339 lines
11 KiB
C++
339 lines
11 KiB
C++
/* -*- C++ -*- ------------------------------------------------------------
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Copyright (c) 2007 Jesse Anders and Demian Nave http://cmldev.net/
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The Configurable Math Library (CML) is distributed under the terms of the
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Boost Software License, v1.0 (see cml/LICENSE for details).
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*-----------------------------------------------------------------------*/
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/** @file
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*
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* Functions for orthonormalizing a set of basis vectors in 3D or 2D, and for
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* constructing an orthonormal basis given various input parameters.
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*/
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#ifndef vector_ortho_h
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#define vector_ortho_h
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#include <cml/mathlib/vector_misc.h>
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#include <cml/mathlib/misc.h>
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namespace cml {
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//////////////////////////////////////////////////////////////////////////////
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// Orthonormalization in 3D and 2D
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//////////////////////////////////////////////////////////////////////////////
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/** Orthonormalize 3 basis vectors in R3.
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*
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* Called with the default values, this function performs a single Gram-
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* Schmidt step to orthonormalize the input vectors. By default, the direction
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* of the 3rd basis vector is unchanged by this operation, but the unaffected
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* axis can be specified via the 'stable_axis' parameter.
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*
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* The arguments 'num_iter' and 's' can be specified to an iterative Gram-
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* Schmidt step. 'num_iter' is the number of iterations applied, and 's' is
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* the fraction applied towards orthonormality each step.
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*
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* In most cases, the default arguments can be ignored, leaving only the three
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* input vectors.
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*/
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template < typename E, class A > void
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orthonormalize(vector<E,A>& v0, vector<E,A>& v1, vector<E,A>& v2,
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size_t stable_axis = 2, size_t num_iter = 0, E s = E(1))
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{
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/* Checking */
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detail::CheckVec3(v0);
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detail::CheckVec3(v1);
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detail::CheckVec3(v2);
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detail::CheckIndex3(stable_axis);
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typedef vector< E, fixed<3> > vector_type;
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typedef typename vector_type::value_type value_type;
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/* Iterative Gram-Schmidt; this step is skipped by default. */
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for (size_t i = 0; i < num_iter; ++i) {
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value_type dot01 = dot(v0,v1);
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value_type dot12 = dot(v1,v2);
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value_type dot20 = dot(v2,v0);
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value_type inv_dot00 = value_type(1) / dot(v0,v0);
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value_type inv_dot11 = value_type(1) / dot(v1,v1);
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value_type inv_dot22 = value_type(1) / dot(v2,v2);
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vector_type temp0 = v0 - s*dot01*inv_dot11*v1 - s*dot20*inv_dot22*v2;
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vector_type temp1 = v1 - s*dot12*inv_dot22*v2 - s*dot01*inv_dot00*v0;
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vector_type temp2 = v2 - s*dot20*inv_dot00*v0 - s*dot12*inv_dot11*v1;
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v0 = temp0;
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v1 = temp1;
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v2 = temp2;
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}
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/* Final Gram-Schmidt step to ensure orthonormality. If no iterations
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* have been requested (num_iter = 0), this is the only step. The step
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* is performed such that the direction of the axis indexed by
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* 'stable_axis' is unchanged.
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*/
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size_t i, j, k;
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cyclic_permutation(stable_axis, i, j, k);
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vector_type v[] = { v0, v1, v2 };
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v[i].normalize();
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v[j] = normalize(project_to_hplane(v[j],v[i]));
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v[k] = normalize(project_to_hplane(project_to_hplane(v[k],v[i]),v[j]));
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v0 = v[0];
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v1 = v[1];
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v2 = v[2];
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}
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/** Orthonormalize 2 basis vectors in R2 */
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template < typename E, class A > void
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orthonormalize(vector<E,A>& v0, vector<E,A>& v1,
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size_t stable_axis = 0, size_t num_iter = 0, E s = E(1))
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{
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typedef vector< E, fixed<2> > vector_type;
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typedef typename vector_type::value_type value_type;
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/* Checking */
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detail::CheckVec2(v0);
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detail::CheckVec2(v1);
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detail::CheckIndex2(stable_axis);
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/* Iterative Gram-Schmidt; this step is skipped by default. */
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for (size_t i = 0; i < num_iter; ++i) {
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value_type dot01 = dot(v0,v1);
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vector_type temp0 = v0 - (s * dot01 * v1) / dot(v1,v1);
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vector_type temp1 = v1 - (s * dot01 * v0) / dot(v0,v0);
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v0 = temp0;
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v1 = temp1;
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}
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/* Final Gram-Schmidt step to ensure orthonormality. If no iterations
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* have been requested (num_iter = 0), this is the only step. The step
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* is performed such that the direction of the axis indexed by
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* 'stable_axis' is unchanged.
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*/
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size_t i, j;
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cyclic_permutation(stable_axis, i, j);
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vector_type v[] = { v0, v1 };
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v[i].normalize();
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v[j] = normalize(project_to_hplane(v[j],v[i]));
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v0 = v[0];
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v1 = v[1];
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}
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//////////////////////////////////////////////////////////////////////////////
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// Orthonormal basis construction in 3D and 2D
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//////////////////////////////////////////////////////////////////////////////
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/** This version of orthonormal_basis() ultimately does the work for all
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* orthonormal_basis_*() functions. Given input vectors 'align' and
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* 'reference', and an order 'axis_order_\<i\>\<j\>\<k\>', it constructs an
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* orthonormal basis such that the i'th basis vector is aligned with (parallel
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* to and pointing in the same direction as) 'align', and the j'th basis
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* vector is maximally aligned with 'reference'. The k'th basis vector is
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* chosen such that the basis has a determinant of +1.
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*
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* @note The algorithm fails when 'align' is nearly parallel to
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* 'reference'; this should be checked for and handled externally if it's a
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* case that may occur.
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*
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* @internal This is an example of the 'non-const argument modification
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* invalidates expression' gotcha. If x, y or z were to be assigned to before
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* we were 'done' with align and reference, and if one of them were the same
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* object as align or reference, then the algorithm could fail. As is the
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* basis vectors are assigned at the end of the function from a temporary
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* array, so all is well.
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*/
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template < class VecT_1, class VecT_2, typename E, class A > void
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orthonormal_basis(
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const VecT_1& align,
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const VecT_2& reference,
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vector<E,A>& x,
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vector<E,A>& y,
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vector<E,A>& z,
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bool normalize_align = true,
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AxisOrder order = axis_order_zyx)
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{
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typedef vector< E,fixed<3> > vector_type;
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typedef typename vector_type::value_type value_type;
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/* Checking handled by cross() and assignment to fixed<3>. */
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size_t i, j, k;
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bool odd;
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detail::unpack_axis_order(order, i, j, k, odd);
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vector_type axis[3];
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axis[i] = normalize_align ? normalize(align) : align;
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axis[k] = unit_cross(axis[i],reference);
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axis[j] = cross(axis[k],axis[i]);
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if (odd) {
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axis[k] = -axis[k];
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}
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x = axis[0];
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y = axis[1];
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z = axis[2];
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}
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/** This version of orthonormal_basis() constructs in arbitrary basis given a
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* vector with which to align the i'th basis vector. To avoid the failure
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* case, the reference vector is always chosen so as to not be parallel to
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* 'align'. This means the algorithm will always generate a valid basis, which
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* can be useful in some circumstances; however, it should be noted that the
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* basis will likely 'pop' as the alignment vector changes, and so may not be
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* suitable for billboarding or other similar applications.
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*/
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template < class VecT, typename E, class A >
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void orthonormal_basis(
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const VecT& align,
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vector<E,A>& x,
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vector<E,A>& y,
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vector<E,A>& z,
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bool normalize_align = true,
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AxisOrder order = axis_order_zyx)
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{
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/* Checking (won't be necessary with index_of_min_abs() member function */
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detail::CheckVec3(align);
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/* @todo: vector member function index_of_min_abs() would clean this up */
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orthonormal_basis(
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align,
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axis_3D(cml::index_of_min_abs(align[0],align[1],align[2])),
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x, y, z, normalize_align, order
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);
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}
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/** orthonormal_basis_axial() generates a basis in which the j'th basis vector
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* is aligned with 'axis' and the i'th basis vector is maximally aligned (as
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* 'aligned as possible') with 'align'. This can be used for e.g. axial
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* billboarding for, say, trees or beam effects.
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*
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* Note that the implementation simply passes off to the 'reference' version
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* of orthonormal_basis(), with the parameters adjusted so that the alignment
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* is axial.
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*
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* @note With this algorithm the failure case is when 'align' and 'axis'
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* are nearly parallel; if this is likely, it should be checked for and
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* handled externally.
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*/
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template < class VecT_1, class VecT_2, typename E, class A >
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void orthonormal_basis_axial(
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const VecT_1& align,
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const VecT_2& axis,
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vector<E,A>& x,
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vector<E,A>& y,
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vector<E,A>& z,
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bool normalize_align = true,
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AxisOrder order = axis_order_zyx)
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{
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orthonormal_basis(
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axis,
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align,
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x,
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y,
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z,
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normalize_align,
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detail::swap_axis_order(order));
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}
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/** orthonormal_basis_viewplane() builds a basis aligned with a viewplane, as
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* extracted from the input view matrix. The function takes into account the
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* handedness of the input view matrix and orients the basis accordingly.
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*
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* @note The generated basis will always be valid.
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*/
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template < class MatT, typename E, class A >
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void orthonormal_basis_viewplane(
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const MatT& view_matrix,
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vector<E,A>& x,
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vector<E,A>& y,
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vector<E,A>& z,
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Handedness handedness,
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AxisOrder order = axis_order_zyx)
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{
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typedef MatT matrix_type;
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typedef typename matrix_type::value_type value_type;
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orthonormal_basis(
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-(handedness == left_handed ? value_type(1) : value_type(-1)) *
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matrix_get_transposed_z_basis_vector(view_matrix),
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matrix_get_transposed_y_basis_vector(view_matrix),
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x, y, z, false, order
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);
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}
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/** Build a viewplane-oriented basis from a left-handedness view matrix. */
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template < class MatT, typename E, class A >
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void orthonormal_basis_viewplane_LH(
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const MatT& view_matrix,
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vector<E,A>& x,
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vector<E,A>& y,
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vector<E,A>& z,
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AxisOrder order = axis_order_zyx)
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{
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orthonormal_basis_viewplane(
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view_matrix,x,y,z,left_handed,order);
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}
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/** Build a viewplane-oriented basis from a right-handedness view matrix. */
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template < class MatT, typename E, class A >
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void orthonormal_basis_viewplane_RH(
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const MatT& view_matrix,
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vector<E,A>& x,
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vector<E,A>& y,
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vector<E,A>& z,
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AxisOrder order = axis_order_zyx)
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{
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orthonormal_basis_viewplane(
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view_matrix,x,y,z,right_handed,order);
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}
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/** Build a 2D orthonormal basis. */
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template < class VecT, typename E, class A >
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void orthonormal_basis_2D(
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const VecT& align,
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vector<E,A>& x,
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vector<E,A>& y,
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bool normalize_align = true,
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AxisOrder2D order = axis_order_xy)
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{
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typedef vector< E,fixed<2> > vector_type;
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/* Checking handled by perp() and assignment to fixed<2>. */
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size_t i, j;
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bool odd;
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detail::unpack_axis_order_2D(order, i, j, odd);
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vector_type axis[2];
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axis[i] = normalize_align ? normalize(align) : align;
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axis[j] = perp(axis[i]);
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if (odd) {
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axis[j] = -axis[j];
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}
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x = axis[0];
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y = axis[1];
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}
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} // namespace cml
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#endif
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