text: /*************************************************************************/
  /*  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)    */
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  /*************************************************************************/
  
  #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, UNIT_EPSILON) && is_orthogonal();
  }
  
  bool Basis::is_symmetric() const {
  
  	if (!Math::is_equal_approx_ratio(elements[0][1], elements[1][0], UNIT_EPSILON))
  		return false;
  	if (!Math::is_equal_approx_ratio(elements[0][2], elements[2][0], UNIT_EPSILON))
  		return false;
  	if (!Math::is_equal_approx_ratio(elements[1][2], elements[2][1], UNIT_EPSILON))
  		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 = SGN(determinant());
  	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 = SGN(determinant());
  	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 {
  
  	/* checking this is a bad idea, because obtaining from scaled transform is a valid use case
  #ifdef MATH_CHECKS
  	ERR_FAIL_COND(!is_rotation());
  #endif
  */
  	// 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;
  
  	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,real_t p_epsilon) 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], p_epsilon))
  				return false;
  		}
  	}
  
  	return true;
  }
  
  bool Basis::is_equal_approx_ratio(const Basis &a, const Basis &b,real_t p_epsilon) const {
  
  	for (int i = 0; i < 3; i++) {
  		for (int j = 0; j < 3; j++) {
  			if (!Math::is_equal_approx_ratio(a.elements[i][j], b.elements[i][j], p_epsilon))
  				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
  	if (!is_rotation()) {
  		ERR_EXPLAIN("Basis must be normalized in order to be casted to a Quaternion. Use get_rotation_quat() or call orthonormalized() instead.");
  		ERR_FAIL_V(Quat());
  	}
  #endif
  	/* Allow getting a quaternion from an unnormalized transform */
  	Basis m = *this;
  	real_t trace = m.elements[0][0] + m.elements[1][1] + m.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] = ((m.elements[2][1] - m.elements[1][2]) * s);
  		temp[1] = ((m.elements[0][2] - m.elements[2][0]) * s);
  		temp[2] = ((m.elements[1][0] - m.elements[0][1]) * s);
  	} else {
  		int i = m.elements[0][0] < m.elements[1][1] ?
  						(m.elements[1][1] < m.elements[2][2] ? 2 : 1) :
  						(m.elements[0][0] < m.elements[2][2] ? 2 : 0);
  		int j = (i + 1) % 3;
  		int k = (i + 2) % 3;
  
  		real_t s = Math::sqrt(m.elements[i][i] - m.elements[j][j] - m.elements[k][k] + 1.0);
  		temp[i] = s * 0.5;
  		s = 0.5 / s;
  
  		temp[3] = (m.elements[k][j] - m.elements[j][k]) * s;
  		temp[j] = (m.elements[j][i] + m.elements[i][j]) * s;
  		temp[k] = (m.elements[k][i] + m.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 {
  	/* checking this is a bad idea, because obtaining from scaled transform is a valid use case
  #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 {
  
  	//consider scale
  	Quat from(*this);
  	Quat to(target);
  
  	Basis b(from.slerp(to, t));
  	b.elements[0] *= Math::lerp(elements[0].length(), target.elements[0].length(), t);
  	b.elements[1] *= Math::lerp(elements[1].length(), target.elements[1].length(), t);
  	b.elements[2] *= Math::lerp(elements[2].length(), target.elements[2].length(), t);
  
  	return b;
  }
  