--- /dev/null
+/* -*- C++ -*- ------------------------------------------------------------
+
+Copyright (c) 2007 Jesse Anders and Demian Nave http://cmldev.net/
+
+The Configurable Math Library (CML) is distributed under the terms of the
+Boost Software License, v1.0 (see cml/LICENSE for details).
+
+ *-----------------------------------------------------------------------*/
+/** @file
+ * @brief
+ */
+
+#ifndef matrix_rotation_h
+#define matrix_rotation_h
+
+#include <cml/mathlib/matrix_misc.h>
+#include <cml/mathlib/vector_ortho.h>
+
+/* Functions related to matrix rotations in 3D and 2D. */
+
+namespace cml {
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D rotation about world axes
+//////////////////////////////////////////////////////////////////////////////
+
+/** Build a matrix representing a 3D rotation about the given world axis */
+template < typename E, class A, class B, class L > void
+matrix_rotation_world_axis( matrix<E,A,B,L>& m, size_t axis, E angle)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+ detail::CheckIndex3(axis);
+
+ size_t i, j, k;
+ cyclic_permutation(axis, i, j, k);
+
+ value_type s = value_type(std::sin(angle));
+ value_type c = value_type(std::cos(angle));
+
+ identity_transform(m);
+
+ m.set_basis_element(j,j, c);
+ m.set_basis_element(j,k, s);
+ m.set_basis_element(k,j,-s);
+ m.set_basis_element(k,k, c);
+}
+
+/** Build a matrix representing a 3D rotation about the world x axis */
+template < typename E, class A, class B, class L > void
+matrix_rotation_world_x(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotation_world_axis(m,0,angle);
+}
+
+/** Build a matrix representing a 3D rotation about the world y axis */
+template < typename E, class A, class B, class L > void
+matrix_rotation_world_y(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotation_world_axis(m,1,angle);
+}
+
+/** Build a matrix representing a 3D rotation about the world z axis */
+template < typename E, class A, class B, class L > void
+matrix_rotation_world_z(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotation_world_axis(m,2,angle);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D rotation from an axis-angle pair
+//////////////////////////////////////////////////////////////////////////////
+
+/** Build a rotation matrix from an axis-angle pair */
+template < typename E, class A, class B, class L, class VecT > void
+matrix_rotation_axis_angle(matrix<E,A,B,L>& m, const VecT& axis, E angle)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+ detail::CheckVec3(axis);
+
+ identity_transform(m);
+
+ value_type s = std::sin(angle);
+ value_type c = std::cos(angle);
+ value_type omc = value_type(1) - c;
+
+ value_type xomc = axis[0] * omc;
+ value_type yomc = axis[1] * omc;
+ value_type zomc = axis[2] * omc;
+
+ value_type xxomc = axis[0] * xomc;
+ value_type yyomc = axis[1] * yomc;
+ value_type zzomc = axis[2] * zomc;
+ value_type xyomc = axis[0] * yomc;
+ value_type yzomc = axis[1] * zomc;
+ value_type zxomc = axis[2] * xomc;
+
+ value_type xs = axis[0] * s;
+ value_type ys = axis[1] * s;
+ value_type zs = axis[2] * s;
+
+ m.set_basis_element(0,0, xxomc + c );
+ m.set_basis_element(0,1, xyomc + zs);
+ m.set_basis_element(0,2, zxomc - ys);
+ m.set_basis_element(1,0, xyomc - zs);
+ m.set_basis_element(1,1, yyomc + c );
+ m.set_basis_element(1,2, yzomc + xs);
+ m.set_basis_element(2,0, zxomc + ys);
+ m.set_basis_element(2,1, yzomc - xs);
+ m.set_basis_element(2,2, zzomc + c );
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D rotation from a quaternion
+//////////////////////////////////////////////////////////////////////////////
+
+/** Build a rotation matrix from a quaternion */
+template < typename E, class A, class B, class L, class QuatT > void
+matrix_rotation_quaternion(matrix<E,A,B,L>& m, const QuatT& q)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef QuatT quaternion_type;
+ typedef typename quaternion_type::order_type order_type;
+ typedef typename matrix_type::value_type value_type;
+
+ enum {
+ W = order_type::W,
+ X = order_type::X,
+ Y = order_type::Y,
+ Z = order_type::Z
+ };
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+ detail::CheckQuat(q);
+
+ identity_transform(m);
+
+ value_type x2 = q[X] + q[X];
+ value_type y2 = q[Y] + q[Y];
+ value_type z2 = q[Z] + q[Z];
+
+ value_type xx2 = q[X] * x2;
+ value_type yy2 = q[Y] * y2;
+ value_type zz2 = q[Z] * z2;
+ value_type xy2 = q[X] * y2;
+ value_type yz2 = q[Y] * z2;
+ value_type zx2 = q[Z] * x2;
+ value_type xw2 = q[W] * x2;
+ value_type yw2 = q[W] * y2;
+ value_type zw2 = q[W] * z2;
+
+ m.set_basis_element(0,0, value_type(1) - yy2 - zz2);
+ m.set_basis_element(0,1, xy2 + zw2);
+ m.set_basis_element(0,2, zx2 - yw2);
+ m.set_basis_element(1,0, xy2 - zw2);
+ m.set_basis_element(1,1, value_type(1) - zz2 - xx2);
+ m.set_basis_element(1,2, yz2 + xw2);
+ m.set_basis_element(2,0, zx2 + yw2);
+ m.set_basis_element(2,1, yz2 - xw2);
+ m.set_basis_element(2,2, value_type(1) - xx2 - yy2);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D rotation from Euler angles
+//////////////////////////////////////////////////////////////////////////////
+
+/** Build a rotation matrix from an Euler-angle triple
+ *
+ * The rotations are applied about the cardinal axes in the order specified by
+ * the 'order' argument, where 'order' is one of the following enumerants:
+ *
+ * euler_order_xyz
+ * euler_order_xzy
+ * euler_order_xyx
+ * euler_order_xzx
+ * euler_order_yzx
+ * euler_order_yxz
+ * euler_order_yzy
+ * euler_order_yxy
+ * euler_order_zxy
+ * euler_order_zyx
+ * euler_order_zxz
+ * euler_order_zyz
+ *
+ * e.g. euler_order_xyz means compute the column-basis rotation matrix
+ * equivalent to R_x * R_y * R_z, where R_i is the rotation matrix above
+ * axis i (the row-basis matrix would be R_z * R_y * R_x).
+ */
+template < typename E, class A, class B, class L > void
+matrix_rotation_euler(matrix<E,A,B,L>& m, E angle_0, E angle_1, E angle_2,
+ EulerOrder order)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+
+ identity_transform(m);
+
+ size_t i, j, k;
+ bool odd, repeat;
+ detail::unpack_euler_order(order, i, j, k, odd, repeat);
+
+ if (odd) {
+ angle_0 = -angle_0;
+ angle_1 = -angle_1;
+ angle_2 = -angle_2;
+ }
+
+ value_type s0 = std::sin(angle_0);
+ value_type c0 = std::cos(angle_0);
+ value_type s1 = std::sin(angle_1);
+ value_type c1 = std::cos(angle_1);
+ value_type s2 = std::sin(angle_2);
+ value_type c2 = std::cos(angle_2);
+
+ value_type s0s2 = s0 * s2;
+ value_type s0c2 = s0 * c2;
+ value_type c0s2 = c0 * s2;
+ value_type c0c2 = c0 * c2;
+
+ if (repeat) {
+ m.set_basis_element(i,i, c1 );
+ m.set_basis_element(i,j, s1 * s2 );
+ m.set_basis_element(i,k,-s1 * c2 );
+ m.set_basis_element(j,i, s0 * s1 );
+ m.set_basis_element(j,j,-c1 * s0s2 + c0c2);
+ m.set_basis_element(j,k, c1 * s0c2 + c0s2);
+ m.set_basis_element(k,i, c0 * s1 );
+ m.set_basis_element(k,j,-c1 * c0s2 - s0c2);
+ m.set_basis_element(k,k, c1 * c0c2 - s0s2);
+ } else {
+ m.set_basis_element(i,i, c1 * c2 );
+ m.set_basis_element(i,j, c1 * s2 );
+ m.set_basis_element(i,k,-s1 );
+ m.set_basis_element(j,i, s1 * s0c2 - c0s2);
+ m.set_basis_element(j,j, s1 * s0s2 + c0c2);
+ m.set_basis_element(j,k, s0 * c1 );
+ m.set_basis_element(k,i, s1 * c0c2 + s0s2);
+ m.set_basis_element(k,j, s1 * c0s2 - s0c2);
+ m.set_basis_element(k,k, c0 * c1 );
+ }
+}
+
+/** Build a matrix of derivatives of Euler angles about the specified axis.
+ *
+ * The rotation derivatives are applied about the cardinal axes in the
+ * order specified by the 'order' argument, where 'order' is one of the
+ * following enumerants:
+ *
+ * euler_order_xyz
+ * euler_order_xzy
+ * euler_order_yzx
+ * euler_order_yxz
+ * euler_order_zxy
+ * euler_order_zyx
+ *
+ * e.g. euler_order_xyz means compute the column-basis rotation matrix
+ * equivalent to R_x * R_y * R_z, where R_i is the rotation matrix above
+ * axis i (the row-basis matrix would be R_z * R_y * R_x).
+ *
+ * The derivative is taken with respect to the specified 'axis', which is
+ * the position of the axis in the triple; e.g. if order = euler_order_xyz,
+ * then axis = 0 would mean take the derivative with respect to x. Note
+ * that repeated axes are not currently supported.
+ */
+template < typename E, class A, class B, class L > void
+matrix_rotation_euler_derivatives(
+ matrix<E,A,B,L>& m, int axis, E angle_0, E angle_1, E angle_2,
+ EulerOrder order)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+
+ size_t i, j, k;
+ bool odd, repeat;
+ detail::unpack_euler_order(order, i, j, k, odd, repeat);
+ if(repeat) throw std::invalid_argument(
+ "matrix_rotation_euler_derivatives does not support repeated axes");
+
+ if (odd) {
+ angle_0 = -angle_0;
+ angle_1 = -angle_1;
+ angle_2 = -angle_2;
+ }
+
+ value_type s0 = std::sin(angle_0);
+ value_type c0 = std::cos(angle_0);
+ value_type s1 = std::sin(angle_1);
+ value_type c1 = std::cos(angle_1);
+ value_type s2 = std::sin(angle_2);
+ value_type c2 = std::cos(angle_2);
+
+ value_type s0s2 = s0 * s2;
+ value_type s0c2 = s0 * c2;
+ value_type c0s2 = c0 * s2;
+ value_type c0c2 = c0 * c2;
+
+ if(axis == 0) {
+ m.set_basis_element(i,i, 0. );
+ m.set_basis_element(i,j, 0. );
+ m.set_basis_element(i,k, 0. );
+ m.set_basis_element(j,i, s1 * c0*c2 + s0*s2);
+ m.set_basis_element(j,j, s1 * c0*s2 - s0*c2);
+ m.set_basis_element(j,k, c0 * c1 );
+ m.set_basis_element(k,i,-s1 * s0*c2 + c0*s2);
+ m.set_basis_element(k,j,-s1 * s0*s2 - c0*c2);
+ m.set_basis_element(k,k,-s0 * c1 );
+ } else if(axis == 1) {
+ m.set_basis_element(i,i,-s1 * c2 );
+ m.set_basis_element(i,j,-s1 * s2 );
+ m.set_basis_element(i,k,-c1 );
+ m.set_basis_element(j,i, c1 * s0*c2 );
+ m.set_basis_element(j,j, c1 * s0*s2 );
+ m.set_basis_element(j,k,-s0 * s1 );
+ m.set_basis_element(k,i, c1 * c0*c2 );
+ m.set_basis_element(k,j, c1 * c0*s2 );
+ m.set_basis_element(k,k,-c0 * s1 );
+ } else if(axis == 2) {
+ m.set_basis_element(i,i,-c1 * s2 );
+ m.set_basis_element(i,j, c1 * c2 );
+ m.set_basis_element(i,k, 0. );
+ m.set_basis_element(j,i,-s1 * s0*s2 - c0*c2);
+ m.set_basis_element(j,j, s1 * s0*c2 - c0*s2);
+ m.set_basis_element(j,k, 0. );
+ m.set_basis_element(k,i,-s1 * c0*s2 + s0*c2);
+ m.set_basis_element(k,j, s1 * c0*c2 + s0*s2);
+ m.set_basis_element(k,k, 0. );
+ }
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D rotation to align with a vector, multiple vectors, or the view plane
+//////////////////////////////////////////////////////////////////////////////
+
+/** See vector_ortho.h for details */
+template < typename E,class A,class B,class L,class VecT_1,class VecT_2 > void
+matrix_rotation_align(
+ matrix<E,A,B,L>& m,
+ const VecT_1& align,
+ const VecT_2& reference,
+ bool normalize = true,
+ AxisOrder order = axis_order_zyx)
+{
+ typedef vector< E,fixed<3> > vector_type;
+
+ identity_transform(m);
+
+ vector_type x, y, z;
+
+ orthonormal_basis(align, reference, x, y, z, normalize, order);
+ matrix_set_basis_vectors(m, x, y, z);
+}
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L, class VecT > void
+matrix_rotation_align(matrix<E,A,B,L>& m, const VecT& align,
+ bool normalize = true, AxisOrder order = axis_order_zyx)
+{
+ typedef vector< E,fixed<3> > vector_type;
+
+ identity_transform(m);
+
+ vector_type x, y, z;
+
+ orthonormal_basis(align, x, y, z, normalize, order);
+ matrix_set_basis_vectors(m, x, y, z);
+}
+
+/** See vector_ortho.h for details */
+template < typename E,class A,class B,class L,class VecT_1,class VecT_2 > void
+matrix_rotation_align_axial(matrix<E,A,B,L>& m, const VecT_1& align,
+ const VecT_2& axis, bool normalize = true,
+ AxisOrder order = axis_order_zyx)
+{
+ typedef vector< E,fixed<3> > vector_type;
+
+ identity_transform(m);
+
+ vector_type x, y, z;
+
+ orthonormal_basis_axial(align, axis, x, y, z, normalize, order);
+ matrix_set_basis_vectors(m, x, y, z);
+}
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L, class MatT > void
+matrix_rotation_align_viewplane(
+ matrix<E,A,B,L>& m,
+ const MatT& view_matrix,
+ Handedness handedness,
+ AxisOrder order = axis_order_zyx)
+{
+ typedef vector< E, fixed<3> > vector_type;
+
+ identity_transform(m);
+
+ vector_type x, y, z;
+
+ orthonormal_basis_viewplane(view_matrix, x, y, z, handedness, order);
+ matrix_set_basis_vectors(m, x, y, z);
+}
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L, class MatT > void
+matrix_rotation_align_viewplane_LH(
+ matrix<E,A,B,L>& m,
+ const MatT& view_matrix,
+ AxisOrder order = axis_order_zyx)
+{
+ matrix_rotation_align_viewplane(
+ m,view_matrix,left_handed,order);
+}
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L, class MatT > void
+matrix_rotation_align_viewplane_RH(
+ matrix<E,A,B,L>& m,
+ const MatT& view_matrix,
+ AxisOrder order = axis_order_zyx)
+{
+ matrix_rotation_align_viewplane(
+ m,view_matrix,right_handed,order);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D rotation to aim at a target
+//////////////////////////////////////////////////////////////////////////////
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L,
+ class VecT_1, class VecT_2, class VecT_3 > void
+matrix_rotation_aim_at(
+ matrix<E,A,B,L>& m,
+ const VecT_1& pos,
+ const VecT_2& target,
+ const VecT_3& reference,
+ AxisOrder order = axis_order_zyx)
+{
+ matrix_rotation_align(m, target - pos, reference, true, order);
+}
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L,
+ class VecT_1, class VecT_2 > void
+matrix_rotation_aim_at(
+ matrix<E,A,B,L>& m,
+ const VecT_1& pos,
+ const VecT_2& target,
+ AxisOrder order = axis_order_zyx)
+{
+ matrix_rotation_align(m, target - pos, true, order);
+}
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L,
+ class VecT_1, class VecT_2, class VecT_3 > void
+matrix_rotation_aim_at_axial(
+ matrix<E,A,B,L>& m,
+ const VecT_1& pos,
+ const VecT_2& target,
+ const VecT_3& axis,
+ AxisOrder order = axis_order_zyx)
+{
+ matrix_rotation_align_axial(m, target - pos, axis, true, order);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 2D rotation
+//////////////////////////////////////////////////////////////////////////////
+
+/** Build a matrix representing a 2D rotation */
+template < typename E, class A, class B, class L > void
+matrix_rotation_2D( matrix<E,A,B,L>& m, E angle)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear2D(m);
+
+ value_type s = value_type(std::sin(angle));
+ value_type c = value_type(std::cos(angle));
+
+ identity_transform(m);
+
+ m.set_basis_element(0,0, c);
+ m.set_basis_element(0,1, s);
+ m.set_basis_element(1,0,-s);
+ m.set_basis_element(1,1, c);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 2D rotation to align with a vector
+//////////////////////////////////////////////////////////////////////////////
+
+/** See vector_ortho.h for details */
+template < typename E, class A, class B, class L, class VecT > void
+matrix_rotation_align_2D(matrix<E,A,B,L>& m, const VecT& align,
+ bool normalize = true, AxisOrder2D order = axis_order_xy)
+{
+ typedef vector< E, fixed<2> > vector_type;
+
+ identity_transform(m);
+
+ vector_type x, y;
+
+ orthonormal_basis_2D(align, x, y, normalize, order);
+ matrix_set_basis_vectors_2D(m, x, y);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D relative rotation about world axes
+//////////////////////////////////////////////////////////////////////////////
+
+/** Rotate a rotation matrix about the given world axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_world_axis(matrix<E,A,B,L>& m, size_t axis, E angle)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+ detail::CheckIndex3(axis);
+
+ size_t i, j, k;
+ cyclic_permutation(axis, i, j, k);
+
+ value_type s = value_type(std::sin(angle));
+ value_type c = value_type(std::cos(angle));
+
+ value_type ij = c * m.basis_element(i,j) - s * m.basis_element(i,k);
+ value_type jj = c * m.basis_element(j,j) - s * m.basis_element(j,k);
+ value_type kj = c * m.basis_element(k,j) - s * m.basis_element(k,k);
+
+ m.set_basis_element(i,k, s*m.basis_element(i,j) + c*m.basis_element(i,k));
+ m.set_basis_element(j,k, s*m.basis_element(j,j) + c*m.basis_element(j,k));
+ m.set_basis_element(k,k, s*m.basis_element(k,j) + c*m.basis_element(k,k));
+
+ m.set_basis_element(i,j,ij);
+ m.set_basis_element(j,j,jj);
+ m.set_basis_element(k,j,kj);
+}
+
+/** Rotate a rotation matrix about the world x axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_world_x(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotate_about_world_axis(m,0,angle);
+}
+
+/** Rotate a rotation matrix about the world y axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_world_y(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotate_about_world_axis(m,1,angle);
+}
+
+/** Rotate a rotation matrix about the world z axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_world_z(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotate_about_world_axis(m,2,angle);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 3D relative rotation about local axes
+//////////////////////////////////////////////////////////////////////////////
+
+/** Rotate a rotation matrix about the given local axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_local_axis(matrix<E,A,B,L>& m, size_t axis, E angle)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+ detail::CheckIndex3(axis);
+
+ size_t i, j, k;
+ cyclic_permutation(axis, i, j, k);
+
+ value_type s = value_type(std::sin(angle));
+ value_type c = value_type(std::cos(angle));
+
+ value_type j0 = c * m.basis_element(j,0) + s * m.basis_element(k,0);
+ value_type j1 = c * m.basis_element(j,1) + s * m.basis_element(k,1);
+ value_type j2 = c * m.basis_element(j,2) + s * m.basis_element(k,2);
+
+ m.set_basis_element(k,0, c*m.basis_element(k,0) - s*m.basis_element(j,0));
+ m.set_basis_element(k,1, c*m.basis_element(k,1) - s*m.basis_element(j,1));
+ m.set_basis_element(k,2, c*m.basis_element(k,2) - s*m.basis_element(j,2));
+
+ m.set_basis_element(j,0,j0);
+ m.set_basis_element(j,1,j1);
+ m.set_basis_element(j,2,j2);
+}
+
+/** Rotate a rotation matrix about its local x axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_local_x(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotate_about_local_axis(m,0,angle);
+}
+
+/** Rotate a rotation matrix about its local y axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_local_y(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotate_about_local_axis(m,1,angle);
+}
+
+/** Rotate a rotation matrix about its local z axis */
+template < typename E, class A, class B, class L > void
+matrix_rotate_about_local_z(matrix<E,A,B,L>& m, E angle) {
+ matrix_rotate_about_local_axis(m,2,angle);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// 2D relative rotation
+//////////////////////////////////////////////////////////////////////////////
+
+template < typename E, class A, class B, class L > void
+matrix_rotate_2D(matrix<E,A,B,L>& m, E angle)
+{
+ typedef matrix<E,A,B,L> matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear2D(m);
+
+ value_type s = value_type(std::sin(angle));
+ value_type c = value_type(std::cos(angle));
+
+ value_type m00 = c * m.basis_element(0,0) - s * m.basis_element(0,1);
+ value_type m10 = c * m.basis_element(1,0) - s * m.basis_element(1,1);
+
+ m.set_basis_element(0,1, s*m.basis_element(0,0) + c*m.basis_element(0,1));
+ m.set_basis_element(1,1, s*m.basis_element(1,0) + c*m.basis_element(1,1));
+
+ m.set_basis_element(0,0,m00);
+ m.set_basis_element(1,0,m10);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// Rotation from vector to vector
+//////////////////////////////////////////////////////////////////////////////
+
+/** Build a rotation matrix to rotate from one vector to another
+ *
+ * Note: The quaternion algorithm is more stable than the matrix algorithm, so
+ * we simply pass off to the quaternion function here.
+ */
+template < class E,class A,class B,class L,class VecT_1,class VecT_2 > void
+matrix_rotation_vec_to_vec(
+ matrix<E,A,B,L>& m,
+ const VecT_1& v1,
+ const VecT_2& v2,
+ bool unit_length_vectors = false)
+{
+ typedef quaternion< E,fixed<>,vector_first,positive_cross >
+ quaternion_type;
+
+ quaternion_type q;
+ quaternion_rotation_vec_to_vec(q,v1,v2,unit_length_vectors);
+ matrix_rotation_quaternion(m,q);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// Scale the angle of a rotation matrix
+//////////////////////////////////////////////////////////////////////////////
+
+/** Scale the angle of a 3D rotation matrix */
+template < typename E, class A, class B, class L > void
+matrix_scale_rotation_angle(matrix<E,A,B,L>& m, E t,
+ E tolerance = epsilon<E>::placeholder())
+{
+ typedef vector< E,fixed<3> > vector_type;
+ typedef typename vector_type::value_type value_type;
+
+ vector_type axis;
+ value_type angle;
+ matrix_to_axis_angle(m, axis, angle, tolerance);
+ matrix_rotation_axis_angle(m, axis, angle * t);
+}
+
+/** Scale the angle of a 2D rotation matrix */
+template < typename E, class A, class B, class L > void
+matrix_scale_rotation_angle_2D(
+ matrix<E,A,B,L>& m, E t, E tolerance = epsilon<E>::placeholder())
+{
+ typedef vector< E,fixed<2> > vector_type;
+ typedef typename vector_type::value_type value_type;
+
+ value_type angle = matrix_to_rotation_2D(m);
+ matrix_rotation_2D(m, angle * t);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// Support functions for uniform handling of row- and column-basis matrices
+//////////////////////////////////////////////////////////////////////////////
+
+/* Note: The matrix rotation slerp, difference and concatenation functions do
+ * not use et::MatrixPromote<M1,M2>::temporary_type as the return type, even
+ * though that is the return type of the underlying matrix multiplication.
+ * This is because the sizes of these matrices are known at compile time (3x3
+ * and 2x2), and using fixed<> obviates the need for resizing of intermediate
+ * temporaries.
+ *
+ * Also, no size- or type-checking is done on the arguments to these
+ * functions, as any such errors will be caught by the matrix multiplication
+ * and assignment to the 3x3 temporary.
+ */
+
+/** A fixed-size temporary 3x3 matrix */
+#define MAT_TEMP_3X3 matrix< \
+ typename et::ScalarPromote< \
+ typename MatT_1::value_type, \
+ typename MatT_2::value_type \
+ >::type, \
+ fixed<3,3>, \
+ typename MatT_1::basis_orient, \
+ row_major \
+>
+
+/** A fixed-size temporary 2x2 matrix */
+#define MAT_TEMP_2X2 matrix< \
+ typename et::ScalarPromote< \
+ typename MatT_1::value_type, \
+ typename MatT_2::value_type \
+ >::type, \
+ fixed<2,2>, \
+ typename MatT_1::basis_orient, \
+ row_major \
+>
+
+namespace detail {
+
+/** Concatenate two 3D row-basis rotation matrices in the order m1->m2 */
+template < class MatT_1, class MatT_2 > MAT_TEMP_3X3
+matrix_concat_rotations(const MatT_1& m1, const MatT_2& m2, row_basis) {
+ return m1*m2;
+}
+
+/** Concatenate two 3D col-basis rotation matrices in the order m1->m2 */
+template < class MatT_1, class MatT_2 > MAT_TEMP_3X3
+matrix_concat_rotations(const MatT_1& m1, const MatT_2& m2, col_basis) {
+ return m2*m1;
+}
+
+/** Concatenate two 3D rotation matrices in the order m1->m2 */
+template < class MatT_1, class MatT_2 > MAT_TEMP_3X3
+matrix_concat_rotations(const MatT_1& m1, const MatT_2& m2) {
+ return matrix_concat_rotations(m1,m2,typename MatT_1::basis_orient());
+}
+
+/** Concatenate two 2D row-basis rotation matrices in the order m1->m2 */
+template < class MatT_1, class MatT_2 > MAT_TEMP_2X2
+matrix_concat_rotations_2D(const MatT_1& m1, const MatT_2& m2, row_basis) {
+ return m1*m2;
+}
+
+/** Concatenate two 2D col-basis rotation matrices in the order m1->m2 */
+template < class MatT_1, class MatT_2 > MAT_TEMP_2X2
+matrix_concat_rotations_2D(const MatT_1& m1, const MatT_2& m2, col_basis) {
+ return m2*m1;
+}
+
+/** Concatenate two 2D rotation matrices in the order m1->m2 */
+template < class MatT_1, class MatT_2 > MAT_TEMP_2X2
+matrix_concat_rotations_2D(const MatT_1& m1, const MatT_2& m2) {
+ return matrix_concat_rotations_2D(m1,m2,typename MatT_1::basis_orient());
+}
+
+} // namespace detail
+
+//////////////////////////////////////////////////////////////////////////////
+// Matrix rotation difference
+//////////////////////////////////////////////////////////////////////////////
+
+/** Return the rotational 'difference' between two 3D rotation matrices */
+template < class MatT_1, class MatT_2 > MAT_TEMP_3X3
+matrix_rotation_difference(const MatT_1& m1, const MatT_2& m2) {
+ return detail::matrix_concat_rotations(transpose(m1),m2);
+}
+
+/** Return the rotational 'difference' between two 2D rotation matrices */
+template < class MatT_1, class MatT_2 > MAT_TEMP_2X2
+matrix_rotation_difference_2D(const MatT_1& m1, const MatT_2& m2) {
+ return detail::matrix_concat_rotations_2D(transpose(m1),m2);
+}
+
+//////////////////////////////////////////////////////////////////////////////
+// Spherical linear interpolation of rotation matrices
+//////////////////////////////////////////////////////////////////////////////
+
+/* @todo: It might be as fast or faster to simply convert the matrices to
+ * quaternions, interpolate, and convert back.
+ *
+ * @todo: The behavior of matrix slerp is currently a little different than
+ * for quaternions: in the matrix function, when the two matrices are close
+ * to identical the first is returned, while in the quaternion function the
+ * quaternions are nlerp()'d in this case.
+ *
+ * I still need to do the equivalent of nlerp() for matrices, in which case
+ * these functions could be revised to pass off to nlerp() when the matrices
+ * are nearly aligned.
+*/
+
+/** Spherical linear interpolation of two 3D rotation matrices */
+template < class MatT_1, class MatT_2, typename E > MAT_TEMP_3X3
+matrix_slerp(const MatT_1& m1, const MatT_2& m2, E t,
+ E tolerance = epsilon<E>::placeholder())
+{
+ typedef MAT_TEMP_3X3 temporary_type;
+
+ temporary_type m = matrix_rotation_difference(m1,m2);
+ matrix_scale_rotation_angle(m,t,tolerance);
+ return detail::matrix_concat_rotations(m1,m);
+}
+
+/** Spherical linear interpolation of two 2D rotation matrices */
+template < class MatT_1, class MatT_2, typename E > MAT_TEMP_2X2
+matrix_slerp_2D(const MatT_1& m1, const MatT_2& m2, E t,
+ E tolerance = epsilon<E>::placeholder())
+{
+ typedef MAT_TEMP_2X2 temporary_type;
+
+ temporary_type m = matrix_rotation_difference_2D(m1,m2);
+ matrix_scale_rotation_angle_2D(m,t,tolerance);
+ return detail::matrix_concat_rotations_2D(m1,m);
+}
+
+#undef MAT_TEMP_3X3
+#undef MAT_TEMP_2X2
+
+//////////////////////////////////////////////////////////////////////////////
+// Conversions
+//////////////////////////////////////////////////////////////////////////////
+
+/** Convert a 3D rotation matrix to an axis-angle pair */
+template < class MatT, typename E, class A > void
+matrix_to_axis_angle(
+ const MatT& m,
+ vector<E,A >& axis,
+ E& angle,
+ E tolerance = epsilon<E>::placeholder())
+{
+ typedef MatT matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+
+ axis.set(
+ m.basis_element(1,2) - m.basis_element(2,1),
+ m.basis_element(2,0) - m.basis_element(0,2),
+ m.basis_element(0,1) - m.basis_element(1,0)
+ );
+ value_type l = length(axis);
+ value_type tmo = trace_3x3(m) - value_type(1);
+
+ if (l > tolerance) {
+ axis /= l;
+ angle = std::atan2(l, tmo); // l=2sin(theta),tmo=2cos(theta)
+ } else if (tmo > value_type(0)) {
+ axis.zero();
+ angle = value_type(0);
+ } else {
+ size_t largest_diagonal_element =
+ index_of_max(
+ m.basis_element(0,0),
+ m.basis_element(1,1),
+ m.basis_element(2,2)
+ );
+ size_t i, j, k;
+ cyclic_permutation(largest_diagonal_element, i, j, k);
+ axis[i] =
+ std::sqrt(
+ m.basis_element(i,i) -
+ m.basis_element(j,j) -
+ m.basis_element(k,k) +
+ value_type(1)
+ ) * value_type(.5);
+ value_type s = value_type(.5) / axis[i];
+ axis[j] = m.basis_element(i,j) * s;
+ axis[k] = m.basis_element(i,k) * s;
+ angle = constants<value_type>::pi();
+ }
+}
+
+/** Convert a 3D rotation matrix to an Euler-angle triple */
+template < class MatT, typename Real >
+void matrix_to_euler(
+ const MatT& m,
+ Real& angle_0,
+ Real& angle_1,
+ Real& angle_2,
+ EulerOrder order,
+ Real tolerance = epsilon<Real>::placeholder())
+{
+ typedef MatT matrix_type;
+ typedef typename matrix_type::value_type value_type;
+
+ /* Checking */
+ detail::CheckMatLinear3D(m);
+
+ size_t i, j, k;
+ bool odd, repeat;
+ detail::unpack_euler_order(order, i, j, k, odd, repeat);
+
+ if (repeat) {
+ value_type s1 = length(m.basis_element(j,i),m.basis_element(k,i));
+ value_type c1 = m.basis_element(i,i);
+
+ angle_1 = std::atan2(s1, c1);
+ if (s1 > tolerance) {
+ angle_0 = std::atan2(m.basis_element(j,i),m.basis_element(k,i));
+ angle_2 = std::atan2(m.basis_element(i,j),-m.basis_element(i,k));
+ } else {
+ angle_0 = value_type(0);
+ angle_2 = sign(c1) *
+ std::atan2(-m.basis_element(k,j),m.basis_element(j,j));
+ }
+ } else {
+ value_type s1 = -m.basis_element(i,k);
+ value_type c1 = length(m.basis_element(i,i),m.basis_element(i,j));
+
+ angle_1 = std::atan2(s1, c1);
+ if (c1 > tolerance) {
+ angle_0 = std::atan2(m.basis_element(j,k),m.basis_element(k,k));
+ angle_2 = std::atan2(m.basis_element(i,j),m.basis_element(i,i));
+ } else {
+ angle_0 = value_type(0);
+ angle_2 = -sign(s1) *
+ std::atan2(-m.basis_element(k,j),m.basis_element(j,j));
+ }
+ }
+
+ if (odd) {
+ angle_0 = -angle_0;
+ angle_1 = -angle_1;
+ angle_2 = -angle_2;
+ }
+}
+
+/** Convenience function to return a 3D vector containing the Euler angles
+ * in the requested order.
+ */
+template < class MatT, typename Real > vector< Real, fixed<3> >
+matrix_to_euler(
+ const MatT& m,
+ EulerOrder order,
+ Real tolerance = epsilon<Real>::placeholder())
+{
+ Real e0, e1, e2;
+ matrix_to_euler(m, e0, e1, e2, order, tolerance);
+ return vector< Real, fixed<3> >(e0, e1, e2);
+}
+
+/** Convert a 2D rotation matrix to a rotation angle */
+template < class MatT > typename MatT::value_type
+matrix_to_rotation_2D(const MatT& m)
+{
+ /* Checking */
+ detail::CheckMatLinear2D(m);
+
+ return std::atan2(m.basis_element(0,1),m.basis_element(0,0));
+}
+
+} // namespace cml
+
+#endif
projects / chaz / yoink / blobdiff