+++ /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 vector_ortho_h
-#define vector_ortho_h
-
-#include <cml/mathlib/vector_misc.h>
-#include <cml/mathlib/misc.h>
-
-/* Functions for orthonormalizing a set of basis vector in 3D or 2D, and for
- * constructing an orthonormal basis given various input parameters.
- */
-
-namespace cml {
-
-/* Orthonormalize 3 basis vectors in R3.
- *
- * Called with the default values, this function performs a single Gram-
- * Schmidt step to orthonormalize the input vectors. By default, the direction
- * of the 3rd basis vector is unchanged by this operation, but the unaffected
- * axis can be specified via the 'stable_axis' parameter.
- *
- * The arguments 'num_iter' and 's' can be specified to an iterative Gram-
- * Schmidt step. 'num_iter' is the number of iterations applied, and 's' is
- * the fraction applied towards orthonormality each step.
- *
- * In most cases, the default arguments can be ignored, leaving only the three
- * input vectors.
- */
-
-//////////////////////////////////////////////////////////////////////////////
-// Orthonormalization in 3D and 2D
-//////////////////////////////////////////////////////////////////////////////
-
-template < typename E, class A > void
-orthonormalize(vector<E,A>& v0, vector<E,A>& v1, vector<E,A>& v2,
- size_t stable_axis = 2, size_t num_iter = 0, E s = E(1))
-{
- /* Checking */
- detail::CheckVec3(v0);
- detail::CheckVec3(v1);
- detail::CheckVec3(v2);
- detail::CheckIndex3(stable_axis);
-
- typedef vector< E, fixed<3> > vector_type;
- typedef typename vector_type::value_type value_type;
-
- /* Iterative Gram-Schmidt; this step is skipped by default. */
-
- for (size_t i = 0; i < num_iter; ++i) {
- value_type dot01 = dot(v0,v1);
- value_type dot12 = dot(v1,v2);
- value_type dot20 = dot(v2,v0);
- value_type inv_dot00 = value_type(1) / dot(v0,v0);
- value_type inv_dot11 = value_type(1) / dot(v1,v1);
- value_type inv_dot22 = value_type(1) / dot(v2,v2);
-
- vector_type temp0 = v0 - s*dot01*inv_dot11*v1 - s*dot20*inv_dot22*v2;
- vector_type temp1 = v1 - s*dot12*inv_dot22*v2 - s*dot01*inv_dot00*v0;
- vector_type temp2 = v2 - s*dot20*inv_dot00*v0 - s*dot12*inv_dot11*v1;
-
- v0 = temp0;
- v1 = temp1;
- v2 = temp2;
- }
-
- /* Final Gram-Schmidt step to ensure orthonormality. If no iterations
- * have been requested (num_iter = 0), this is the only step. The step
- * is performed such that the direction of the axis indexed by
- * 'stable_axis' is unchanged.
- */
-
- size_t i, j, k;
- cyclic_permutation(stable_axis, i, j, k);
- vector_type v[] = { v0, v1, v2 };
-
- v[i].normalize();
- v[j] = normalize(project_to_hplane(v[j],v[i]));
- v[k] = normalize(project_to_hplane(project_to_hplane(v[k],v[i]),v[j]));
-
- v0 = v[0];
- v1 = v[1];
- v2 = v[2];
-}
-
-/* Orthonormalize 2 basis vectors in R2 */
-template < typename E, class A > void
-orthonormalize(vector<E,A>& v0, vector<E,A>& v1,
- size_t stable_axis = 0, size_t num_iter = 0, E s = E(1))
-{
- typedef vector< E, fixed<2> > vector_type;
- typedef typename vector_type::value_type value_type;
-
- /* Checking */
- detail::CheckVec2(v0);
- detail::CheckVec2(v1);
- detail::CheckIndex2(stable_axis);
-
- /* Iterative Gram-Schmidt; this step is skipped by default. */
-
- for (size_t i = 0; i < num_iter; ++i) {
- value_type dot01 = dot(v0,v1);
-
- vector_type temp0 = v0 - (s * dot01 * v1) / dot(v1,v1);
- vector_type temp1 = v1 - (s * dot01 * v0) / dot(v0,v0);
-
- v0 = temp0;
- v1 = temp1;
- }
-
- /* Final Gram-Schmidt step to ensure orthonormality. If no iterations
- * have been requested (num_iter = 0), this is the only step. The step
- * is performed such that the direction of the axis indexed by
- * 'stable_axis' is unchanged.
- */
-
- size_t i, j;
- cyclic_permutation(stable_axis, i, j);
- vector_type v[] = { v0, v1 };
-
- v[i].normalize();
- v[j] = normalize(project_to_hplane(v[j],v[i]));
-
- v0 = v[0];
- v1 = v[1];
-}
-
-//////////////////////////////////////////////////////////////////////////////
-// Orthonormal basis construction in 3D and 2D
-//////////////////////////////////////////////////////////////////////////////
-
-/* This version of orthonormal_basis() ultimately does the work for all
- * orthonormal_basis_*() functions. Given input vectors 'align' and
- * 'reference', and an order 'axis_order_<i><j><k>', it constructs an
- * orthonormal basis such that the i'th basis vector is aligned with (parallel
- * to and pointing in the same direction as) 'align', and the j'th basis
- * vector is maximally aligned with 'reference'. The k'th basis vector is
- * chosen such that the basis has a determinant of +1.
- *
- * Note that the algorithm fails when 'align' is nearly parallel to
- * 'reference'; this should be checked for and handled externally if it's a
- * case that may occur.
- */
-
-/* Note: This is an example of the 'non-const argument modification
- * invalidates expression' gotcha. If x, y or z were to be assigned to before
- * we were 'done' with align and reference, and if one of them were the same
- * object as align or reference, then the algorithm could fail. As is the
- * basis vectors are assigned at the end of the function from a temporary
- * array, so all is well.
- */
-
-template < class VecT_1, class VecT_2, typename E, class A > void
-orthonormal_basis(
- const VecT_1& align,
- const VecT_2& reference,
- vector<E,A>& x,
- vector<E,A>& y,
- vector<E,A>& z,
- bool normalize_align = true,
- AxisOrder order = axis_order_zyx)
-{
- typedef vector< E,fixed<3> > vector_type;
- typedef typename vector_type::value_type value_type;
-
- /* Checking handled by cross() and assignment to fixed<3>. */
-
- size_t i, j, k;
- bool odd;
- detail::unpack_axis_order(order, i, j, k, odd);
-
- vector_type axis[3];
-
- axis[i] = normalize_align ? normalize(align) : align;
- axis[k] = unit_cross(axis[i],reference);
- axis[j] = cross(axis[k],axis[i]);
-
- if (odd) {
- axis[k] = -axis[k];
- }
-
- x = axis[0];
- y = axis[1];
- z = axis[2];
-}
-
-/* This version of orthonormal_basis() constructs in arbitrary basis given a
- * vector with which to align the i'th basis vector. To avoid the failure
- * case, the reference vector is always chosen so as to not be parallel to
- * 'align'. This means the algorithm will always generate a valid basis, which
- * can be useful in some circumstances; however, it should be noted that the
- * basis will likely 'pop' as the alignment vector changes, and so may not be
- * suitable for billboarding or other similar applications.
- */
-
-template < class VecT, typename E, class A >
-void orthonormal_basis(
- const VecT& align,
- vector<E,A>& x,
- vector<E,A>& y,
- vector<E,A>& z,
- bool normalize_align = true,
- AxisOrder order = axis_order_zyx)
-{
- /* Checking (won't be necessary with index_of_min_abs() member function */
- detail::CheckVec3(align);
-
- /* @todo: vector member function index_of_min_abs() would clean this up */
-
- orthonormal_basis(
- align,
- axis_3D(cml::index_of_min_abs(align[0],align[1],align[2])),
- x, y, z, normalize_align, order
- );
-}
-
-/* orthonormal_basis_axial() generates a basis in which the j'th basis vector
- * is aligned with 'axis' and the i'th basis vector is maximally aligned (as
- * 'aligned as possible') with 'align'. This can be used for e.g. axial
- * billboarding for, say, trees or beam effects.
- *
- * Note that the implementation simply passes off to the 'reference' version
- * of orthonormal_basis(), with the parameters adjusted so that the alignment
- * is axial.
- *
- * With this algorithm the failure case is when 'align' and 'axis' are nearly
- * parallel; if this is likely, it should be checked for and handled
- * externally.
- */
-
-template < class VecT_1, class VecT_2, typename E, class A >
-void orthonormal_basis_axial(
- const VecT_1& align,
- const VecT_2& axis,
- vector<E,A>& x,
- vector<E,A>& y,
- vector<E,A>& z,
- bool normalize_align = true,
- AxisOrder order = axis_order_zyx)
-{
- orthonormal_basis(
- axis,
- align,
- x,
- y,
- z,
- normalize_align,
- detail::swap_axis_order(order));
-}
-
-/* orthonormal_basis_viewplane() builds a basis aligned with a viewplane, as
- * extracted from the input view matrix. The function takes into account the
- * handedness of the input view matrix and orients the basis accordingly.
- *
- * The generated basis will always be valid.
- */
-template < class MatT, typename E, class A >
-void orthonormal_basis_viewplane(
- const MatT& view_matrix,
- vector<E,A>& x,
- vector<E,A>& y,
- vector<E,A>& z,
- Handedness handedness,
- AxisOrder order = axis_order_zyx)
-{
- typedef MatT matrix_type;
- typedef typename matrix_type::value_type value_type;
-
- orthonormal_basis(
- -(handedness == left_handed ? value_type(1) : value_type(-1)) *
- matrix_get_transposed_z_basis_vector(view_matrix),
- matrix_get_transposed_y_basis_vector(view_matrix),
- x, y, z, false, order
- );
-}
-
-/** Build a viewplane-oriented basis from a left-handedness view matrix. */
-template < class MatT, typename E, class A >
-void orthonormal_basis_viewplane_LH(
- const MatT& view_matrix,
- vector<E,A>& x,
- vector<E,A>& y,
- vector<E,A>& z,
- AxisOrder order = axis_order_zyx)
-{
- orthonormal_basis_viewplane(
- view_matrix,x,y,z,left_handed,order);
-}
-
-/** Build a viewplane-oriented basis from a right-handedness view matrix. */
-template < class MatT, typename E, class A >
-void orthonormal_basis_viewplane_RH(
- const MatT& view_matrix,
- vector<E,A>& x,
- vector<E,A>& y,
- vector<E,A>& z,
- AxisOrder order = axis_order_zyx)
-{
- orthonormal_basis_viewplane(
- view_matrix,x,y,z,right_handed,order);
-}
-
-/* Build a 2D orthonormal basis. */
-template < class VecT, typename E, class A >
-void orthonormal_basis_2D(
- const VecT& align,
- vector<E,A>& x,
- vector<E,A>& y,
- bool normalize_align = true,
- AxisOrder2D order = axis_order_xy)
-{
- typedef vector< E,fixed<2> > vector_type;
-
- /* Checking handled by perp() and assignment to fixed<2>. */
-
- size_t i, j;
- bool odd;
- detail::unpack_axis_order_2D(order, i, j, odd);
-
- vector_type axis[2];
-
- axis[i] = normalize_align ? normalize(align) : align;
- axis[j] = perp(axis[i]);
-
- if (odd) {
- axis[j] = -axis[j];
- }
-
- x = axis[0];
- y = axis[1];
-}
-
-} // namespace cml
-
-#endif
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