/* -*- 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 #include /* 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& v0, vector& v1, vector& 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& v0, vector& 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_', 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& x, vector& y, vector& 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& x, vector& y, vector& 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& x, vector& y, vector& 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& x, vector& y, vector& 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& x, vector& y, vector& 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& x, vector& y, vector& 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& x, vector& 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