625 lines
18 KiB
C++
625 lines
18 KiB
C++
/*************************************************************************/
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/* matrix3.cpp */
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/*************************************************************************/
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/* This file is part of: */
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/* GODOT ENGINE */
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/* http://www.godotengine.org */
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/*************************************************************************/
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/* Copyright (c) 2007-2017 Juan Linietsky, Ariel Manzur. */
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/* */
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/* Permission is hereby granted, free of charge, to any person obtaining */
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/* a copy of this software and associated documentation files (the */
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/* "Software"), to deal in the Software without restriction, including */
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/* without limitation the rights to use, copy, modify, merge, publish, */
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/* distribute, sublicense, and/or sell copies of the Software, and to */
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/* permit persons to whom the Software is furnished to do so, subject to */
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/* the following conditions: */
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/* */
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/* The above copyright notice and this permission notice shall be */
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/* included in all copies or substantial portions of the Software. */
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/* */
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/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, */
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/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF */
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/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/
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/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY */
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/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, */
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/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE */
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/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */
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/*************************************************************************/
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#include "matrix3.h"
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#include "math_funcs.h"
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#include "os/copymem.h"
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#define cofac(row1,col1, row2, col2)\
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(elements[row1][col1] * elements[row2][col2] - elements[row1][col2] * elements[row2][col1])
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void Matrix3::from_z(const Vector3& p_z) {
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if (Math::abs(p_z.z) > Math_SQRT12 ) {
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// choose p in y-z plane
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real_t a = p_z[1]*p_z[1] + p_z[2]*p_z[2];
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real_t k = 1.0/Math::sqrt(a);
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elements[0]=Vector3(0,-p_z[2]*k,p_z[1]*k);
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elements[1]=Vector3(a*k,-p_z[0]*elements[0][2],p_z[0]*elements[0][1]);
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} else {
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// choose p in x-y plane
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real_t a = p_z.x*p_z.x + p_z.y*p_z.y;
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real_t k = 1.0/Math::sqrt(a);
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elements[0]=Vector3(-p_z.y*k,p_z.x*k,0);
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elements[1]=Vector3(-p_z.z*elements[0].y,p_z.z*elements[0].x,a*k);
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}
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elements[2]=p_z;
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}
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void Matrix3::invert() {
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real_t co[3]={
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cofac(1, 1, 2, 2), cofac(1, 2, 2, 0), cofac(1, 0, 2, 1)
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};
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real_t det = elements[0][0] * co[0]+
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elements[0][1] * co[1]+
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elements[0][2] * co[2];
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ERR_FAIL_COND( det == 0 );
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real_t s = 1.0/det;
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set( co[0]*s, cofac(0, 2, 2, 1) * s, cofac(0, 1, 1, 2) * s,
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co[1]*s, cofac(0, 0, 2, 2) * s, cofac(0, 2, 1, 0) * s,
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co[2]*s, cofac(0, 1, 2, 0) * s, cofac(0, 0, 1, 1) * s );
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}
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void Matrix3::orthonormalize() {
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ERR_FAIL_COND(determinant() == 0);
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// Gram-Schmidt Process
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Vector3 x=get_axis(0);
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Vector3 y=get_axis(1);
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Vector3 z=get_axis(2);
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x.normalize();
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y = (y-x*(x.dot(y)));
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y.normalize();
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z = (z-x*(x.dot(z))-y*(y.dot(z)));
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z.normalize();
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set_axis(0,x);
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set_axis(1,y);
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set_axis(2,z);
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}
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Matrix3 Matrix3::orthonormalized() const {
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Matrix3 c = *this;
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c.orthonormalize();
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return c;
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}
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bool Matrix3::is_orthogonal() const {
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Matrix3 id;
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Matrix3 m = (*this)*transposed();
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return isequal_approx(id,m);
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}
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bool Matrix3::is_rotation() const {
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return Math::isequal_approx(determinant(), 1) && is_orthogonal();
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}
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bool Matrix3::is_symmetric() const {
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if (Math::abs(elements[0][1] - elements[1][0]) > CMP_EPSILON)
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return false;
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if (Math::abs(elements[0][2] - elements[2][0]) > CMP_EPSILON)
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return false;
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if (Math::abs(elements[1][2] - elements[2][1]) > CMP_EPSILON)
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return false;
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return true;
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}
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Matrix3 Matrix3::diagonalize() {
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//NOTE: only implemented for symmetric matrices
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//with the Jacobi iterative method method
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ERR_FAIL_COND_V(!is_symmetric(), Matrix3());
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const int ite_max = 1024;
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real_t off_matrix_norm_2 = elements[0][1] * elements[0][1] + elements[0][2] * elements[0][2] + elements[1][2] * elements[1][2];
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int ite = 0;
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Matrix3 acc_rot;
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while (off_matrix_norm_2 > CMP_EPSILON2 && ite++ < ite_max ) {
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real_t el01_2 = elements[0][1] * elements[0][1];
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real_t el02_2 = elements[0][2] * elements[0][2];
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real_t el12_2 = elements[1][2] * elements[1][2];
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// Find the pivot element
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int i, j;
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if (el01_2 > el02_2) {
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if (el12_2 > el01_2) {
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i = 1;
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j = 2;
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} else {
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i = 0;
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j = 1;
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}
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} else {
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if (el12_2 > el02_2) {
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i = 1;
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j = 2;
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} else {
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i = 0;
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j = 2;
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}
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}
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// Compute the rotation angle
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real_t angle;
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if (Math::abs(elements[j][j] - elements[i][i]) < CMP_EPSILON) {
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angle = Math_PI / 4;
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} else {
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angle = 0.5 * Math::atan(2 * elements[i][j] / (elements[j][j] - elements[i][i]));
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}
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// Compute the rotation matrix
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Matrix3 rot;
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rot.elements[i][i] = rot.elements[j][j] = Math::cos(angle);
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rot.elements[i][j] = - (rot.elements[j][i] = Math::sin(angle));
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// Update the off matrix norm
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off_matrix_norm_2 -= elements[i][j] * elements[i][j];
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// Apply the rotation
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*this = rot * *this * rot.transposed();
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acc_rot = rot * acc_rot;
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}
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return acc_rot;
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}
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Matrix3 Matrix3::inverse() const {
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Matrix3 inv=*this;
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inv.invert();
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return inv;
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}
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void Matrix3::transpose() {
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SWAP(elements[0][1],elements[1][0]);
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SWAP(elements[0][2],elements[2][0]);
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SWAP(elements[1][2],elements[2][1]);
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}
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Matrix3 Matrix3::transposed() const {
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Matrix3 tr=*this;
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tr.transpose();
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return tr;
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}
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// Multiplies the matrix from left by the scaling matrix: M -> S.M
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// See the comment for Matrix3::rotated for further explanation.
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void Matrix3::scale(const Vector3& p_scale) {
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elements[0][0]*=p_scale.x;
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elements[0][1]*=p_scale.x;
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elements[0][2]*=p_scale.x;
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elements[1][0]*=p_scale.y;
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elements[1][1]*=p_scale.y;
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elements[1][2]*=p_scale.y;
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elements[2][0]*=p_scale.z;
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elements[2][1]*=p_scale.z;
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elements[2][2]*=p_scale.z;
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}
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Matrix3 Matrix3::scaled( const Vector3& p_scale ) const {
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Matrix3 m = *this;
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m.scale(p_scale);
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return m;
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}
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Vector3 Matrix3::get_scale() const {
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// We are assuming M = R.S, and performing a polar decomposition to extract R and S.
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// FIXME: We eventually need a proper polar decomposition.
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// As a cheap workaround until then, to ensure that R is a proper rotation matrix with determinant +1
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// (such that it can be represented by a Quat or Euler angles), we absorb the sign flip into the scaling matrix.
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// As such, it works in conjuction with get_rotation().
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real_t det_sign = determinant() > 0 ? 1 : -1;
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return det_sign*Vector3(
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Vector3(elements[0][0],elements[1][0],elements[2][0]).length(),
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Vector3(elements[0][1],elements[1][1],elements[2][1]).length(),
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Vector3(elements[0][2],elements[1][2],elements[2][2]).length()
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);
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}
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// Multiplies the matrix from left by the rotation matrix: M -> R.M
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// Note that this does *not* rotate the matrix itself.
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//
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// The main use of Matrix3 is as Transform.basis, which is used a the transformation matrix
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// of 3D object. Rotate here refers to rotation of the object (which is R * (*this)),
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// not the matrix itself (which is R * (*this) * R.transposed()).
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Matrix3 Matrix3::rotated(const Vector3& p_axis, real_t p_phi) const {
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return Matrix3(p_axis, p_phi) * (*this);
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}
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void Matrix3::rotate(const Vector3& p_axis, real_t p_phi) {
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*this = rotated(p_axis, p_phi);
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}
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Matrix3 Matrix3::rotated(const Vector3& p_euler) const {
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return Matrix3(p_euler) * (*this);
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}
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void Matrix3::rotate(const Vector3& p_euler) {
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*this = rotated(p_euler);
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}
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Vector3 Matrix3::get_rotation() const {
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// Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
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// and returns the Euler angles corresponding to the rotation part, complementing get_scale().
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// See the comment in get_scale() for further information.
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Matrix3 m = orthonormalized();
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real_t det = m.determinant();
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if (det < 0) {
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// Ensure that the determinant is 1, such that result is a proper rotation matrix which can be represented by Euler angles.
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m.scale(Vector3(-1,-1,-1));
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}
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return m.get_euler();
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}
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// get_euler returns a vector containing the Euler angles in the format
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// (a1,a2,a3), where a3 is the angle of the first rotation, and a1 is the last
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// (following the convention they are commonly defined in the literature).
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//
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// The current implementation uses XYZ convention (Z is the first rotation),
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// so euler.z is the angle of the (first) rotation around Z axis and so on,
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//
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// And thus, assuming the matrix is a rotation matrix, this function returns
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// the angles in the decomposition R = X(a1).Y(a2).Z(a3) where Z(a) rotates
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// around the z-axis by a and so on.
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Vector3 Matrix3::get_euler() const {
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// Euler angles in XYZ convention.
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// See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
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//
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// rot = cy*cz -cy*sz sy
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// cz*sx*sy+cx*sz cx*cz-sx*sy*sz -cy*sx
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// -cx*cz*sy+sx*sz cz*sx+cx*sy*sz cx*cy
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Vector3 euler;
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ERR_FAIL_COND_V(is_rotation() == false, euler);
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euler.y = Math::asin(elements[0][2]);
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if ( euler.y < Math_PI*0.5) {
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if ( euler.y > -Math_PI*0.5) {
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euler.x = Math::atan2(-elements[1][2],elements[2][2]);
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euler.z = Math::atan2(-elements[0][1],elements[0][0]);
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} else {
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real_t r = Math::atan2(elements[1][0],elements[1][1]);
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euler.z = 0.0;
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euler.x = euler.z - r;
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}
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} else {
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real_t r = Math::atan2(elements[0][1],elements[1][1]);
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euler.z = 0;
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euler.x = r - euler.z;
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}
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return euler;
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}
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// set_euler expects a vector containing the Euler angles in the format
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// (c,b,a), where a is the angle of the first rotation, and c is the last.
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// The current implementation uses XYZ convention (Z is the first rotation).
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void Matrix3::set_euler(const Vector3& p_euler) {
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real_t c, s;
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c = Math::cos(p_euler.x);
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s = Math::sin(p_euler.x);
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Matrix3 xmat(1.0,0.0,0.0,0.0,c,-s,0.0,s,c);
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c = Math::cos(p_euler.y);
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s = Math::sin(p_euler.y);
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Matrix3 ymat(c,0.0,s,0.0,1.0,0.0,-s,0.0,c);
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c = Math::cos(p_euler.z);
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s = Math::sin(p_euler.z);
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Matrix3 zmat(c,-s,0.0,s,c,0.0,0.0,0.0,1.0);
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//optimizer will optimize away all this anyway
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*this = xmat*(ymat*zmat);
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}
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bool Matrix3::isequal_approx(const Matrix3& a, const Matrix3& b) const {
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for (int i=0;i<3;i++) {
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for (int j=0;j<3;j++) {
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if (Math::isequal_approx(a.elements[i][j],b.elements[i][j]) == false)
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return false;
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}
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}
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return true;
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}
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bool Matrix3::operator==(const Matrix3& p_matrix) const {
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for (int i=0;i<3;i++) {
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for (int j=0;j<3;j++) {
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if (elements[i][j] != p_matrix.elements[i][j])
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return false;
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}
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}
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return true;
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}
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bool Matrix3::operator!=(const Matrix3& p_matrix) const {
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return (!(*this==p_matrix));
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}
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Matrix3::operator String() const {
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String mtx;
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for (int i=0;i<3;i++) {
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for (int j=0;j<3;j++) {
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if (i!=0 || j!=0)
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mtx+=", ";
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mtx+=rtos( elements[i][j] );
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}
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}
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return mtx;
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}
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Matrix3::operator Quat() const {
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ERR_FAIL_COND_V(is_rotation() == false, Quat());
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real_t trace = elements[0][0] + elements[1][1] + elements[2][2];
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real_t temp[4];
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if (trace > 0.0)
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{
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real_t s = Math::sqrt(trace + 1.0);
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temp[3]=(s * 0.5);
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s = 0.5 / s;
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temp[0]=((elements[2][1] - elements[1][2]) * s);
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temp[1]=((elements[0][2] - elements[2][0]) * s);
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temp[2]=((elements[1][0] - elements[0][1]) * s);
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}
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else
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{
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int i = elements[0][0] < elements[1][1] ?
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(elements[1][1] < elements[2][2] ? 2 : 1) :
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(elements[0][0] < elements[2][2] ? 2 : 0);
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int j = (i + 1) % 3;
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int k = (i + 2) % 3;
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real_t s = Math::sqrt(elements[i][i] - elements[j][j] - elements[k][k] + 1.0);
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temp[i] = s * 0.5;
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s = 0.5 / s;
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temp[3] = (elements[k][j] - elements[j][k]) * s;
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temp[j] = (elements[j][i] + elements[i][j]) * s;
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temp[k] = (elements[k][i] + elements[i][k]) * s;
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}
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return Quat(temp[0],temp[1],temp[2],temp[3]);
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}
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static const Matrix3 _ortho_bases[24]={
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Matrix3(1, 0, 0, 0, 1, 0, 0, 0, 1),
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Matrix3(0, -1, 0, 1, 0, 0, 0, 0, 1),
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Matrix3(-1, 0, 0, 0, -1, 0, 0, 0, 1),
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Matrix3(0, 1, 0, -1, 0, 0, 0, 0, 1),
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Matrix3(1, 0, 0, 0, 0, -1, 0, 1, 0),
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Matrix3(0, 0, 1, 1, 0, 0, 0, 1, 0),
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Matrix3(-1, 0, 0, 0, 0, 1, 0, 1, 0),
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Matrix3(0, 0, -1, -1, 0, 0, 0, 1, 0),
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Matrix3(1, 0, 0, 0, -1, 0, 0, 0, -1),
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Matrix3(0, 1, 0, 1, 0, 0, 0, 0, -1),
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Matrix3(-1, 0, 0, 0, 1, 0, 0, 0, -1),
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Matrix3(0, -1, 0, -1, 0, 0, 0, 0, -1),
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Matrix3(1, 0, 0, 0, 0, 1, 0, -1, 0),
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Matrix3(0, 0, -1, 1, 0, 0, 0, -1, 0),
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Matrix3(-1, 0, 0, 0, 0, -1, 0, -1, 0),
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Matrix3(0, 0, 1, -1, 0, 0, 0, -1, 0),
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Matrix3(0, 0, 1, 0, 1, 0, -1, 0, 0),
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Matrix3(0, -1, 0, 0, 0, 1, -1, 0, 0),
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Matrix3(0, 0, -1, 0, -1, 0, -1, 0, 0),
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Matrix3(0, 1, 0, 0, 0, -1, -1, 0, 0),
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Matrix3(0, 0, 1, 0, -1, 0, 1, 0, 0),
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Matrix3(0, 1, 0, 0, 0, 1, 1, 0, 0),
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Matrix3(0, 0, -1, 0, 1, 0, 1, 0, 0),
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Matrix3(0, -1, 0, 0, 0, -1, 1, 0, 0)
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};
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int Matrix3::get_orthogonal_index() const {
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//could be sped up if i come up with a way
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Matrix3 orth=*this;
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for(int i=0;i<3;i++) {
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for(int j=0;j<3;j++) {
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real_t v = orth[i][j];
|
|
if (v>0.5)
|
|
v=1.0;
|
|
else if (v<-0.5)
|
|
v=-1.0;
|
|
else
|
|
v=0;
|
|
|
|
orth[i][j]=v;
|
|
}
|
|
}
|
|
|
|
for(int i=0;i<24;i++) {
|
|
|
|
if (_ortho_bases[i]==orth)
|
|
return i;
|
|
|
|
|
|
}
|
|
|
|
return 0;
|
|
}
|
|
|
|
void Matrix3::set_orthogonal_index(int p_index){
|
|
|
|
//there only exist 24 orthogonal bases in r3
|
|
ERR_FAIL_INDEX(p_index,24);
|
|
|
|
|
|
*this=_ortho_bases[p_index];
|
|
|
|
}
|
|
|
|
|
|
void Matrix3::get_axis_and_angle(Vector3 &r_axis,real_t& r_angle) const {
|
|
ERR_FAIL_COND(is_rotation() == false);
|
|
|
|
|
|
double angle,x,y,z; // variables for result
|
|
double epsilon = 0.01; // margin to allow for rounding errors
|
|
double epsilon2 = 0.1; // margin to distinguish between 0 and 180 degrees
|
|
|
|
if ( (Math::abs(elements[1][0]-elements[0][1])< epsilon)
|
|
&& (Math::abs(elements[2][0]-elements[0][2])< epsilon)
|
|
&& (Math::abs(elements[2][1]-elements[1][2])< epsilon)) {
|
|
// singularity found
|
|
// first check for identity matrix which must have +1 for all terms
|
|
// in leading diagonaland zero in other terms
|
|
if ((Math::abs(elements[1][0]+elements[0][1]) < epsilon2)
|
|
&& (Math::abs(elements[2][0]+elements[0][2]) < epsilon2)
|
|
&& (Math::abs(elements[2][1]+elements[1][2]) < epsilon2)
|
|
&& (Math::abs(elements[0][0]+elements[1][1]+elements[2][2]-3) < epsilon2)) {
|
|
// this singularity is identity matrix so angle = 0
|
|
r_axis=Vector3(0,1,0);
|
|
r_angle=0;
|
|
return;
|
|
}
|
|
// otherwise this singularity is angle = 180
|
|
angle = Math_PI;
|
|
double xx = (elements[0][0]+1)/2;
|
|
double yy = (elements[1][1]+1)/2;
|
|
double zz = (elements[2][2]+1)/2;
|
|
double xy = (elements[1][0]+elements[0][1])/4;
|
|
double xz = (elements[2][0]+elements[0][2])/4;
|
|
double yz = (elements[2][1]+elements[1][2])/4;
|
|
if ((xx > yy) && (xx > zz)) { // elements[0][0] is the largest diagonal term
|
|
if (xx< epsilon) {
|
|
x = 0;
|
|
y = 0.7071;
|
|
z = 0.7071;
|
|
} else {
|
|
x = Math::sqrt(xx);
|
|
y = xy/x;
|
|
z = xz/x;
|
|
}
|
|
} else if (yy > zz) { // elements[1][1] is the largest diagonal term
|
|
if (yy< epsilon) {
|
|
x = 0.7071;
|
|
y = 0;
|
|
z = 0.7071;
|
|
} else {
|
|
y = Math::sqrt(yy);
|
|
x = xy/y;
|
|
z = yz/y;
|
|
}
|
|
} else { // elements[2][2] is the largest diagonal term so base result on this
|
|
if (zz< epsilon) {
|
|
x = 0.7071;
|
|
y = 0.7071;
|
|
z = 0;
|
|
} else {
|
|
z = Math::sqrt(zz);
|
|
x = xz/z;
|
|
y = yz/z;
|
|
}
|
|
}
|
|
r_axis=Vector3(x,y,z);
|
|
r_angle=angle;
|
|
return;
|
|
}
|
|
// as we have reached here there are no singularities so we can handle normally
|
|
double s = Math::sqrt((elements[1][2] - elements[2][1])*(elements[1][2] - elements[2][1])
|
|
+(elements[2][0] - elements[0][2])*(elements[2][0] - elements[0][2])
|
|
+(elements[0][1] - elements[1][0])*(elements[0][1] - elements[1][0])); // s=|axis||sin(angle)|, used to normalise
|
|
|
|
angle = Math::acos(( elements[0][0] + elements[1][1] + elements[2][2] - 1)/2);
|
|
if (angle < 0) s = -s;
|
|
x = (elements[2][1] - elements[1][2])/s;
|
|
y = (elements[0][2] - elements[2][0])/s;
|
|
z = (elements[1][0] - elements[0][1])/s;
|
|
|
|
r_axis=Vector3(x,y,z);
|
|
r_angle=angle;
|
|
}
|
|
|
|
Matrix3::Matrix3(const Vector3& p_euler) {
|
|
|
|
set_euler( p_euler );
|
|
|
|
}
|
|
|
|
Matrix3::Matrix3(const Quat& p_quat) {
|
|
|
|
real_t d = p_quat.length_squared();
|
|
real_t s = 2.0 / d;
|
|
real_t xs = p_quat.x * s, ys = p_quat.y * s, zs = p_quat.z * s;
|
|
real_t wx = p_quat.w * xs, wy = p_quat.w * ys, wz = p_quat.w * zs;
|
|
real_t xx = p_quat.x * xs, xy = p_quat.x * ys, xz = p_quat.x * zs;
|
|
real_t yy = p_quat.y * ys, yz = p_quat.y * zs, zz = p_quat.z * zs;
|
|
set( 1.0 - (yy + zz), xy - wz, xz + wy,
|
|
xy + wz, 1.0 - (xx + zz), yz - wx,
|
|
xz - wy, yz + wx, 1.0 - (xx + yy)) ;
|
|
|
|
}
|
|
|
|
Matrix3::Matrix3(const Vector3& p_axis, real_t p_phi) {
|
|
// Rotation matrix from axis and angle, see https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
|
|
|
|
Vector3 axis_sq(p_axis.x*p_axis.x,p_axis.y*p_axis.y,p_axis.z*p_axis.z);
|
|
|
|
real_t cosine= Math::cos(p_phi);
|
|
real_t sine= Math::sin(p_phi);
|
|
|
|
elements[0][0] = axis_sq.x + cosine * ( 1.0 - axis_sq.x );
|
|
elements[0][1] = p_axis.x * p_axis.y * ( 1.0 - cosine ) - p_axis.z * sine;
|
|
elements[0][2] = p_axis.z * p_axis.x * ( 1.0 - cosine ) + p_axis.y * sine;
|
|
|
|
elements[1][0] = p_axis.x * p_axis.y * ( 1.0 - cosine ) + p_axis.z * sine;
|
|
elements[1][1] = axis_sq.y + cosine * ( 1.0 - axis_sq.y );
|
|
elements[1][2] = p_axis.y * p_axis.z * ( 1.0 - cosine ) - p_axis.x * sine;
|
|
|
|
elements[2][0] = p_axis.z * p_axis.x * ( 1.0 - cosine ) - p_axis.y * sine;
|
|
elements[2][1] = p_axis.y * p_axis.z * ( 1.0 - cosine ) + p_axis.x * sine;
|
|
elements[2][2] = axis_sq.z + cosine * ( 1.0 - axis_sq.z );
|
|
|
|
}
|
|
|