[Core] Rename Matrix3 file to Basis

The code already referred to "Basis", it's just the file name that was different for some reason.

1 files changed, 0 insertions, 848 deletions

diff --git a/core/math/matrix3.cpp b/core/math/matrix3.cpp
deleted file mode 100644
index 0aa67078fb..0000000000
--- a/core/math/matrix3.cpp
+++ /dev/null
@@ -1,848 +0,0 @@
-/*************************************************************************/
-/*  matrix3.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 "matrix3.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));
-}