 770 | |
---|
| 771 | // build "right" matrix R |
---|
| 772 | Matrix3 kR; |
---|
| 773 | kR[0][0] = kQ[0][0]*m[0][0] + kQ[1][0]*m[1][0] + |
---|
| 774 | kQ[2][0]*m[2][0]; |
---|
| 775 | kR[0][1] = kQ[0][0]*m[0][1] + kQ[1][0]*m[1][1] + |
---|
| 776 | kQ[2][0]*m[2][1]; |
---|
| 777 | kR[1][1] = kQ[0][1]*m[0][1] + kQ[1][1]*m[1][1] + |
---|
| 778 | kQ[2][1]*m[2][1]; |
---|
| 779 | kR[0][2] = kQ[0][0]*m[0][2] + kQ[1][0]*m[1][2] + |
---|
| 780 | kQ[2][0]*m[2][2]; |
---|
| 781 | kR[1][2] = kQ[0][1]*m[0][2] + kQ[1][1]*m[1][2] + |
---|
| 782 | kQ[2][1]*m[2][2]; |
---|
| 783 | kR[2][2] = kQ[0][2]*m[0][2] + kQ[1][2]*m[1][2] + |
---|
| 784 | kQ[2][2]*m[2][2]; |
---|
| 785 | |
---|
| 786 | // the scaling component |
---|
| 787 | kD[0] = kR[0][0]; |
---|
| 788 | kD[1] = kR[1][1]; |
---|
| 789 | kD[2] = kR[2][2]; |
---|
| 790 | |
---|
| 791 | // the shear component |
---|
| 792 | Real fInvD0 = 1.0/kD[0]; |
---|
| 793 | kU[0] = kR[0][1]*fInvD0; |
---|
| 794 | kU[1] = kR[0][2]*fInvD0; |
---|
| 795 | kU[2] = kR[1][2]/kD[1]; |
---|
| 796 | } |
---|
| 797 | //----------------------------------------------------------------------- |
---|
| 798 | Real Matrix3::MaxCubicRoot (Real afCoeff[3]) |
---|
| 799 | { |
---|
| 800 | // Spectral norm is for A^T*A, so characteristic polynomial |
---|
| 801 | // P(x) = c[0]+c[1]*x+c[2]*x^2+x^3 has three positive real roots. |
---|
| 802 | // This yields the assertions c[0] < 0 and c[2]*c[2] >= 3*c[1]. |
---|
| 803 | |
---|
| 804 | // quick out for uniform scale (triple root) |
---|
| 805 | const Real fOneThird = 1.0/3.0; |
---|
| 806 | const Real fEpsilon = 1e-06; |
---|
| 807 | Real fDiscr = afCoeff[2]*afCoeff[2] - 3.0*afCoeff[1]; |
---|
| 808 | if ( fDiscr <= fEpsilon ) |
---|
| 809 | return -fOneThird*afCoeff[2]; |
---|
| 810 | |
---|
| 811 | // Compute an upper bound on roots of P(x). This assumes that A^T*A |
---|
| 812 | // has been scaled by its largest entry. |
---|
| 813 | Real fX = 1.0; |
---|
| 814 | Real fPoly = afCoeff[0]+fX*(afCoeff[1]+fX*(afCoeff[2]+fX)); |
---|
| 815 | if ( fPoly < 0.0 ) |
---|
| 816 | { |
---|
| 817 | // uses a matrix norm to find an upper bound on maximum root |
---|
| 818 | fX = Math::Abs(afCoeff[0]); |
---|
| 819 | Real fTmp = 1.0+Math::Abs(afCoeff[1]); |
---|
| 820 | if ( fTmp > fX ) |
---|
| 821 | fX = fTmp; |
---|
| 822 | fTmp = 1.0+Math::Abs(afCoeff[2]); |
---|
| 823 | if ( fTmp > fX ) |
---|
| 824 | fX = fTmp; |
---|
| 825 | } |
---|
| 826 | |
---|
| 827 | // Newton's method to find root |
---|
| 828 | Real fTwoC2 = 2.0*afCoeff[2]; |
---|
| 829 | for (int i = 0; i < 16; i++) |
---|
| 830 | { |
---|
| 831 | fPoly = afCoeff[0]+fX*(afCoeff[1]+fX*(afCoeff[2]+fX)); |
---|
| 832 | if ( Math::Abs(fPoly) <= fEpsilon ) |
---|
| 833 | return fX; |
---|
| 834 | |
---|
| 835 | Real fDeriv = afCoeff[1]+fX*(fTwoC2+3.0*fX); |
---|
| 836 | fX -= fPoly/fDeriv; |
---|
| 837 | } |
---|
| 838 | |
---|
| 839 | return fX; |
---|
| 840 | } |
---|
| 841 | //----------------------------------------------------------------------- |
---|
| 842 | Real Matrix3::SpectralNorm () const |
---|
| 843 | { |
---|
| 844 | Matrix3 kP; |
---|
| 845 | size_t iRow, iCol; |
---|
| 846 | Real fPmax = 0.0; |
---|
| 847 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 848 | { |
---|
| 849 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 850 | { |
---|
| 851 | kP[iRow][iCol] = 0.0; |
---|
| 852 | for (int iMid = 0; iMid < 3; iMid++) |
---|
| 853 | { |
---|
| 854 | kP[iRow][iCol] += |
---|
| 855 | m[iMid][iRow]*m[iMid][iCol]; |
---|
| 856 | } |
---|
| 857 | if ( kP[iRow][iCol] > fPmax ) |
---|
| 858 | fPmax = kP[iRow][iCol]; |
---|
| 859 | } |
---|
| 860 | } |
---|
| 861 | |
---|
| 862 | Real fInvPmax = 1.0/fPmax; |
---|
| 863 | for (iRow = 0; iRow < 3; iRow++) |
---|
| 864 | { |
---|
| 865 | for (iCol = 0; iCol < 3; iCol++) |
---|
| 866 | kP[iRow][iCol] *= fInvPmax; |
---|
| 867 | } |
---|
| 868 | |
---|
| 869 | Real afCoeff[3]; |
---|
| 870 | afCoeff[0] = -(kP[0][0]*(kP[1][1]*kP[2][2]-kP[1][2]*kP[2][1]) + |
---|
| 871 | kP[0][1]*(kP[2][0]*kP[1][2]-kP[1][0]*kP[2][2]) + |
---|
| 872 | kP[0][2]*(kP[1][0]*kP[2][1]-kP[2][0]*kP[1][1])); |
---|
| 873 | afCoeff[1] = kP[0][0]*kP[1][1]-kP[0][1]*kP[1][0] + |
---|
| 874 | kP[0][0]*kP[2][2]-kP[0][2]*kP[2][0] + |
---|
| 875 | kP[1][1]*kP[2][2]-kP[1][2]*kP[2][1]; |
---|
| 876 | afCoeff[2] = -(kP[0][0]+kP[1][1]+kP[2][2]); |
---|
| 877 | |
---|
| 878 | Real fRoot = MaxCubicRoot(afCoeff); |
---|
| 879 | Real fNorm = Math::Sqrt(fPmax*fRoot); |
---|
| 880 | return fNorm; |
---|
| 881 | } |
---|
| 882 | //----------------------------------------------------------------------- |
---|
| 883 | void Matrix3::ToAxisAngle (Vector3& rkAxis, Radian& rfRadians) const |
---|
| 884 | { |
---|
| 885 | // Let (x,y,z) be the unit-length axis and let A be an angle of rotation. |
---|
| 886 | // The rotation matrix is R = I + sin(A)*P + (1-cos(A))*P^2 where |
---|
| 887 | // I is the identity and |
---|
| 888 | // |
---|
| 889 | // +- -+ |
---|
| 890 | // P = | 0 -z +y | |
---|
| 891 | // | +z 0 -x | |
---|
| 892 | // | -y +x 0 | |
---|
| 893 | // +- -+ |
---|
| 894 | // |
---|
| 895 | // If A > 0, R represents a counterclockwise rotation about the axis in |
---|
| 896 | // the sense of looking from the tip of the axis vector towards the |
---|
| 897 | // origin. Some algebra will show that |
---|
| 898 | // |
---|
| 899 | // cos(A) = (trace(R)-1)/2 and R - R^t = 2*sin(A)*P |
---|
| 900 | // |
---|
| 901 | // In the event that A = pi, R-R^t = 0 which prevents us from extracting |
---|
| 902 | // the axis through P. Instead note that R = I+2*P^2 when A = pi, so |
---|
| 903 | // P^2 = (R-I)/2. The diagonal entries of P^2 are x^2-1, y^2-1, and |
---|
| 904 | // z^2-1. We can solve these for axis (x,y,z). Because the angle is pi, |
---|
| 905 | // it does not matter which sign you choose on the square roots. |
---|
| 906 | |
---|
| 907 | Real fTrace = m[0][0] + m[1][1] + m[2][2]; |
---|
| 908 | Real fCos = 0.5*(fTrace-1.0); |
---|
| 909 | rfRadians = Math::ACos(fCos); // in [0,PI] |
---|
| 910 | |
---|
| 911 | if ( rfRadians > Radian(0.0) ) |
---|
| 912 | { |
---|
| 913 | if ( rfRadians < Radian(Math::PI) ) |
---|
| 914 | { |
---|
| 915 | rkAxis.x = m[2][1]-m[1][2]; |
---|
| 916 | rkAxis.y = m[0][2]-m[2][0]; |
---|
| 917 | rkAxis.z = m[1][0]-m[0][1]; |
---|
| 918 | rkAxis.normalise(); |
---|
| 919 | } |
---|
| 920 | else |
---|
| 921 | { |
---|
| 922 | // angle is PI |
---|
| 923 | float fHalfInverse; |
---|
| 924 | if ( m[0][0] >= m[1][1] ) |
---|
| 925 | { |
---|
| 926 | // r00 >= r11 |
---|
| 927 | if ( m[0][0] >= m[2][2] ) |
---|
| 928 | { |
---|
| 929 | // r00 is maximum diagonal term |
---|
| 930 | rkAxis.x = 0.5*Math::Sqrt(m[0][0] - |
---|
| 931 | m[1][1] - m[2][2] + 1.0); |
---|
| 932 | fHalfInverse = 0.5/rkAxis.x; |
---|
| 933 | rkAxis.y = fHalfInverse*m[0][1]; |
---|
| 934 | rkAxis.z = fHalfInverse*m[0][2]; |
---|
| 935 | } |
---|
| 936 | else |
---|
| 937 | { |
---|
| 938 | // r22 is maximum diagonal term |
---|
| 939 | rkAxis.z = 0.5*Math::Sqrt(m[2][2] - |
---|
| 940 | m[0][0] - m[1][1] + 1.0); |
---|
| 941 | fHalfInverse = 0.5/rkAxis.z; |
---|
| 942 | rkAxis.x = fHalfInverse*m[0][2]; |
---|
| 943 | rkAxis.y = fHalfInverse*m[1][2]; |
---|
| 944 | } |
---|
| 945 | } |
---|
| 946 | else |
---|
| 947 | { |
---|
| 948 | // r11 > r00 |
---|
| 949 | if ( m[1][1] >= m[2][2] ) |
---|
| 950 | { |
---|
| 951 | // r11 is maximum diagonal term |
---|
| 952 | rkAxis.y = 0.5*Math::Sqrt(m[1][1] - |
---|
| 953 | m[0][0] - m[2][2] + 1.0); |
---|
| 954 | fHalfInverse = 0.5/rkAxis.y; |
---|
| 955 | rkAxis.x = fHalfInverse*m[0][1]; |
---|
| 956 | rkAxis.z = fHalfInverse*m[1][2]; |
---|
| 957 | } |
---|
| 958 | else |
---|
| 959 | { |
---|
| 960 | // r22 is maximum diagonal term |
---|
| 961 | rkAxis.z = 0.5*Math::Sqrt(m[2][2] - |
---|
| 962 | m[0][0] - m[1][1] + 1.0); |
---|
| 963 | fHalfInverse = 0.5/rkAxis.z; |
---|
| 964 | rkAxis.x = fHalfInverse*m[0][2]; |
---|
| 965 | rkAxis.y = fHalfInverse*m[1][2]; |
---|
| 966 | } |
---|
| 967 | } |
---|
| 968 | } |
---|
| 969 | } |
---|
| 970 | else |
---|
| 971 | { |
---|
| 972 | // The angle is 0 and the matrix is the identity. Any axis will |
---|
| 973 | // work, so just use the x-axis. |
---|
| 974 | rkAxis.x = 1.0; |
---|
| 975 | rkAxis.y = 0.0; |
---|
| 976 | rkAxis.z = 0.0; |
---|
| 977 | } |
---|
| 978 | } |
---|
| 979 | //----------------------------------------------------------------------- |
---|
| 980 | void Matrix3::FromAxisAngle (const Vector3& rkAxis, const Radian& fRadians) |
---|
| 981 | { |
---|
| 982 | Real fCos = Math::Cos(fRadians); |
---|
| 983 | Real fSin = Math::Sin(fRadians); |
---|
| 984 | Real fOneMinusCos = 1.0-fCos; |
---|
| 985 | Real fX2 = rkAxis.x*rkAxis.x; |
---|
| 986 | Real fY2 = rkAxis.y*rkAxis.y; |
---|
| 987 | Real fZ2 = rkAxis.z*rkAxis.z; |
---|
| 988 | Real fXYM = rkAxis.x*rkAxis.y*fOneMinusCos; |
---|
| 989 | Real fXZM = rkAxis.x*rkAxis.z*fOneMinusCos; |
---|
| 990 | Real fYZM = rkAxis.y*rkAxis.z*fOneMinusCos; |
---|
| 991 | Real fXSin = rkAxis.x*fSin; |
---|
| 992 | Real fYSin = rkAxis.y*fSin; |
---|
| 993 | Real fZSin = rkAxis.z*fSin; |
---|
| 994 | |
---|
| 995 | m[0][0] = fX2*fOneMinusCos+fCos; |
---|
| 996 | m[0][1] = fXYM-fZSin; |
---|
| 997 | m[0][2] = fXZM+fYSin; |
---|
| 998 | m[1][0] = fXYM+fZSin; |
---|
| 999 | m[1][1] = fY2*fOneMinusCos+fCos; |
---|
| 1000 | m[1][2] = fYZM-fXSin; |
---|
| 1001 | m[2][0] = fXZM-fYSin; |
---|
| 1002 | m[2][1] = fYZM+fXSin; |
---|
| 1003 | m[2][2] = fZ2*fOneMinusCos+fCos; |
---|
| 1004 | } |
---|
| 1005 | //----------------------------------------------------------------------- |
---|
| 1006 | bool Matrix3::ToEulerAnglesXYZ (Radian& rfYAngle, Radian& rfPAngle, |
---|
| 1007 | Radian& rfRAngle) const |
---|
| 1008 | { |
---|
| 1009 | // rot = cy*cz -cy*sz sy |
---|
| 1010 | // cz*sx*sy+cx*sz cx*cz-sx*sy*sz -cy*sx |
---|
| 1011 | // -cx*cz*sy+sx*sz cz*sx+cx*sy*sz cx*cy |
---|
| 1012 | |
---|
| 1013 | rfPAngle = Radian(Math::ASin(m[0][2])); |
---|
| 1014 | if ( rfPAngle < Radian(Math::HALF_PI) ) |
---|
| 1015 | { |
---|
| 1016 | if ( rfPAngle > Radian(-Math::HALF_PI) ) |
---|
| 1017 | { |
---|
| 1018 | rfYAngle = Math::ATan2(-m[1][2],m[2][2]); |
---|
| 1019 | rfRAngle = Math::ATan2(-m[0][1],m[0][0]); |
---|
| 1020 | return true; |
---|
| 1021 | } |
---|
| 1022 | else |
---|
| 1023 | { |
---|
| 1024 | // WARNING. Not a unique solution. |
---|
| 1025 | Radian fRmY = Math::ATan2(m[1][0],m[1][1]); |
---|
| 1026 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1027 | rfYAngle = rfRAngle - fRmY; |
---|
| 1028 | return false; |
---|
| 1029 | } |
---|
| 1030 | } |
---|
| 1031 | else |
---|
| 1032 | { |
---|
| 1033 | // WARNING. Not a unique solution. |
---|
| 1034 | Radian fRpY = Math::ATan2(m[1][0],m[1][1]); |
---|
| 1035 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1036 | rfYAngle = fRpY - rfRAngle; |
---|
| 1037 | return false; |
---|
| 1038 | } |
---|
| 1039 | } |
---|
| 1040 | //----------------------------------------------------------------------- |
---|
| 1041 | bool Matrix3::ToEulerAnglesXZY (Radian& rfYAngle, Radian& rfPAngle, |
---|
| 1042 | Radian& rfRAngle) const |
---|
| 1043 | { |
---|
| 1044 | // rot = cy*cz -sz cz*sy |
---|
| 1045 | // sx*sy+cx*cy*sz cx*cz -cy*sx+cx*sy*sz |
---|
| 1046 | // -cx*sy+cy*sx*sz cz*sx cx*cy+sx*sy*sz |
---|
| 1047 | |
---|
| 1048 | rfPAngle = Math::ASin(-m[0][1]); |
---|
| 1049 | if ( rfPAngle < Radian(Math::HALF_PI) ) |
---|
| 1050 | { |
---|
| 1051 | if ( rfPAngle > Radian(-Math::HALF_PI) ) |
---|
| 1052 | { |
---|
| 1053 | rfYAngle = Math::ATan2(m[2][1],m[1][1]); |
---|
| 1054 | rfRAngle = Math::ATan2(m[0][2],m[0][0]); |
---|
| 1055 | return true; |
---|
| 1056 | } |
---|
| 1057 | else |
---|
| 1058 | { |
---|
| 1059 | // WARNING. Not a unique solution. |
---|
| 1060 | Radian fRmY = Math::ATan2(-m[2][0],m[2][2]); |
---|
| 1061 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1062 | rfYAngle = rfRAngle - fRmY; |
---|
| 1063 | return false; |
---|
| 1064 | } |
---|
| 1065 | } |
---|
| 1066 | else |
---|
| 1067 | { |
---|
| 1068 | // WARNING. Not a unique solution. |
---|
| 1069 | Radian fRpY = Math::ATan2(-m[2][0],m[2][2]); |
---|
| 1070 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1071 | rfYAngle = fRpY - rfRAngle; |
---|
| 1072 | return false; |
---|
| 1073 | } |
---|
| 1074 | } |
---|
| 1075 | //----------------------------------------------------------------------- |
---|
| 1076 | bool Matrix3::ToEulerAnglesYXZ (Radian& rfYAngle, Radian& rfPAngle, |
---|
| 1077 | Radian& rfRAngle) const |
---|
| 1078 | { |
---|
| 1079 | // rot = cy*cz+sx*sy*sz cz*sx*sy-cy*sz cx*sy |
---|
| 1080 | // cx*sz cx*cz -sx |
---|
| 1081 | // -cz*sy+cy*sx*sz cy*cz*sx+sy*sz cx*cy |
---|
| 1082 | |
---|
| 1083 | rfPAngle = Math::ASin(-m[1][2]); |
---|
| 1084 | if ( rfPAngle < Radian(Math::HALF_PI) ) |
---|
| 1085 | { |
---|
| 1086 | if ( rfPAngle > Radian(-Math::HALF_PI) ) |
---|
| 1087 | { |
---|
| 1088 | rfYAngle = Math::ATan2(m[0][2],m[2][2]); |
---|
| 1089 | rfRAngle = Math::ATan2(m[1][0],m[1][1]); |
---|
| 1090 | return true; |
---|
| 1091 | } |
---|
| 1092 | else |
---|
| 1093 | { |
---|
| 1094 | // WARNING. Not a unique solution. |
---|
| 1095 | Radian fRmY = Math::ATan2(-m[0][1],m[0][0]); |
---|
| 1096 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1097 | rfYAngle = rfRAngle - fRmY; |
---|
| 1098 | return false; |
---|
| 1099 | } |
---|
| 1100 | } |
---|
| 1101 | else |
---|
| 1102 | { |
---|
| 1103 | // WARNING. Not a unique solution. |
---|
| 1104 | Radian fRpY = Math::ATan2(-m[0][1],m[0][0]); |
---|
| 1105 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1106 | rfYAngle = fRpY - rfRAngle; |
---|
| 1107 | return false; |
---|
| 1108 | } |
---|
| 1109 | } |
---|
| 1110 | //----------------------------------------------------------------------- |
---|
| 1111 | bool Matrix3::ToEulerAnglesYZX (Radian& rfYAngle, Radian& rfPAngle, |
---|
| 1112 | Radian& rfRAngle) const |
---|
| 1113 | { |
---|
| 1114 | // rot = cy*cz sx*sy-cx*cy*sz cx*sy+cy*sx*sz |
---|
| 1115 | // sz cx*cz -cz*sx |
---|
| 1116 | // -cz*sy cy*sx+cx*sy*sz cx*cy-sx*sy*sz |
---|
| 1117 | |
---|
| 1118 | rfPAngle = Math::ASin(m[1][0]); |
---|
| 1119 | if ( rfPAngle < Radian(Math::HALF_PI) ) |
---|
| 1120 | { |
---|
| 1121 | if ( rfPAngle > Radian(-Math::HALF_PI) ) |
---|
| 1122 | { |
---|
| 1123 | rfYAngle = Math::ATan2(-m[2][0],m[0][0]); |
---|
| 1124 | rfRAngle = Math::ATan2(-m[1][2],m[1][1]); |
---|
| 1125 | return true; |
---|
| 1126 | } |
---|
| 1127 | else |
---|
| 1128 | { |
---|
| 1129 | // WARNING. Not a unique solution. |
---|
| 1130 | Radian fRmY = Math::ATan2(m[2][1],m[2][2]); |
---|
| 1131 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1132 | rfYAngle = rfRAngle - fRmY; |
---|
| 1133 | return false; |
---|
| 1134 | } |
---|
| 1135 | } |
---|
| 1136 | else |
---|
| 1137 | { |
---|
| 1138 | // WARNING. Not a unique solution. |
---|
| 1139 | Radian fRpY = Math::ATan2(m[2][1],m[2][2]); |
---|
| 1140 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1141 | rfYAngle = fRpY - rfRAngle; |
---|
| 1142 | return false; |
---|
| 1143 | } |
---|
| 1144 | } |
---|
| 1145 | //----------------------------------------------------------------------- |
---|
| 1146 | bool Matrix3::ToEulerAnglesZXY (Radian& rfYAngle, Radian& rfPAngle, |
---|
| 1147 | Radian& rfRAngle) const |
---|
| 1148 | { |
---|
| 1149 | // rot = cy*cz-sx*sy*sz -cx*sz cz*sy+cy*sx*sz |
---|
| 1150 | // cz*sx*sy+cy*sz cx*cz -cy*cz*sx+sy*sz |
---|
| 1151 | // -cx*sy sx cx*cy |
---|
| 1152 | |
---|
| 1153 | rfPAngle = Math::ASin(m[2][1]); |
---|
| 1154 | if ( rfPAngle < Radian(Math::HALF_PI) ) |
---|
| 1155 | { |
---|
| 1156 | if ( rfPAngle > Radian(-Math::HALF_PI) ) |
---|
| 1157 | { |
---|
| 1158 | rfYAngle = Math::ATan2(-m[0][1],m[1][1]); |
---|
| 1159 | rfRAngle = Math::ATan2(-m[2][0],m[2][2]); |
---|
| 1160 | return true; |
---|
| 1161 | } |
---|
| 1162 | else |
---|
| 1163 | { |
---|
| 1164 | // WARNING. Not a unique solution. |
---|
| 1165 | Radian fRmY = Math::ATan2(m[0][2],m[0][0]); |
---|
| 1166 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1167 | rfYAngle = rfRAngle - fRmY; |
---|
| 1168 | return false; |
---|
| 1169 | } |
---|
| 1170 | } |
---|
| 1171 | else |
---|
| 1172 | { |
---|
| 1173 | // WARNING. Not a unique solution. |
---|
| 1174 | Radian fRpY = Math::ATan2(m[0][2],m[0][0]); |
---|
| 1175 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1176 | rfYAngle = fRpY - rfRAngle; |
---|
| 1177 | return false; |
---|
| 1178 | } |
---|
| 1179 | } |
---|
| 1180 | //----------------------------------------------------------------------- |
---|
| 1181 | bool Matrix3::ToEulerAnglesZYX (Radian& rfYAngle, Radian& rfPAngle, |
---|
| 1182 | Radian& rfRAngle) const |
---|
| 1183 | { |
---|
| 1184 | // rot = cy*cz cz*sx*sy-cx*sz cx*cz*sy+sx*sz |
---|
| 1185 | // cy*sz cx*cz+sx*sy*sz -cz*sx+cx*sy*sz |
---|
| 1186 | // -sy cy*sx cx*cy |
---|
| 1187 | |
---|
| 1188 | rfPAngle = Math::ASin(-m[2][0]); |
---|
| 1189 | if ( rfPAngle < Radian(Math::HALF_PI) ) |
---|
| 1190 | { |
---|
| 1191 | if ( rfPAngle > Radian(-Math::HALF_PI) ) |
---|
| 1192 | { |
---|
| 1193 | rfYAngle = Math::ATan2(m[1][0],m[0][0]); |
---|
| 1194 | rfRAngle = Math::ATan2(m[2][1],m[2][2]); |
---|
| 1195 | return true; |
---|
| 1196 | } |
---|
| 1197 | else |
---|
| 1198 | { |
---|
| 1199 | // WARNING. Not a unique solution. |
---|
| 1200 | Radian fRmY = Math::ATan2(-m[0][1],m[0][2]); |
---|
| 1201 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1202 | rfYAngle = rfRAngle - fRmY; |
---|
| 1203 | return false; |
---|
| 1204 | } |
---|
| 1205 | } |
---|
| 1206 | else |
---|
| 1207 | { |
---|
| 1208 | // WARNING. Not a unique solution. |
---|
| 1209 | Radian fRpY = Math::ATan2(-m[0][1],m[0][2]); |
---|
| 1210 | rfRAngle = Radian(0.0); // any angle works |
---|
| 1211 | rfYAngle = fRpY - rfRAngle; |
---|
| 1212 | return false; |
---|
| 1213 | } |
---|
| 1214 | } |
---|
| 1215 | //----------------------------------------------------------------------- |
---|
| 1216 | void Matrix3::FromEulerAnglesXYZ (const Radian& fYAngle, const Radian& fPAngle, |
---|
| 1217 | const Radian& fRAngle) |
---|
| 1218 | { |
---|
| 1219 | Real fCos, fSin; |
---|
| 1220 | |
---|
| 1221 | fCos = Math::Cos(fYAngle); |
---|
| 1222 | fSin = Math::Sin(fYAngle); |
---|
| 1223 | Matrix3 kXMat(1.0,0.0,0.0,0.0,fCos,-fSin,0.0,fSin,fCos); |
---|
| 1224 | |
---|
| 1225 | fCos = Math::Cos(fPAngle); |
---|
| 1226 | fSin = Math::Sin(fPAngle); |
---|
| 1227 | Matrix3 kYMat(fCos,0.0,fSin,0.0,1.0,0.0,-fSin,0.0,fCos); |
---|
| 1228 | |
---|
| 1229 | fCos = Math::Cos(fRAngle); |
---|
| 1230 | fSin = Math::Sin(fRAngle); |
---|
| 1231 | Matrix3 kZMat(fCos,-fSin,0.0,fSin,fCos,0.0,0.0,0.0,1.0); |
---|
| 1232 | |
---|
| 1233 | *this = kXMat*(kYMat*kZMat); |
---|
| 1234 | } |
---|
| 1235 | //----------------------------------------------------------------------- |
---|
| 1236 | void Matrix3::FromEulerAnglesXZY (const Radian& fYAngle, const Radian& fPAngle, |
---|
| 1237 | const Radian& fRAngle) |
---|
| 1238 | { |
---|
| 1239 | Real fCos, fSin; |
---|
| 1240 | |
---|
| 1241 | fCos = Math::Cos(fYAngle); |
---|
| 1242 | fSin = Math::Sin(fYAngle); |
---|
| 1243 | Matrix3 kXMat(1.0,0.0,0.0,0.0,fCos,-fSin,0.0,fSin,fCos); |
---|
| 1244 | |
---|
| 1245 | fCos = Math::Cos(fPAngle); |
---|
| 1246 | fSin = Math::Sin(fPAngle); |
---|
| 1247 | Matrix3 kZMat(fCos,-fSin,0.0,fSin,fCos,0.0,0.0,0.0,1.0); |
---|
| 1248 | |
---|
| 1249 | fCos = Math::Cos(fRAngle); |
---|
| 1250 | fSin = Math::Sin(fRAngle); |
---|
| 1251 | Matrix3 kYMat(fCos,0.0,fSin,0.0,1.0,0.0,-fSin,0.0,fCos); |
---|
| 1252 | |
---|
| 1253 | *this = kXMat*(kZMat*kYMat); |
---|
| 1254 | } |
---|
| 1255 | //----------------------------------------------------------------------- |
---|
| 1256 | void Matrix3::FromEulerAnglesYXZ (const Radian& fYAngle, const Radian& fPAngle, |
---|
| 1257 | const Radian& fRAngle) |
---|
| 1258 | { |
---|
| 1259 | Real fCos, fSin; |
---|
| 1260 | |
---|
| 1261 | fCos = Math::Cos(fYAngle); |
---|
| 1262 | fSin = Math::Sin(fYAngle); |
---|
| 1263 | Matrix3 kYMat(fCos,0.0,fSin,0.0,1.0,0.0,-fSin,0.0,fCos); |
---|
| 1264 | |
---|
| 1265 | fCos = Math::Cos(fPAngle); |
---|
| 1266 | fSin = Math::Sin(fPAngle); |
---|
| 1267 | Matrix3 kXMat(1.0,0.0,0.0,0.0,fCos,-fSin,0.0,fSin,fCos); |
---|
| 1268 | |
---|
| 1269 | fCos = Math::Cos(fRAngle); |
---|
| 1270 | fSin = Math::Sin(fRAngle); |
---|
| 1271 | Matrix3 kZMat(fCos,-fSin,0.0,fSin,fCos,0.0,0.0,0.0,1.0); |
---|
| 1272 | |
---|
| 1273 | *this = kYMat*(kXMat*kZMat); |
---|
| 1274 | } |
---|
| 1275 | //----------------------------------------------------------------------- |
---|
| 1276 | void Matrix3::FromEulerAnglesYZX (const Radian& fYAngle, const Radian& fPAngle, |
---|
| 1277 | const Radian& fRAngle) |
---|
| 1278 | { |
---|
| 1279 | Real fCos, fSin; |
---|
| 1280 | |
---|
| 1281 | fCos = Math::Cos(fYAngle); |
---|
| 1282 | fSin = Math::Sin(fYAngle); |
---|
| 1283 | Matrix3 kYMat(fCos,0.0,fSin,0.0,1.0,0.0,-fSin,0.0,fCos); |
---|
| 1284 | |
---|
| 1285 | fCos = Math::Cos(fPAngle); |
---|
| 1286 | fSin = Math::Sin(fPAngle); |
---|
| 1287 | Matrix3 kZMat(fCos,-fSin,0.0,fSin,fCos,0.0,0.0,0.0,1.0); |
---|
| 1288 | |
---|
| 1289 | fCos = Math::Cos(fRAngle); |
---|
| 1290 | fSin = Math::Sin(fRAngle); |
---|
| 1291 | Matrix3 kXMat(1.0,0.0,0.0,0.0,fCos,-fSin,0.0,fSin,fCos); |
---|
| 1292 | |
---|
| 1293 | *this = kYMat*(kZMat*kXMat); |
---|
| 1294 | } |
---|
| 1295 | //----------------------------------------------------------------------- |
---|
| 1296 | void Matrix3::FromEulerAnglesZXY (const Radian& fYAngle, const Radian& fPAngle, |
---|
| 1297 | const Radian& fRAngle) |
---|
| 1298 | { |
---|
| 1299 | Real fCos, fSin; |
---|
| 1300 | |
---|
| 1301 | fCos = Math::Cos(fYAngle); |
---|
| 1302 | fSin = Math::Sin(fYAngle); |
---|
| 1303 | Matrix3 kZMat(fCos,-fSin,0.0,fSin,fCos,0.0,0.0,0.0,1.0); |
---|
| 1304 | |
---|
| 1305 | fCos = Math::Cos(fPAngle); |
---|
| 1306 | fSin = Math::Sin(fPAngle); |
---|
| 1307 | Matrix3 kXMat(1.0,0.0,0.0,0.0,fCos,-fSin,0.0,fSin,fCos); |
---|
| 1308 | |
---|
| 1309 | fCos = Math::Cos(fRAngle); |
---|
| 1310 | fSin = Math::Sin(fRAngle); |
---|
| 1311 | Matrix3 kYMat(fCos,0.0,fSin,0.0,1.0,0.0,-fSin,0.0,fCos); |
---|
| 1312 | |
---|
| 1313 | *this = kZMat*(kXMat*kYMat); |
---|
| 1314 | } |
---|
| 1315 | //----------------------------------------------------------------------- |
---|
| 1316 | void Matrix3::FromEulerAnglesZYX (const Radian& fYAngle, const Radian& fPAngle, |
---|
| 1317 | const Radian& fRAngle) |
---|
| 1318 | { |
---|
| 1319 | Real fCos, fSin; |
---|
| 1320 | |
---|
| 1321 | fCos = Math::Cos(fYAngle); |
---|
| 1322 | fSin = Math::Sin(fYAngle); |
---|
| 1323 | Matrix3 kZMat(fCos,-fSin,0.0,fSin,fCos,0.0,0.0,0.0,1.0); |
---|
| 1324 | |
---|
| 1325 | fCos = Math::Cos(fPAngle); |
---|
| 1326 | fSin = Math::Sin(fPAngle); |
---|
| 1327 | Matrix3 kYMat(fCos,0.0,fSin,0.0,1.0,0.0,-fSin,0.0,fCos); |
---|
| 1328 | |
---|
| 1329 | fCos = Math::Cos(fRAngle); |
---|
| 1330 | fSin = Math::Sin(fRAngle); |
---|
| 1331 | Matrix3 kXMat(1.0,0.0,0.0,0.0,fCos,-fSin,0.0,fSin,fCos); |
---|
| 1332 | |
---|
| 1333 | *this = kZMat*(kYMat*kXMat); |
---|
| 1334 | } |
---|
| 1335 | //----------------------------------------------------------------------- |
---|
| 1336 | void Matrix3::Tridiagonal (Real afDiag[3], Real afSubDiag[3]) |
---|
| 1337 | { |
---|
| 1338 | // Householder reduction T = Q^t M Q |
---|
| 1339 | // Input: |
---|
| 1340 | // mat, symmetric 3x3 matrix M |
---|
| 1341 | // Output: |
---|
| 1342 | // mat, orthogonal matrix Q |
---|
| 1343 | // diag, diagonal entries of T |
---|
| 1344 | // subd, subdiagonal entries of T (T is symmetric) |
---|
| 1345 | |
---|
| 1346 | Real fA = m[0][0]; |
---|
| 1347 | Real fB = m[0][1]; |
---|
| 1348 | Real fC = m[0][2]; |
---|
| 1349 | Real fD = m[1][1]; |
---|
| 1350 | Real fE = m[1][2]; |
---|
| 1351 | Real fF = m[2][2]; |
---|
| 1352 | |
---|
| 1353 | afDiag[0] = fA; |
---|
| 1354 | afSubDiag[2] = 0.0; |
---|
| 1355 | if ( Math::Abs(fC) >= EPSILON ) |
---|
| 1356 | { |
---|
| 1357 | Real fLength = Math::Sqrt(fB*fB+fC*fC); |
---|
| 1358 | Real fInvLength = 1.0/fLength; |
---|
| 1359 | fB *= fInvLength; |
---|
| 1360 | fC *= fInvLength; |
---|
| 1361 | Real fQ = 2.0*fB*fE+fC*(fF-fD); |
---|
| 1362 | afDiag[1] = fD+fC*fQ; |
---|
| 1363 | afDiag[2] = fF-fC*fQ; |
---|
| 1364 | afSubDiag[0] = fLength; |
---|
| 1365 | afSubDiag[1] = fE-fB*fQ; |
---|
| 1366 | m[0][0] = 1.0; |
---|
| 1367 | m[0][1] = 0.0; |
---|
| 1368 | m[0][2] = 0.0; |
---|
| 1369 | m[1][0] = 0.0; |
---|
| 1370 | m[1][1] = fB; |
---|
| 1371 | m[1][2] = fC; |
---|
| 1372 | m[2][0] = 0.0; |
---|
| 1373 | m[2][1] = fC; |
---|
| 1374 | m[2][2] = -fB; |
---|
| 1375 | } |
---|
| 1376 | else |
---|
| 1377 | { |
---|
| 1378 | afDiag[1] = fD; |
---|
| 1379 | afDiag[2] = fF; |
---|
| 1380 | afSubDiag[0] = fB; |
---|
| 1381 | afSubDiag[1] = fE; |
---|
| 1382 | m[0][0] = 1.0; |
---|
| 1383 | m[0][1] = 0.0; |
---|
| 1384 | m[0][2] = 0.0; |
---|
| 1385 | m[1][0] = 0.0; |
---|
| 1386 | m[1][1] = 1.0; |
---|
| 1387 | m[1][2] = 0.0; |
---|
| 1388 | m[2][0] = 0.0; |
---|
| 1389 | m[2][1] = 0.0; |
---|
| 1390 | m[2][2] = 1.0; |
---|
| 1391 | } |
---|
| 1392 | } |
---|
| 1393 | //----------------------------------------------------------------------- |
---|
| 1394 | bool Matrix3::QLAlgorithm (Real afDiag[3], Real afSubDiag[3]) |
---|
| 1395 | { |
---|
| 1396 | // QL iteration with implicit shifting to reduce matrix from tridiagonal |
---|
| 1397 | // to diagonal |
---|
| 1398 | |
---|
| 1399 | for (int i0 = 0; i0 < 3; i0++) |
---|
| 1400 | { |
---|
| 1401 | const unsigned int iMaxIter = 32; |
---|
| 1402 | unsigned int iIter; |
---|
| 1403 | for (iIter = 0; iIter < iMaxIter; iIter++) |
---|
| 1404 | { |
---|
| 1405 | int i1; |
---|
| 1406 | for (i1 = i0; i1 <= 1; i1++) |
---|
| 1407 | { |
---|
| 1408 | Real fSum = Math::Abs(afDiag[i1]) + |
---|
| 1409 | Math::Abs(afDiag[i1+1]); |
---|
| 1410 | if ( Math::Abs(afSubDiag[i1]) + fSum == fSum ) |
---|
| 1411 | break; |
---|
| 1412 | } |
---|
| 1413 | if ( i1 == i0 ) |
---|
| 1414 | break; |
---|
| 1415 | |
---|
| 1416 | Real fTmp0 = (afDiag[i0+1]-afDiag[i0])/(2.0*afSubDiag[i0]); |
---|
| 1417 | Real fTmp1 = Math::Sqrt(fTmp0*fTmp0+1.0); |
---|
| 1418 | if ( fTmp0 < 0.0 ) |
---|
| 1419 | fTmp0 = afDiag[i1]-afDiag[i0]+afSubDiag[i0]/(fTmp0-fTmp1); |
---|
| 1420 | else |
---|
| 1421 | fTmp0 = afDiag[i1]-afDiag[i0]+afSubDiag[i0]/(fTmp0+fTmp1); |
---|
| 1422 | Real fSin = 1.0; |
---|
| 1423 | Real fCos = 1.0; |
---|
| 1424 | Real fTmp2 = 0.0; |
---|
| 1425 | for (int i2 = i1-1; i2 >= i0; i2--) |
---|
| 1426 | { |
---|
| 1427 | Real fTmp3 = fSin*afSubDiag[i2]; |
---|
| 1428 | Real fTmp4 = fCos*afSubDiag[i2]; |
---|
| 1429 | if ( Math::Abs(fTmp3) >= Math::Abs(fTmp0) ) |
---|
| 1430 | { |
---|
| 1431 | fCos = fTmp0/fTmp3; |
---|
| 1432 | fTmp1 = Math::Sqrt(fCos*fCos+1.0); |
---|
| 1433 | afSubDiag[i2+1] = fTmp3*fTmp1; |
---|
| 1434 | fSin = 1.0/fTmp1; |
---|
| 1435 | fCos *= fSin; |
---|
| 1436 | } |
---|
| 1437 | else |
---|
| 1438 | { |
---|
| 1439 | fSin = fTmp3/fTmp0; |
---|
| 1440 | fTmp1 = Math::Sqrt(fSin*fSin+1.0); |
---|
| 1441 | afSubDiag[i2+1] = fTmp0*fTmp1; |
---|
| 1442 | fCos = 1.0/fTmp1; |
---|
| 1443 | fSin *= fCos; |
---|
| 1444 | } |
---|
| 1445 | fTmp0 = afDiag[i2+1]-fTmp2; |
---|
| 1446 | fTmp1 = (afDiag[i2]-fTmp0)*fSin+2.0*fTmp4*fCos; |
---|
| 1447 | fTmp2 = fSin*fTmp1; |
---|
| 1448 | afDiag[i2+1] = fTmp0+fTmp2; |
---|
| 1449 | fTmp0 = fCos*fTmp1-fTmp4; |
---|
| 1450 | |
---|
| 1451 | for (int iRow = 0; iRow < 3; iRow++) |
---|
| 1452 | { |
---|
| 1453 | fTmp3 = m[iRow][i2+1]; |
---|
| 1454 | m[iRow][i2+1] = fSin*m[iRow][i2] + |
---|
| 1455 | fCos*fTmp3; |
---|
| 1456 | m[iRow][i2] = fCos*m[iRow][i2] - |
---|
| 1457 | fSin*fTmp3; |
---|
| 1458 | } |
---|
| 1459 | } |
---|
| 1460 | afDiag[i0] -= fTmp2; |
---|
| 1461 | afSubDiag[i0] = fTmp0; |
---|
| 1462 | afSubDiag[i1] = 0.0; |
---|
| 1463 | } |
---|
| 1464 | |
---|
| 1465 | if ( iIter == iMaxIter ) |
---|
| 1466 | { |
---|
| 1467 | // should not get here under normal circumstances |
---|
| 1468 | return false; |
---|
| 1469 | } |
---|
| 1470 | } |
---|
| 1471 | |
---|
| 1472 | return true; |
---|
| 1473 | } |
---|
| 1474 | //----------------------------------------------------------------------- |
---|
| 1475 | void Matrix3::EigenSolveSymmetric (Real afEigenvalue[3], |
---|
| 1476 | Vector3 akEigenvector[3]) const |
---|
| 1477 | { |
---|
| 1478 | Matrix3 kMatrix = *this; |
---|
| 1479 | Real afSubDiag[3]; |
---|
| 1480 | kMatrix.Tridiagonal(afEigenvalue,afSubDiag); |
---|
| 1481 | kMatrix.QLAlgorithm(afEigenvalue,afSubDiag); |
---|
| 1482 | |
---|
| 1483 | for (size_t i = 0; i < 3; i++) |
---|
| 1484 | { |
---|
| 1485 | akEigenvector[i][0] = kMatrix[0][i]; |
---|
| 1486 | akEigenvector[i][1] = kMatrix[1][i]; |
---|
| 1487 | akEigenvector[i][2] = kMatrix[2][i]; |
---|
| 1488 | } |
---|
| 1489 | |
---|
| 1490 | // make eigenvectors form a right--handed system |
---|
| 1491 | Vector3 kCross = akEigenvector[1].crossProduct(akEigenvector[2]); |
---|
| 1492 | Real fDet = akEigenvector[0].dotProduct(kCross); |
---|
| 1493 | if ( fDet < 0.0 ) |
---|
| 1494 | { |
---|
| 1495 | akEigenvector[2][0] = - akEigenvector[2][0]; |
---|
| 1496 | akEigenvector[2][1] = - akEigenvector[2][1]; |
---|
| 1497 | akEigenvector[2][2] = - akEigenvector[2][2]; |
---|
| 1498 | } |
---|
| 1499 | } |
---|
| 1500 | //----------------------------------------------------------------------- |
---|
| 1501 | void Matrix3::TensorProduct (const Vector3& rkU, const Vector3& rkV, |
---|
| 1502 | Matrix3& rkProduct) |
---|
| 1503 | { |
---|
| 1504 | for (size_t iRow = 0; iRow < 3; iRow++) |
---|
| 1505 | { |
---|
| 1506 | for (size_t iCol = 0; iCol < 3; iCol++) |
---|
| 1507 | rkProduct[iRow][iCol] = rkU[iRow]*rkV[iCol]; |
---|
| 1508 | } |
---|
| 1509 | } |
---|
| 1510 | //----------------------------------------------------------------------- |
---|
| 1511 | } |
---|