--- /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
projects / chaz / yoink / blobdiff