/src/FreeImage/Source/OpenEXR/Imath/ImathMatrixAlgo.h
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- ///////////////////////////////////////////////////////////////////////////
- //
- // Copyright (c) 2002, Industrial Light & Magic, a division of Lucas
- // Digital Ltd. LLC
- //
- // All rights reserved.
- //
- // Redistribution and use in source and binary forms, with or without
- // modification, are permitted provided that the following conditions are
- // met:
- // * Redistributions of source code must retain the above copyright
- // notice, this list of conditions and the following disclaimer.
- // * Redistributions in binary form must reproduce the above
- // copyright notice, this list of conditions and the following disclaimer
- // in the documentation and/or other materials provided with the
- // distribution.
- // * Neither the name of Industrial Light & Magic nor the names of
- // its contributors may be used to endorse or promote products derived
- // from this software without specific prior written permission.
- //
- // THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
- // "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
- // LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
- // A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
- // OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
- // SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
- // LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
- // DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
- // THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
- // (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
- // OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
- //
- ///////////////////////////////////////////////////////////////////////////
- #ifndef INCLUDED_IMATHMATRIXALGO_H
- #define INCLUDED_IMATHMATRIXALGO_H
- //-------------------------------------------------------------------------
- //
- // This file contains algorithms applied to or in conjunction with
- // transformation matrices (Imath::Matrix33 and Imath::Matrix44).
- // The assumption made is that these functions are called much less
- // often than the basic point functions or these functions require
- // more support classes.
- //
- // This file also defines a few predefined constant matrices.
- //
- //-------------------------------------------------------------------------
- #include "ImathMatrix.h"
- #include "ImathQuat.h"
- #include "ImathEuler.h"
- #include "ImathExc.h"
- #include "ImathVec.h"
- #include <math.h>
- #ifdef OPENEXR_DLL
- #ifdef IMATH_EXPORTS
- #define IMATH_EXPORT_CONST extern __declspec(dllexport)
- #else
- #define IMATH_EXPORT_CONST extern __declspec(dllimport)
- #endif
- #else
- #define IMATH_EXPORT_CONST extern const
- #endif
- namespace Imath {
- //------------------
- // Identity matrices
- //------------------
- IMATH_EXPORT_CONST M33f identity33f;
- IMATH_EXPORT_CONST M44f identity44f;
- IMATH_EXPORT_CONST M33d identity33d;
- IMATH_EXPORT_CONST M44d identity44d;
- //----------------------------------------------------------------------
- // Extract scale, shear, rotation, and translation values from a matrix:
- //
- // Notes:
- //
- // This implementation follows the technique described in the paper by
- // Spencer W. Thomas in the Graphics Gems II article: "Decomposing a
- // Matrix into Simple Transformations", p. 320.
- //
- // - Some of the functions below have an optional exc parameter
- // that determines the functions' behavior when the matrix'
- // scaling is very close to zero:
- //
- // If exc is true, the functions throw an Imath::ZeroScale exception.
- //
- // If exc is false:
- //
- // extractScaling (m, s) returns false, s is invalid
- // sansScaling (m) returns m
- // removeScaling (m) returns false, m is unchanged
- // sansScalingAndShear (m) returns m
- // removeScalingAndShear (m) returns false, m is unchanged
- // extractAndRemoveScalingAndShear (m, s, h)
- // returns false, m is unchanged,
- // (sh) are invalid
- // checkForZeroScaleInRow () returns false
- // extractSHRT (m, s, h, r, t) returns false, (shrt) are invalid
- //
- // - Functions extractEuler(), extractEulerXYZ() and extractEulerZYX()
- // assume that the matrix does not include shear or non-uniform scaling,
- // but they do not examine the matrix to verify this assumption.
- // Matrices with shear or non-uniform scaling are likely to produce
- // meaningless results. Therefore, you should use the
- // removeScalingAndShear() routine, if necessary, prior to calling
- // extractEuler...() .
- //
- // - All functions assume that the matrix does not include perspective
- // transformation(s), but they do not examine the matrix to verify
- // this assumption. Matrices with perspective transformations are
- // likely to produce meaningless results.
- //
- //----------------------------------------------------------------------
- //
- // Declarations for 4x4 matrix.
- //
- template <class T> bool extractScaling
- (const Matrix44<T> &mat,
- Vec3<T> &scl,
- bool exc = true);
-
- template <class T> Matrix44<T> sansScaling (const Matrix44<T> &mat,
- bool exc = true);
- template <class T> bool removeScaling
- (Matrix44<T> &mat,
- bool exc = true);
- template <class T> bool extractScalingAndShear
- (const Matrix44<T> &mat,
- Vec3<T> &scl,
- Vec3<T> &shr,
- bool exc = true);
-
- template <class T> Matrix44<T> sansScalingAndShear
- (const Matrix44<T> &mat,
- bool exc = true);
- template <class T> bool removeScalingAndShear
- (Matrix44<T> &mat,
- bool exc = true);
- template <class T> bool extractAndRemoveScalingAndShear
- (Matrix44<T> &mat,
- Vec3<T> &scl,
- Vec3<T> &shr,
- bool exc = true);
- template <class T> void extractEulerXYZ
- (const Matrix44<T> &mat,
- Vec3<T> &rot);
- template <class T> void extractEulerZYX
- (const Matrix44<T> &mat,
- Vec3<T> &rot);
- template <class T> Quat<T> extractQuat (const Matrix44<T> &mat);
- template <class T> bool extractSHRT
- (const Matrix44<T> &mat,
- Vec3<T> &s,
- Vec3<T> &h,
- Vec3<T> &r,
- Vec3<T> &t,
- bool exc /*= true*/,
- typename Euler<T>::Order rOrder);
- template <class T> bool extractSHRT
- (const Matrix44<T> &mat,
- Vec3<T> &s,
- Vec3<T> &h,
- Vec3<T> &r,
- Vec3<T> &t,
- bool exc = true);
- template <class T> bool extractSHRT
- (const Matrix44<T> &mat,
- Vec3<T> &s,
- Vec3<T> &h,
- Euler<T> &r,
- Vec3<T> &t,
- bool exc = true);
- //
- // Internal utility function.
- //
- template <class T> bool checkForZeroScaleInRow
- (const T &scl,
- const Vec3<T> &row,
- bool exc = true);
- //
- // Returns a matrix that rotates "fromDirection" vector to "toDirection"
- // vector.
- //
- template <class T> Matrix44<T> rotationMatrix (const Vec3<T> &fromDirection,
- const Vec3<T> &toDirection);
- //
- // Returns a matrix that rotates the "fromDir" vector
- // so that it points towards "toDir". You may also
- // specify that you want the up vector to be pointing
- // in a certain direction "upDir".
- //
- template <class T> Matrix44<T> rotationMatrixWithUpDir
- (const Vec3<T> &fromDir,
- const Vec3<T> &toDir,
- const Vec3<T> &upDir);
- //
- // Returns a matrix that rotates the z-axis so that it
- // points towards "targetDir". You must also specify
- // that you want the up vector to be pointing in a
- // certain direction "upDir".
- //
- // Notes: The following degenerate cases are handled:
- // (a) when the directions given by "toDir" and "upDir"
- // are parallel or opposite;
- // (the direction vectors must have a non-zero cross product)
- // (b) when any of the given direction vectors have zero length
- //
- template <class T> Matrix44<T> alignZAxisWithTargetDir
- (Vec3<T> targetDir,
- Vec3<T> upDir);
- //----------------------------------------------------------------------
- //
- // Declarations for 3x3 matrix.
- //
-
- template <class T> bool extractScaling
- (const Matrix33<T> &mat,
- Vec2<T> &scl,
- bool exc = true);
-
- template <class T> Matrix33<T> sansScaling (const Matrix33<T> &mat,
- bool exc = true);
- template <class T> bool removeScaling
- (Matrix33<T> &mat,
- bool exc = true);
- template <class T> bool extractScalingAndShear
- (const Matrix33<T> &mat,
- Vec2<T> &scl,
- T &h,
- bool exc = true);
-
- template <class T> Matrix33<T> sansScalingAndShear
- (const Matrix33<T> &mat,
- bool exc = true);
- template <class T> bool removeScalingAndShear
- (Matrix33<T> &mat,
- bool exc = true);
- template <class T> bool extractAndRemoveScalingAndShear
- (Matrix33<T> &mat,
- Vec2<T> &scl,
- T &shr,
- bool exc = true);
- template <class T> void extractEuler
- (const Matrix33<T> &mat,
- T &rot);
- template <class T> bool extractSHRT (const Matrix33<T> &mat,
- Vec2<T> &s,
- T &h,
- T &r,
- Vec2<T> &t,
- bool exc = true);
- template <class T> bool checkForZeroScaleInRow
- (const T &scl,
- const Vec2<T> &row,
- bool exc = true);
- //-----------------------------------------------------------------------------
- // Implementation for 4x4 Matrix
- //------------------------------
- template <class T>
- bool
- extractScaling (const Matrix44<T> &mat, Vec3<T> &scl, bool exc)
- {
- Vec3<T> shr;
- Matrix44<T> M (mat);
- if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
- return false;
-
- return true;
- }
- template <class T>
- Matrix44<T>
- sansScaling (const Matrix44<T> &mat, bool exc)
- {
- Vec3<T> scl;
- Vec3<T> shr;
- Vec3<T> rot;
- Vec3<T> tran;
- if (! extractSHRT (mat, scl, shr, rot, tran, exc))
- return mat;
- Matrix44<T> M;
-
- M.translate (tran);
- M.rotate (rot);
- M.shear (shr);
- return M;
- }
- template <class T>
- bool
- removeScaling (Matrix44<T> &mat, bool exc)
- {
- Vec3<T> scl;
- Vec3<T> shr;
- Vec3<T> rot;
- Vec3<T> tran;
- if (! extractSHRT (mat, scl, shr, rot, tran, exc))
- return false;
- mat.makeIdentity ();
- mat.translate (tran);
- mat.rotate (rot);
- mat.shear (shr);
- return true;
- }
- template <class T>
- bool
- extractScalingAndShear (const Matrix44<T> &mat,
- Vec3<T> &scl, Vec3<T> &shr, bool exc)
- {
- Matrix44<T> M (mat);
- if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
- return false;
-
- return true;
- }
- template <class T>
- Matrix44<T>
- sansScalingAndShear (const Matrix44<T> &mat, bool exc)
- {
- Vec3<T> scl;
- Vec3<T> shr;
- Matrix44<T> M (mat);
- if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
- return mat;
-
- return M;
- }
- template <class T>
- bool
- removeScalingAndShear (Matrix44<T> &mat, bool exc)
- {
- Vec3<T> scl;
- Vec3<T> shr;
- if (! extractAndRemoveScalingAndShear (mat, scl, shr, exc))
- return false;
-
- return true;
- }
- template <class T>
- bool
- extractAndRemoveScalingAndShear (Matrix44<T> &mat,
- Vec3<T> &scl, Vec3<T> &shr, bool exc)
- {
- //
- // This implementation follows the technique described in the paper by
- // Spencer W. Thomas in the Graphics Gems II article: "Decomposing a
- // Matrix into Simple Transformations", p. 320.
- //
- Vec3<T> row[3];
- row[0] = Vec3<T> (mat[0][0], mat[0][1], mat[0][2]);
- row[1] = Vec3<T> (mat[1][0], mat[1][1], mat[1][2]);
- row[2] = Vec3<T> (mat[2][0], mat[2][1], mat[2][2]);
-
- T maxVal = 0;
- for (int i=0; i < 3; i++)
- for (int j=0; j < 3; j++)
- if (Imath::abs (row[i][j]) > maxVal)
- maxVal = Imath::abs (row[i][j]);
- //
- // We normalize the 3x3 matrix here.
- // It was noticed that this can improve numerical stability significantly,
- // especially when many of the upper 3x3 matrix's coefficients are very
- // close to zero; we correct for this step at the end by multiplying the
- // scaling factors by maxVal at the end (shear and rotation are not
- // affected by the normalization).
- if (maxVal != 0)
- {
- for (int i=0; i < 3; i++)
- if (! checkForZeroScaleInRow (maxVal, row[i], exc))
- return false;
- else
- row[i] /= maxVal;
- }
- // Compute X scale factor.
- scl.x = row[0].length ();
- if (! checkForZeroScaleInRow (scl.x, row[0], exc))
- return false;
- // Normalize first row.
- row[0] /= scl.x;
- // An XY shear factor will shear the X coord. as the Y coord. changes.
- // There are 6 combinations (XY, XZ, YZ, YX, ZX, ZY), although we only
- // extract the first 3 because we can effect the last 3 by shearing in
- // XY, XZ, YZ combined rotations and scales.
- //
- // shear matrix < 1, YX, ZX, 0,
- // XY, 1, ZY, 0,
- // XZ, YZ, 1, 0,
- // 0, 0, 0, 1 >
- // Compute XY shear factor and make 2nd row orthogonal to 1st.
- shr[0] = row[0].dot (row[1]);
- row[1] -= shr[0] * row[0];
- // Now, compute Y scale.
- scl.y = row[1].length ();
- if (! checkForZeroScaleInRow (scl.y, row[1], exc))
- return false;
- // Normalize 2nd row and correct the XY shear factor for Y scaling.
- row[1] /= scl.y;
- shr[0] /= scl.y;
- // Compute XZ and YZ shears, orthogonalize 3rd row.
- shr[1] = row[0].dot (row[2]);
- row[2] -= shr[1] * row[0];
- shr[2] = row[1].dot (row[2]);
- row[2] -= shr[2] * row[1];
- // Next, get Z scale.
- scl.z = row[2].length ();
- if (! checkForZeroScaleInRow (scl.z, row[2], exc))
- return false;
- // Normalize 3rd row and correct the XZ and YZ shear factors for Z scaling.
- row[2] /= scl.z;
- shr[1] /= scl.z;
- shr[2] /= scl.z;
- // At this point, the upper 3x3 matrix in mat is orthonormal.
- // Check for a coordinate system flip. If the determinant
- // is less than zero, then negate the matrix and the scaling factors.
- if (row[0].dot (row[1].cross (row[2])) < 0)
- for (int i=0; i < 3; i++)
- {
- scl[i] *= -1;
- row[i] *= -1;
- }
- // Copy over the orthonormal rows into the returned matrix.
- // The upper 3x3 matrix in mat is now a rotation matrix.
- for (int i=0; i < 3; i++)
- {
- mat[i][0] = row[i][0];
- mat[i][1] = row[i][1];
- mat[i][2] = row[i][2];
- }
- // Correct the scaling factors for the normalization step that we
- // performed above; shear and rotation are not affected by the
- // normalization.
- scl *= maxVal;
- return true;
- }
- template <class T>
- void
- extractEulerXYZ (const Matrix44<T> &mat, Vec3<T> &rot)
- {
- //
- // Normalize the local x, y and z axes to remove scaling.
- //
- Vec3<T> i (mat[0][0], mat[0][1], mat[0][2]);
- Vec3<T> j (mat[1][0], mat[1][1], mat[1][2]);
- Vec3<T> k (mat[2][0], mat[2][1], mat[2][2]);
- i.normalize();
- j.normalize();
- k.normalize();
- Matrix44<T> M (i[0], i[1], i[2], 0,
- j[0], j[1], j[2], 0,
- k[0], k[1], k[2], 0,
- 0, 0, 0, 1);
- //
- // Extract the first angle, rot.x.
- //
- rot.x = Math<T>::atan2 (M[1][2], M[2][2]);
- //
- // Remove the rot.x rotation from M, so that the remaining
- // rotation, N, is only around two axes, and gimbal lock
- // cannot occur.
- //
- Matrix44<T> N;
- N.rotate (Vec3<T> (-rot.x, 0, 0));
- N = N * M;
- // Extract the other two angles, rot.y and rot.z, from N.
- //
- T cy = Math<T>::sqrt (N[0][0]*N[0][0] + N[0][1]*N[0][1]);
- rot.y = Math<T>::atan2 (-N[0][2], cy);
- rot.z = Math<T>::atan2 (-N[1][0], N[1][1]);
- }
- template <class T>
- void
- extractEulerZYX (const Matrix44<T> &mat, Vec3<T> &rot)
- {
- //
- // Normalize the local x, y and z axes to remove scaling.
- //
- Vec3<T> i (mat[0][0], mat[0][1], mat[0][2]);
- Vec3<T> j (mat[1][0], mat[1][1], mat[1][2]);
- Vec3<T> k (mat[2][0], mat[2][1], mat[2][2]);
- i.normalize();
- j.normalize();
- k.normalize();
- Matrix44<T> M (i[0], i[1], i[2], 0,
- j[0], j[1], j[2], 0,
- k[0], k[1], k[2], 0,
- 0, 0, 0, 1);
- //
- // Extract the first angle, rot.x.
- //
- rot.x = -Math<T>::atan2 (M[1][0], M[0][0]);
- //
- // Remove the x rotation from M, so that the remaining
- // rotation, N, is only around two axes, and gimbal lock
- // cannot occur.
- //
- Matrix44<T> N;
- N.rotate (Vec3<T> (0, 0, -rot.x));
- N = N * M;
- //
- // Extract the other two angles, rot.y and rot.z, from N.
- //
- T cy = Math<T>::sqrt (N[2][2]*N[2][2] + N[2][1]*N[2][1]);
- rot.y = -Math<T>::atan2 (-N[2][0], cy);
- rot.z = -Math<T>::atan2 (-N[1][2], N[1][1]);
- }
- template <class T>
- Quat<T>
- extractQuat (const Matrix44<T> &mat)
- {
- Matrix44<T> rot;
- T tr, s;
- T q[4];
- int i, j, k;
- Quat<T> quat;
- int nxt[3] = {1, 2, 0};
- tr = mat[0][0] + mat[1][1] + mat[2][2];
- // check the diagonal
- if (tr > 0.0) {
- s = Math<T>::sqrt (tr + 1.0);
- quat.r = s / 2.0;
- s = 0.5 / s;
- quat.v.x = (mat[1][2] - mat[2][1]) * s;
- quat.v.y = (mat[2][0] - mat[0][2]) * s;
- quat.v.z = (mat[0][1] - mat[1][0]) * s;
- }
- else {
- // diagonal is negative
- i = 0;
- if (mat[1][1] > mat[0][0])
- i=1;
- if (mat[2][2] > mat[i][i])
- i=2;
-
- j = nxt[i];
- k = nxt[j];
- s = Math<T>::sqrt ((mat[i][i] - (mat[j][j] + mat[k][k])) + 1.0);
-
- q[i] = s * 0.5;
- if (s != 0.0)
- s = 0.5 / s;
- q[3] = (mat[j][k] - mat[k][j]) * s;
- q[j] = (mat[i][j] + mat[j][i]) * s;
- q[k] = (mat[i][k] + mat[k][i]) * s;
- quat.v.x = q[0];
- quat.v.y = q[1];
- quat.v.z = q[2];
- quat.r = q[3];
- }
- return quat;
- }
- template <class T>
- bool
- extractSHRT (const Matrix44<T> &mat,
- Vec3<T> &s,
- Vec3<T> &h,
- Vec3<T> &r,
- Vec3<T> &t,
- bool exc /* = true */ ,
- typename Euler<T>::Order rOrder /* = Euler<T>::XYZ */ )
- {
- Matrix44<T> rot;
- rot = mat;
- if (! extractAndRemoveScalingAndShear (rot, s, h, exc))
- return false;
- extractEulerXYZ (rot, r);
- t.x = mat[3][0];
- t.y = mat[3][1];
- t.z = mat[3][2];
- if (rOrder != Euler<T>::XYZ)
- {
- Imath::Euler<T> eXYZ (r, Imath::Euler<T>::XYZ);
- Imath::Euler<T> e (eXYZ, rOrder);
- r = e.toXYZVector ();
- }
- return true;
- }
- template <class T>
- bool
- extractSHRT (const Matrix44<T> &mat,
- Vec3<T> &s,
- Vec3<T> &h,
- Vec3<T> &r,
- Vec3<T> &t,
- bool exc)
- {
- return extractSHRT(mat, s, h, r, t, exc, Imath::Euler<T>::XYZ);
- }
- template <class T>
- bool
- extractSHRT (const Matrix44<T> &mat,
- Vec3<T> &s,
- Vec3<T> &h,
- Euler<T> &r,
- Vec3<T> &t,
- bool exc /* = true */)
- {
- return extractSHRT (mat, s, h, r, t, exc, r.order ());
- }
- template <class T>
- bool
- checkForZeroScaleInRow (const T& scl,
- const Vec3<T> &row,
- bool exc /* = true */ )
- {
- for (int i = 0; i < 3; i++)
- {
- if ((abs (scl) < 1 && abs (row[i]) >= limits<T>::max() * abs (scl)))
- {
- if (exc)
- throw Imath::ZeroScaleExc ("Cannot remove zero scaling "
- "from matrix.");
- else
- return false;
- }
- }
- return true;
- }
- template <class T>
- Matrix44<T>
- rotationMatrix (const Vec3<T> &from, const Vec3<T> &to)
- {
- Quat<T> q;
- q.setRotation(from, to);
- return q.toMatrix44();
- }
- template <class T>
- Matrix44<T>
- rotationMatrixWithUpDir (const Vec3<T> &fromDir,
- const Vec3<T> &toDir,
- const Vec3<T> &upDir)
- {
- //
- // The goal is to obtain a rotation matrix that takes
- // "fromDir" to "toDir". We do this in two steps and
- // compose the resulting rotation matrices;
- // (a) rotate "fromDir" into the z-axis
- // (b) rotate the z-axis into "toDir"
- //
- // The from direction must be non-zero; but we allow zero to and up dirs.
- if (fromDir.length () == 0)
- return Matrix44<T> ();
- else
- {
- Matrix44<T> zAxis2FromDir = alignZAxisWithTargetDir
- (fromDir, Vec3<T> (0, 1, 0));
- Matrix44<T> fromDir2zAxis = zAxis2FromDir.transposed ();
-
- Matrix44<T> zAxis2ToDir = alignZAxisWithTargetDir (toDir, upDir);
- return fromDir2zAxis * zAxis2ToDir;
- }
- }
- template <class T>
- Matrix44<T>
- alignZAxisWithTargetDir (Vec3<T> targetDir, Vec3<T> upDir)
- {
- //
- // Ensure that the target direction is non-zero.
- //
- if ( targetDir.length () == 0 )
- targetDir = Vec3<T> (0, 0, 1);
- //
- // Ensure that the up direction is non-zero.
- //
- if ( upDir.length () == 0 )
- upDir = Vec3<T> (0, 1, 0);
- //
- // Check for degeneracies. If the upDir and targetDir are parallel
- // or opposite, then compute a new, arbitrary up direction that is
- // not parallel or opposite to the targetDir.
- //
- if (upDir.cross (targetDir).length () == 0)
- {
- upDir = targetDir.cross (Vec3<T> (1, 0, 0));
- if (upDir.length() == 0)
- upDir = targetDir.cross(Vec3<T> (0, 0, 1));
- }
- //
- // Compute the x-, y-, and z-axis vectors of the new coordinate system.
- //
- Vec3<T> targetPerpDir = upDir.cross (targetDir);
- Vec3<T> targetUpDir = targetDir.cross (targetPerpDir);
-
- //
- // Rotate the x-axis into targetPerpDir (row 0),
- // rotate the y-axis into targetUpDir (row 1),
- // rotate the z-axis into targetDir (row 2).
- //
-
- Vec3<T> row[3];
- row[0] = targetPerpDir.normalized ();
- row[1] = targetUpDir .normalized ();
- row[2] = targetDir .normalized ();
-
- Matrix44<T> mat ( row[0][0], row[0][1], row[0][2], 0,
- row[1][0], row[1][1], row[1][2], 0,
- row[2][0], row[2][1], row[2][2], 0,
- 0, 0, 0, 1 );
-
- return mat;
- }
- //-----------------------------------------------------------------------------
- // Implementation for 3x3 Matrix
- //------------------------------
- template <class T>
- bool
- extractScaling (const Matrix33<T> &mat, Vec2<T> &scl, bool exc)
- {
- T shr;
- Matrix33<T> M (mat);
- if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
- return false;
- return true;
- }
- template <class T>
- Matrix33<T>
- sansScaling (const Matrix33<T> &mat, bool exc)
- {
- Vec2<T> scl;
- T shr;
- T rot;
- Vec2<T> tran;
- if (! extractSHRT (mat, scl, shr, rot, tran, exc))
- return mat;
- Matrix33<T> M;
-
- M.translate (tran);
- M.rotate (rot);
- M.shear (shr);
- return M;
- }
- template <class T>
- bool
- removeScaling (Matrix33<T> &mat, bool exc)
- {
- Vec2<T> scl;
- T shr;
- T rot;
- Vec2<T> tran;
- if (! extractSHRT (mat, scl, shr, rot, tran, exc))
- return false;
- mat.makeIdentity ();
- mat.translate (tran);
- mat.rotate (rot);
- mat.shear (shr);
- return true;
- }
- template <class T>
- bool
- extractScalingAndShear (const Matrix33<T> &mat, Vec2<T> &scl, T &shr, bool exc)
- {
- Matrix33<T> M (mat);
- if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
- return false;
- return true;
- }
- template <class T>
- Matrix33<T>
- sansScalingAndShear (const Matrix33<T> &mat, bool exc)
- {
- Vec2<T> scl;
- T shr;
- Matrix33<T> M (mat);
- if (! extractAndRemoveScalingAndShear (M, scl, shr, exc))
- return mat;
-
- return M;
- }
- template <class T>
- bool
- removeScalingAndShear (Matrix33<T> &mat, bool exc)
- {
- Vec2<T> scl;
- T shr;
- if (! extractAndRemoveScalingAndShear (mat, scl, shr, exc))
- return false;
-
- return true;
- }
- template <class T>
- bool
- extractAndRemoveScalingAndShear (Matrix33<T> &mat,
- Vec2<T> &scl, T &shr, bool exc)
- {
- Vec2<T> row[2];
- row[0] = Vec2<T> (mat[0][0], mat[0][1]);
- row[1] = Vec2<T> (mat[1][0], mat[1][1]);
-
- T maxVal = 0;
- for (int i=0; i < 2; i++)
- for (int j=0; j < 2; j++)
- if (Imath::abs (row[i][j]) > maxVal)
- maxVal = Imath::abs (row[i][j]);
- //
- // We normalize the 2x2 matrix here.
- // It was noticed that this can improve numerical stability significantly,
- // especially when many of the upper 2x2 matrix's coefficients are very
- // close to zero; we correct for this step at the end by multiplying the
- // scaling factors by maxVal at the end (shear and rotation are not
- // affected by the normalization).
- if (maxVal != 0)
- {
- for (int i=0; i < 2; i++)
- if (! checkForZeroScaleInRow (maxVal, row[i], exc))
- return false;
- else
- row[i] /= maxVal;
- }
- // Compute X scale factor.
- scl.x = row[0].length ();
- if (! checkForZeroScaleInRow (scl.x, row[0], exc))
- return false;
- // Normalize first row.
- row[0] /= scl.x;
- // An XY shear factor will shear the X coord. as the Y coord. changes.
- // There are 2 combinations (XY, YX), although we only extract the XY
- // shear factor because we can effect the an YX shear factor by
- // shearing in XY combined with rotations and scales.
- //
- // shear matrix < 1, YX, 0,
- // XY, 1, 0,
- // 0, 0, 1 >
- // Compute XY shear factor and make 2nd row orthogonal to 1st.
- shr = row[0].dot (row[1]);
- row[1] -= shr * row[0];
- // Now, compute Y scale.
- scl.y = row[1].length ();
- if (! checkForZeroScaleInRow (scl.y, row[1], exc))
- return false;
- // Normalize 2nd row and correct the XY shear factor for Y scaling.
- row[1] /= scl.y;
- shr /= scl.y;
- // At this point, the upper 2x2 matrix in mat is orthonormal.
- // Check for a coordinate system flip. If the determinant
- // is -1, then flip the rotation matrix and adjust the scale(Y)
- // and shear(XY) factors to compensate.
- if (row[0][0] * row[1][1] - row[0][1] * row[1][0] < 0)
- {
- row[1][0] *= -1;
- row[1][1] *= -1;
- scl[1] *= -1;
- shr *= -1;
- }
- // Copy over the orthonormal rows into the returned matrix.
- // The upper 2x2 matrix in mat is now a rotation matrix.
- for (int i=0; i < 2; i++)
- {
- mat[i][0] = row[i][0];
- mat[i][1] = row[i][1];
- }
- scl *= maxVal;
- return true;
- }
- template <class T>
- void
- extractEuler (const Matrix33<T> &mat, T &rot)
- {
- //
- // Normalize the local x and y axes to remove scaling.
- //
- Vec2<T> i (mat[0][0], mat[0][1]);
- Vec2<T> j (mat[1][0], mat[1][1]);
- i.normalize();
- j.normalize();
- //
- // Extract the angle, rot.
- //
- rot = - Math<T>::atan2 (j[0], i[0]);
- }
- template <class T>
- bool
- extractSHRT (const Matrix33<T> &mat,
- Vec2<T> &s,
- T &h,
- T &r,
- Vec2<T> &t,
- bool exc)
- {
- Matrix33<T> rot;
- rot = mat;
- if (! extractAndRemoveScalingAndShear (rot, s, h, exc))
- return false;
- extractEuler (rot, r);
- t.x = mat[2][0];
- t.y = mat[2][1];
- return true;
- }
- template <class T>
- bool
- checkForZeroScaleInRow (const T& scl,
- const Vec2<T> &row,
- bool exc /* = true */ )
- {
- for (int i = 0; i < 2; i++)
- {
- if ((abs (scl) < 1 && abs (row[i]) >= limits<T>::max() * abs (scl)))
- {
- if (exc)
- throw Imath::ZeroScaleExc ("Cannot remove zero scaling "
- "from matrix.");
- else
- return false;
- }
- }
- return true;
- }
- } // namespace Imath
- #endif