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+/*************************************************************************/
+/* basis.cpp */
+/*************************************************************************/
+/* This file is part of: */
+/* GODOT ENGINE */
+/* https://godotengine.org */
+/*************************************************************************/
+/* Copyright (c) 2007-2019 Juan Linietsky, Ariel Manzur. */
+/* Copyright (c) 2014-2019 Godot Engine contributors (cf. AUTHORS.md) */
+/* */
+/* Permission is hereby granted, free of charge, to any person obtaining */
+/* a copy of this software and associated documentation files (the */
+/* "Software"), to deal in the Software without restriction, including */
+/* without limitation the rights to use, copy, modify, merge, publish, */
+/* distribute, sublicense, and/or sell copies of the Software, and to */
+/* permit persons to whom the Software is furnished to do so, subject to */
+/* the following conditions: */
+/* */
+/* The above copyright notice and this permission notice shall be */
+/* included in all copies or substantial portions of the Software. */
+/* */
+/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, */
+/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF */
+/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/
+/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY */
+/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, */
+/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE */
+/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */
+/*************************************************************************/
+
+#include "basis.h"
+
+#include "core/math/math_funcs.h"
+#include "core/os/copymem.h"
+#include "core/print_string.h"
+
+#define cofac(row1, col1, row2, col2) \
+ (elements[row1][col1] * elements[row2][col2] - elements[row1][col2] * elements[row2][col1])
+
+void Basis::from_z(const Vector3 &p_z) {
+
+ if (Math::abs(p_z.z) > Math_SQRT12) {
+
+ // choose p in y-z plane
+ real_t a = p_z[1] * p_z[1] + p_z[2] * p_z[2];
+ real_t k = 1.0 / Math::sqrt(a);
+ elements[0] = Vector3(0, -p_z[2] * k, p_z[1] * k);
+ elements[1] = Vector3(a * k, -p_z[0] * elements[0][2], p_z[0] * elements[0][1]);
+ } else {
+
+ // choose p in x-y plane
+ real_t a = p_z.x * p_z.x + p_z.y * p_z.y;
+ real_t k = 1.0 / Math::sqrt(a);
+ elements[0] = Vector3(-p_z.y * k, p_z.x * k, 0);
+ elements[1] = Vector3(-p_z.z * elements[0].y, p_z.z * elements[0].x, a * k);
+ }
+ elements[2] = p_z;
+}
+
+void Basis::invert() {
+
+ real_t co[3] = {
+ cofac(1, 1, 2, 2), cofac(1, 2, 2, 0), cofac(1, 0, 2, 1)
+ };
+ real_t det = elements[0][0] * co[0] +
+ elements[0][1] * co[1] +
+ elements[0][2] * co[2];
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND(det == 0);
+#endif
+ real_t s = 1.0 / det;
+
+ set(co[0] * s, cofac(0, 2, 2, 1) * s, cofac(0, 1, 1, 2) * s,
+ co[1] * s, cofac(0, 0, 2, 2) * s, cofac(0, 2, 1, 0) * s,
+ co[2] * s, cofac(0, 1, 2, 0) * s, cofac(0, 0, 1, 1) * s);
+}
+
+void Basis::orthonormalize() {
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND(determinant() == 0);
+#endif
+ // Gram-Schmidt Process
+
+ Vector3 x = get_axis(0);
+ Vector3 y = get_axis(1);
+ Vector3 z = get_axis(2);
+
+ x.normalize();
+ y = (y - x * (x.dot(y)));
+ y.normalize();
+ z = (z - x * (x.dot(z)) - y * (y.dot(z)));
+ z.normalize();
+
+ set_axis(0, x);
+ set_axis(1, y);
+ set_axis(2, z);
+}
+
+Basis Basis::orthonormalized() const {
+
+ Basis c = *this;
+ c.orthonormalize();
+ return c;
+}
+
+bool Basis::is_orthogonal() const {
+ Basis id;
+ Basis m = (*this) * transposed();
+
+ return is_equal_approx(id, m);
+}
+
+bool Basis::is_diagonal() const {
+ return (
+ Math::is_equal_approx(elements[0][1], 0) && Math::is_equal_approx(elements[0][2], 0) &&
+ Math::is_equal_approx(elements[1][0], 0) && Math::is_equal_approx(elements[1][2], 0) &&
+ Math::is_equal_approx(elements[2][0], 0) && Math::is_equal_approx(elements[2][1], 0));
+}
+
+bool Basis::is_rotation() const {
+ return Math::is_equal_approx(determinant(), 1) && is_orthogonal();
+}
+
+bool Basis::is_symmetric() const {
+
+ if (!Math::is_equal_approx(elements[0][1], elements[1][0]))
+ return false;
+ if (!Math::is_equal_approx(elements[0][2], elements[2][0]))
+ return false;
+ if (!Math::is_equal_approx(elements[1][2], elements[2][1]))
+ return false;
+
+ return true;
+}
+
+Basis Basis::diagonalize() {
+
+//NOTE: only implemented for symmetric matrices
+//with the Jacobi iterative method method
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(!is_symmetric(), Basis());
+#endif
+ const int ite_max = 1024;
+
+ real_t off_matrix_norm_2 = elements[0][1] * elements[0][1] + elements[0][2] * elements[0][2] + elements[1][2] * elements[1][2];
+
+ int ite = 0;
+ Basis acc_rot;
+ while (off_matrix_norm_2 > CMP_EPSILON2 && ite++ < ite_max) {
+ real_t el01_2 = elements[0][1] * elements[0][1];
+ real_t el02_2 = elements[0][2] * elements[0][2];
+ real_t el12_2 = elements[1][2] * elements[1][2];
+ // Find the pivot element
+ int i, j;
+ if (el01_2 > el02_2) {
+ if (el12_2 > el01_2) {
+ i = 1;
+ j = 2;
+ } else {
+ i = 0;
+ j = 1;
+ }
+ } else {
+ if (el12_2 > el02_2) {
+ i = 1;
+ j = 2;
+ } else {
+ i = 0;
+ j = 2;
+ }
+ }
+
+ // Compute the rotation angle
+ real_t angle;
+ if (Math::is_equal_approx(elements[j][j], elements[i][i])) {
+ angle = Math_PI / 4;
+ } else {
+ angle = 0.5 * Math::atan(2 * elements[i][j] / (elements[j][j] - elements[i][i]));
+ }
+
+ // Compute the rotation matrix
+ Basis rot;
+ rot.elements[i][i] = rot.elements[j][j] = Math::cos(angle);
+ rot.elements[i][j] = -(rot.elements[j][i] = Math::sin(angle));
+
+ // Update the off matrix norm
+ off_matrix_norm_2 -= elements[i][j] * elements[i][j];
+
+ // Apply the rotation
+ *this = rot * *this * rot.transposed();
+ acc_rot = rot * acc_rot;
+ }
+
+ return acc_rot;
+}
+
+Basis Basis::inverse() const {
+
+ Basis inv = *this;
+ inv.invert();
+ return inv;
+}
+
+void Basis::transpose() {
+
+ SWAP(elements[0][1], elements[1][0]);
+ SWAP(elements[0][2], elements[2][0]);
+ SWAP(elements[1][2], elements[2][1]);
+}
+
+Basis Basis::transposed() const {
+
+ Basis tr = *this;
+ tr.transpose();
+ return tr;
+}
+
+// Multiplies the matrix from left by the scaling matrix: M -> S.M
+// See the comment for Basis::rotated for further explanation.
+void Basis::scale(const Vector3 &p_scale) {
+
+ elements[0][0] *= p_scale.x;
+ elements[0][1] *= p_scale.x;
+ elements[0][2] *= p_scale.x;
+ elements[1][0] *= p_scale.y;
+ elements[1][1] *= p_scale.y;
+ elements[1][2] *= p_scale.y;
+ elements[2][0] *= p_scale.z;
+ elements[2][1] *= p_scale.z;
+ elements[2][2] *= p_scale.z;
+}
+
+Basis Basis::scaled(const Vector3 &p_scale) const {
+ Basis m = *this;
+ m.scale(p_scale);
+ return m;
+}
+
+void Basis::scale_local(const Vector3 &p_scale) {
+ // performs a scaling in object-local coordinate system:
+ // M -> (M.S.Minv).M = M.S.
+ *this = scaled_local(p_scale);
+}
+
+Basis Basis::scaled_local(const Vector3 &p_scale) const {
+ Basis b;
+ b.set_diagonal(p_scale);
+
+ return (*this) * b;
+}
+
+Vector3 Basis::get_scale_abs() const {
+
+ return Vector3(
+ Vector3(elements[0][0], elements[1][0], elements[2][0]).length(),
+ Vector3(elements[0][1], elements[1][1], elements[2][1]).length(),
+ Vector3(elements[0][2], elements[1][2], elements[2][2]).length());
+}
+
+Vector3 Basis::get_scale_local() const {
+ real_t det_sign = determinant() > 0 ? 1 : -1;
+ return det_sign * Vector3(elements[0].length(), elements[1].length(), elements[2].length());
+}
+
+// get_scale works with get_rotation, use get_scale_abs if you need to enforce positive signature.
+Vector3 Basis::get_scale() const {
+ // FIXME: We are assuming M = R.S (R is rotation and S is scaling), and use polar decomposition to extract R and S.
+ // A polar decomposition is M = O.P, where O is an orthogonal matrix (meaning rotation and reflection) and
+ // P is a positive semi-definite matrix (meaning it contains absolute values of scaling along its diagonal).
+ //
+ // Despite being different from what we want to achieve, we can nevertheless make use of polar decomposition
+ // here as follows. We can split O into a rotation and a reflection as O = R.Q, and obtain M = R.S where
+ // we defined S = Q.P. Now, R is a proper rotation matrix and S is a (signed) scaling matrix,
+ // which can involve negative scalings. However, there is a catch: unlike the polar decomposition of M = O.P,
+ // the decomposition of O into a rotation and reflection matrix as O = R.Q is not unique.
+ // Therefore, we are going to do this decomposition by sticking to a particular convention.
+ // This may lead to confusion for some users though.
+ //
+ // The convention we use here is to absorb the sign flip into the scaling matrix.
+ // The same convention is also used in other similar functions such as get_rotation_axis_angle, get_rotation, ...
+ //
+ // A proper way to get rid of this issue would be to store the scaling values (or at least their signs)
+ // as a part of Basis. However, if we go that path, we need to disable direct (write) access to the
+ // matrix elements.
+ //
+ // The rotation part of this decomposition is returned by get_rotation* functions.
+ real_t det_sign = determinant() > 0 ? 1 : -1;
+ return det_sign * Vector3(
+ Vector3(elements[0][0], elements[1][0], elements[2][0]).length(),
+ Vector3(elements[0][1], elements[1][1], elements[2][1]).length(),
+ Vector3(elements[0][2], elements[1][2], elements[2][2]).length());
+}
+
+// Decomposes a Basis into a rotation-reflection matrix (an element of the group O(3)) and a positive scaling matrix as B = O.S.
+// Returns the rotation-reflection matrix via reference argument, and scaling information is returned as a Vector3.
+// This (internal) function is too specific and named too ugly to expose to users, and probably there's no need to do so.
+Vector3 Basis::rotref_posscale_decomposition(Basis &rotref) const {
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(determinant() == 0, Vector3());
+
+ Basis m = transposed() * (*this);
+ ERR_FAIL_COND_V(!m.is_diagonal(), Vector3());
+#endif
+ Vector3 scale = get_scale();
+ Basis inv_scale = Basis().scaled(scale.inverse()); // this will also absorb the sign of scale
+ rotref = (*this) * inv_scale;
+
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(!rotref.is_orthogonal(), Vector3());
+#endif
+ return scale.abs();
+}
+
+// Multiplies the matrix from left by the rotation matrix: M -> R.M
+// Note that this does *not* rotate the matrix itself.
+//
+// The main use of Basis is as Transform.basis, which is used a the transformation matrix
+// of 3D object. Rotate here refers to rotation of the object (which is R * (*this)),
+// not the matrix itself (which is R * (*this) * R.transposed()).
+Basis Basis::rotated(const Vector3 &p_axis, real_t p_phi) const {
+ return Basis(p_axis, p_phi) * (*this);
+}
+
+void Basis::rotate(const Vector3 &p_axis, real_t p_phi) {
+ *this = rotated(p_axis, p_phi);
+}
+
+void Basis::rotate_local(const Vector3 &p_axis, real_t p_phi) {
+ // performs a rotation in object-local coordinate system:
+ // M -> (M.R.Minv).M = M.R.
+ *this = rotated_local(p_axis, p_phi);
+}
+Basis Basis::rotated_local(const Vector3 &p_axis, real_t p_phi) const {
+
+ return (*this) * Basis(p_axis, p_phi);
+}
+
+Basis Basis::rotated(const Vector3 &p_euler) const {
+ return Basis(p_euler) * (*this);
+}
+
+void Basis::rotate(const Vector3 &p_euler) {
+ *this = rotated(p_euler);
+}
+
+Basis Basis::rotated(const Quat &p_quat) const {
+ return Basis(p_quat) * (*this);
+}
+
+void Basis::rotate(const Quat &p_quat) {
+ *this = rotated(p_quat);
+}
+
+Vector3 Basis::get_rotation_euler() const {
+ // Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
+ // and returns the Euler angles corresponding to the rotation part, complementing get_scale().
+ // See the comment in get_scale() for further information.
+ Basis m = orthonormalized();
+ real_t det = m.determinant();
+ if (det < 0) {
+ // Ensure that the determinant is 1, such that result is a proper rotation matrix which can be represented by Euler angles.
+ m.scale(Vector3(-1, -1, -1));
+ }
+
+ return m.get_euler();
+}
+
+Quat Basis::get_rotation_quat() const {
+ // Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
+ // and returns the Euler angles corresponding to the rotation part, complementing get_scale().
+ // See the comment in get_scale() for further information.
+ Basis m = orthonormalized();
+ real_t det = m.determinant();
+ if (det < 0) {
+ // Ensure that the determinant is 1, such that result is a proper rotation matrix which can be represented by Euler angles.
+ m.scale(Vector3(-1, -1, -1));
+ }
+
+ return m.get_quat();
+}
+
+void Basis::get_rotation_axis_angle(Vector3 &p_axis, real_t &p_angle) const {
+ // Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
+ // and returns the Euler angles corresponding to the rotation part, complementing get_scale().
+ // See the comment in get_scale() for further information.
+ Basis m = orthonormalized();
+ real_t det = m.determinant();
+ if (det < 0) {
+ // Ensure that the determinant is 1, such that result is a proper rotation matrix which can be represented by Euler angles.
+ m.scale(Vector3(-1, -1, -1));
+ }
+
+ m.get_axis_angle(p_axis, p_angle);
+}
+
+void Basis::get_rotation_axis_angle_local(Vector3 &p_axis, real_t &p_angle) const {
+ // Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
+ // and returns the Euler angles corresponding to the rotation part, complementing get_scale().
+ // See the comment in get_scale() for further information.
+ Basis m = transposed();
+ m.orthonormalize();
+ real_t det = m.determinant();
+ if (det < 0) {
+ // Ensure that the determinant is 1, such that result is a proper rotation matrix which can be represented by Euler angles.
+ m.scale(Vector3(-1, -1, -1));
+ }
+
+ m.get_axis_angle(p_axis, p_angle);
+ p_angle = -p_angle;
+}
+
+// get_euler_xyz returns a vector containing the Euler angles in the format
+// (a1,a2,a3), where a3 is the angle of the first rotation, and a1 is the last
+// (following the convention they are commonly defined in the literature).
+//
+// The current implementation uses XYZ convention (Z is the first rotation),
+// so euler.z is the angle of the (first) rotation around Z axis and so on,
+//
+// And thus, assuming the matrix is a rotation matrix, this function returns
+// the angles in the decomposition R = X(a1).Y(a2).Z(a3) where Z(a) rotates
+// around the z-axis by a and so on.
+Vector3 Basis::get_euler_xyz() const {
+
+ // Euler angles in XYZ convention.
+ // See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
+ //
+ // rot = cy*cz -cy*sz sy
+ // cz*sx*sy+cx*sz cx*cz-sx*sy*sz -cy*sx
+ // -cx*cz*sy+sx*sz cz*sx+cx*sy*sz cx*cy
+
+ Vector3 euler;
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(!is_rotation(), euler);
+#endif
+ real_t sy = elements[0][2];
+ if (sy < 1.0) {
+ if (sy > -1.0) {
+ // is this a pure Y rotation?
+ if (elements[1][0] == 0.0 && elements[0][1] == 0.0 && elements[1][2] == 0 && elements[2][1] == 0 && elements[1][1] == 1) {
+ // return the simplest form (human friendlier in editor and scripts)
+ euler.x = 0;
+ euler.y = atan2(elements[0][2], elements[0][0]);
+ euler.z = 0;
+ } else {
+ euler.x = Math::atan2(-elements[1][2], elements[2][2]);
+ euler.y = Math::asin(sy);
+ euler.z = Math::atan2(-elements[0][1], elements[0][0]);
+ }
+ } else {
+ euler.x = -Math::atan2(elements[0][1], elements[1][1]);
+ euler.y = -Math_PI / 2.0;
+ euler.z = 0.0;
+ }
+ } else {
+ euler.x = Math::atan2(elements[0][1], elements[1][1]);
+ euler.y = Math_PI / 2.0;
+ euler.z = 0.0;
+ }
+ return euler;
+}
+
+// set_euler_xyz expects a vector containing the Euler angles in the format
+// (ax,ay,az), where ax is the angle of rotation around x axis,
+// and similar for other axes.
+// The current implementation uses XYZ convention (Z is the first rotation).
+void Basis::set_euler_xyz(const Vector3 &p_euler) {
+
+ real_t c, s;
+
+ c = Math::cos(p_euler.x);
+ s = Math::sin(p_euler.x);
+ Basis xmat(1.0, 0.0, 0.0, 0.0, c, -s, 0.0, s, c);
+
+ c = Math::cos(p_euler.y);
+ s = Math::sin(p_euler.y);
+ Basis ymat(c, 0.0, s, 0.0, 1.0, 0.0, -s, 0.0, c);
+
+ c = Math::cos(p_euler.z);
+ s = Math::sin(p_euler.z);
+ Basis zmat(c, -s, 0.0, s, c, 0.0, 0.0, 0.0, 1.0);
+
+ //optimizer will optimize away all this anyway
+ *this = xmat * (ymat * zmat);
+}
+
+// get_euler_yxz returns a vector containing the Euler angles in the YXZ convention,
+// as in first-Z, then-X, last-Y. The angles for X, Y, and Z rotations are returned
+// as the x, y, and z components of a Vector3 respectively.
+Vector3 Basis::get_euler_yxz() const {
+
+ // Euler angles in YXZ convention.
+ // See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
+ //
+ // rot = cy*cz+sy*sx*sz cz*sy*sx-cy*sz cx*sy
+ // cx*sz cx*cz -sx
+ // cy*sx*sz-cz*sy cy*cz*sx+sy*sz cy*cx
+
+ Vector3 euler;
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(!is_rotation(), euler);
+#endif
+ real_t m12 = elements[1][2];
+
+ if (m12 < 1) {
+ if (m12 > -1) {
+ // is this a pure X rotation?
+ if (elements[1][0] == 0 && elements[0][1] == 0 && elements[0][2] == 0 && elements[2][0] == 0 && elements[0][0] == 1) {
+ // return the simplest form (human friendlier in editor and scripts)
+ euler.x = atan2(-m12, elements[1][1]);
+ euler.y = 0;
+ euler.z = 0;
+ } else {
+ euler.x = asin(-m12);
+ euler.y = atan2(elements[0][2], elements[2][2]);
+ euler.z = atan2(elements[1][0], elements[1][1]);
+ }
+ } else { // m12 == -1
+ euler.x = Math_PI * 0.5;
+ euler.y = -atan2(-elements[0][1], elements[0][0]);
+ euler.z = 0;
+ }
+ } else { // m12 == 1
+ euler.x = -Math_PI * 0.5;
+ euler.y = -atan2(-elements[0][1], elements[0][0]);
+ euler.z = 0;
+ }
+
+ return euler;
+}
+
+// set_euler_yxz expects a vector containing the Euler angles in the format
+// (ax,ay,az), where ax is the angle of rotation around x axis,
+// and similar for other axes.
+// The current implementation uses YXZ convention (Z is the first rotation).
+void Basis::set_euler_yxz(const Vector3 &p_euler) {
+
+ real_t c, s;
+
+ c = Math::cos(p_euler.x);
+ s = Math::sin(p_euler.x);
+ Basis xmat(1.0, 0.0, 0.0, 0.0, c, -s, 0.0, s, c);
+
+ c = Math::cos(p_euler.y);
+ s = Math::sin(p_euler.y);
+ Basis ymat(c, 0.0, s, 0.0, 1.0, 0.0, -s, 0.0, c);
+
+ c = Math::cos(p_euler.z);
+ s = Math::sin(p_euler.z);
+ Basis zmat(c, -s, 0.0, s, c, 0.0, 0.0, 0.0, 1.0);
+
+ //optimizer will optimize away all this anyway
+ *this = ymat * xmat * zmat;
+}
+
+bool Basis::is_equal_approx(const Basis &a, const Basis &b) const {
+
+ for (int i = 0; i < 3; i++) {
+ for (int j = 0; j < 3; j++) {
+ if (!Math::is_equal_approx(a.elements[i][j], b.elements[i][j]))
+ return false;
+ }
+ }
+
+ return true;
+}
+
+bool Basis::operator==(const Basis &p_matrix) const {
+
+ for (int i = 0; i < 3; i++) {
+ for (int j = 0; j < 3; j++) {
+ if (elements[i][j] != p_matrix.elements[i][j])
+ return false;
+ }
+ }
+
+ return true;
+}
+
+bool Basis::operator!=(const Basis &p_matrix) const {
+
+ return (!(*this == p_matrix));
+}
+
+Basis::operator String() const {
+
+ String mtx;
+ for (int i = 0; i < 3; i++) {
+
+ for (int j = 0; j < 3; j++) {
+
+ if (i != 0 || j != 0)
+ mtx += ", ";
+
+ mtx += rtos(elements[i][j]);
+ }
+ }
+
+ return mtx;
+}
+
+Quat Basis::get_quat() const {
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(!is_rotation(), Quat());
+#endif
+ real_t trace = elements[0][0] + elements[1][1] + elements[2][2];
+ real_t temp[4];
+
+ if (trace > 0.0) {
+ real_t s = Math::sqrt(trace + 1.0);
+ temp[3] = (s * 0.5);
+ s = 0.5 / s;
+
+ temp[0] = ((elements[2][1] - elements[1][2]) * s);
+ temp[1] = ((elements[0][2] - elements[2][0]) * s);
+ temp[2] = ((elements[1][0] - elements[0][1]) * s);
+ } else {
+ int i = elements[0][0] < elements[1][1] ?
+ (elements[1][1] < elements[2][2] ? 2 : 1) :
+ (elements[0][0] < elements[2][2] ? 2 : 0);
+ int j = (i + 1) % 3;
+ int k = (i + 2) % 3;
+
+ real_t s = Math::sqrt(elements[i][i] - elements[j][j] - elements[k][k] + 1.0);
+ temp[i] = s * 0.5;
+ s = 0.5 / s;
+
+ temp[3] = (elements[k][j] - elements[j][k]) * s;
+ temp[j] = (elements[j][i] + elements[i][j]) * s;
+ temp[k] = (elements[k][i] + elements[i][k]) * s;
+ }
+
+ return Quat(temp[0], temp[1], temp[2], temp[3]);
+}
+
+static const Basis _ortho_bases[24] = {
+ Basis(1, 0, 0, 0, 1, 0, 0, 0, 1),
+ Basis(0, -1, 0, 1, 0, 0, 0, 0, 1),
+ Basis(-1, 0, 0, 0, -1, 0, 0, 0, 1),
+ Basis(0, 1, 0, -1, 0, 0, 0, 0, 1),
+ Basis(1, 0, 0, 0, 0, -1, 0, 1, 0),
+ Basis(0, 0, 1, 1, 0, 0, 0, 1, 0),
+ Basis(-1, 0, 0, 0, 0, 1, 0, 1, 0),
+ Basis(0, 0, -1, -1, 0, 0, 0, 1, 0),
+ Basis(1, 0, 0, 0, -1, 0, 0, 0, -1),
+ Basis(0, 1, 0, 1, 0, 0, 0, 0, -1),
+ Basis(-1, 0, 0, 0, 1, 0, 0, 0, -1),
+ Basis(0, -1, 0, -1, 0, 0, 0, 0, -1),
+ Basis(1, 0, 0, 0, 0, 1, 0, -1, 0),
+ Basis(0, 0, -1, 1, 0, 0, 0, -1, 0),
+ Basis(-1, 0, 0, 0, 0, -1, 0, -1, 0),
+ Basis(0, 0, 1, -1, 0, 0, 0, -1, 0),
+ Basis(0, 0, 1, 0, 1, 0, -1, 0, 0),
+ Basis(0, -1, 0, 0, 0, 1, -1, 0, 0),
+ Basis(0, 0, -1, 0, -1, 0, -1, 0, 0),
+ Basis(0, 1, 0, 0, 0, -1, -1, 0, 0),
+ Basis(0, 0, 1, 0, -1, 0, 1, 0, 0),
+ Basis(0, 1, 0, 0, 0, 1, 1, 0, 0),
+ Basis(0, 0, -1, 0, 1, 0, 1, 0, 0),
+ Basis(0, -1, 0, 0, 0, -1, 1, 0, 0)
+};
+
+int Basis::get_orthogonal_index() const {
+
+ //could be sped up if i come up with a way
+ Basis orth = *this;
+ for (int i = 0; i < 3; i++) {
+ for (int j = 0; j < 3; j++) {
+
+ 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 Basis::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 Basis::get_axis_angle(Vector3 &r_axis, real_t &r_angle) const {
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND(!is_rotation());
+#endif
+ real_t angle, x, y, z; // variables for result
+ real_t epsilon = 0.01; // margin to allow for rounding errors
+ real_t 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;
+ real_t xx = (elements[0][0] + 1) / 2;
+ real_t yy = (elements[1][1] + 1) / 2;
+ real_t zz = (elements[2][2] + 1) / 2;
+ real_t xy = (elements[1][0] + elements[0][1]) / 4;
+ real_t xz = (elements[2][0] + elements[0][2]) / 4;
+ real_t 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
+ real_t 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;
+}
+
+void Basis::set_quat(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));
+}
+
+void Basis::set_axis_angle(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_angle
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND(!p_axis.is_normalized());
+#endif
+ 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);
+}
+
+void Basis::set_axis_angle_scale(const Vector3 &p_axis, real_t p_phi, const Vector3 &p_scale) {
+ set_diagonal(p_scale);
+ rotate(p_axis, p_phi);
+}
+
+void Basis::set_euler_scale(const Vector3 &p_euler, const Vector3 &p_scale) {
+ set_diagonal(p_scale);
+ rotate(p_euler);
+}
+
+void Basis::set_quat_scale(const Quat &p_quat, const Vector3 &p_scale) {
+ set_diagonal(p_scale);
+ rotate(p_quat);
+}
+
+void Basis::set_diagonal(const Vector3 p_diag) {
+ elements[0][0] = p_diag.x;
+ elements[0][1] = 0;
+ elements[0][2] = 0;
+
+ elements[1][0] = 0;
+ elements[1][1] = p_diag.y;
+ elements[1][2] = 0;
+
+ elements[2][0] = 0;
+ elements[2][1] = 0;
+ elements[2][2] = p_diag.z;
+}
+
+Basis Basis::slerp(const Basis &target, const real_t &t) const {
+// TODO: implement this directly without using quaternions to make it more efficient
+#ifdef MATH_CHECKS
+ ERR_FAIL_COND_V(!is_rotation(), Basis());
+ ERR_FAIL_COND_V(!target.is_rotation(), Basis());
+#endif
+
+ Quat from(*this);
+ Quat to(target);
+
+ return Basis(from.slerp(to, t));
+}
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