/*************************************************************************/
/*                       This file is part of:                           */
/*                           BLENDOT ENGINE                              */
/*                      https://blendot.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.                */
/*************************************************************************/

import "basis.h"
import "math_funcs.h"
import "copymem.h"
import "print_string.h"

#define cofac(row1, col1, row2, col2) (elements[row1][col1] * elements[row2][col2] - elements[row1][col2] * elements[row2][col1])

def 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


def 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)


def 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)

@const
def Basis::orthonormalized() -> Basis:
	Basis c = *this
	c.orthonormalize()
	return c

@const
def Basis::is_orthogonal() -> bool:
	Basis id
	Basis m = (*this) * transposed()
	return is_equal_approx(id, m)

@const
def Basis::is_diagonal() -> bool:
	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))

@const
def Basis::is_rotation() -> bool:
	return Math::is_equal_approx(determinant(), 1, UNIT_EPSILON) && is_orthogonal()

@const
def Basis::is_symmetric() -> bool:
	if not Math::is_equal_approx_ratio(elements[0][1], elements[1][0], UNIT_EPSILON):
		return false
	if not Math::is_equal_approx_ratio(elements[0][2], elements[2][0], UNIT_EPSILON):
		return false
	if not Math::is_equal_approx_ratio(elements[1][2], elements[2][1], UNIT_EPSILON):
		return false
	return true


def Basis::diagonalize() -> Basis:
	//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

@const
def Basis::inverse() ->Basis:
	Basis inv = *this
	inv.invert()
	return inv


def Basis::transpose():
	SWAP(elements[0][1], elements[1][0])
	SWAP(elements[0][2], elements[2][0])
	SWAP(elements[1][2], elements[2][1])

@const
def Basis::transposed() ->Basis:
	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.
def 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

@const
def Basis::scaled(const Vector3 &p_scale) ->Basis:
	Basis m = *this
	m.scale(p_scale)
	return m


def 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)

@const
def Basis::scaled_local(const Vector3 &p_scale) ->Basis:
	Basis b
	b.set_diagonal(p_scale)
	return (*this) * b

@const
def Basis::get_scale_abs() -> Vector3:
	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())

@const
def Basis::get_scale_local() -> Vector3:
	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.
@const
def Basis::get_scale() -> Vector3:
	// 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.
@const
def Basis::rotref_posscale_decomposition(Basis &rotref) ->Vector3:
	#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()).
@const
def Basis::rotated(const Vector3 &p_axis, real_t p_phi) ->Basis:
	return Basis(p_axis, p_phi) * (*this)


def Basis::rotate(const Vector3 &p_axis, real_t p_phi):
	*this = rotated(p_axis, p_phi)


def 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)

@const
def Basis::rotated_local(const Vector3 &p_axis, real_t p_phi) ->Basis:
	return (*this) * Basis(p_axis, p_phi)

@const
def Basis::rotated(const Vector3 &p_euler) ->Basis:
	return Basis(p_euler) * (*this)

def Basis::rotate(const Vector3 &p_euler):
	*this = rotated(p_euler)

@const
def Basis::rotated(const Quat &p_quat) ->Basis:
	return Basis(p_quat) * (*this)


def Basis::rotate(const Quat &p_quat):
	*this = rotated(p_quat)

@const
def Basis::get_rotation_euler() ->Vector3:
	// 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()

@const
def Basis::get_rotation_quat() ->Quat:
	// 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()

@const
def Basis::get_rotation_axis_angle(Vector3 &p_axis, real_t &p_angle):
	// 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)

@const
def Basis::get_rotation_axis_angle_local(Vector3 &p_axis, real_t &p_angle):
	// 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.
@const
def Basis::get_euler_xyz() ->Vector3:
	// 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).
def 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.
@const
def Basis::get_euler_yxz() ->Vector3:
	// 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).
def 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

@const
def Basis::is_equal_approx(const Basis &a, const Basis &b, real_t p_epsilon) ->bool:
	for (int i = 0; i < 3; i++):
		for (int j = 0; j < 3; j++):
			if not Math::is_equal_approx(a.elements[i][j], b.elements[i][j], p_epsilon):
				return false
	return true

@const
def Basis::is_equal_approx_ratio(const Basis &a, const Basis &b, real_t p_epsilon) ->bool:
	for (int i = 0; i < 3; i++):
		for (int j = 0; j < 3; j++):
			if not Math::is_equal_approx_ratio(a.elements[i][j], b.elements[i][j], p_epsilon):
				return false
	return true

@const
def Basis::operator==(const Basis &p_matrix) ->bool:
	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

@const
def Basis::operator!=(const Basis &p_matrix) ->bool:
	return (!(*this == p_matrix))


@const
def Basis::operator String():
	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

@const
def Basis::get_quat() ->Quat:
	#ifdef MATH_CHECKS
	  ERR_FAIL_COND_V_MSG(!is_rotation(), Quat(), "Basis must be normalized in order to be casted to a Quaternion. Use get_rotation_quat() or call orthonormalized() instead.");
	#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)};

@const
def Basis::get_orthogonal_index() ->int:
	//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
			elif 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


def 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]

@const
def Basis::get_axis_angle(Vector3 &r_axis, real_t &r_angle):
	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
		elif 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
	// s=|axis||sin(angle)|, used to normalise
	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]))
	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


def 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))


def 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)
	elements[0][0] = axis_sq.x + cosine * (1.0 - axis_sq.x)
	elements[1][1] = axis_sq.y + cosine * (1.0 - axis_sq.y)
	elements[2][2] = axis_sq.z + cosine * (1.0 - axis_sq.z)
	real_t sine = Math::sin(p_phi)
	real_t t = 1 - cosine
	real_t xyzt = p_axis.x * p_axis.y * t
	real_t zyxs = p_axis.z * sine
	elements[0][1] = xyzt - zyxs
	elements[1][0] = xyzt + zyxs
	xyzt = p_axis.x * p_axis.z * t
	zyxs = p_axis.y * sine
	elements[0][2] = xyzt + zyxs
	elements[2][0] = xyzt - zyxs
	xyzt = p_axis.y * p_axis.z * t
	zyxs = p_axis.x * sine
	elements[1][2] = xyzt - zyxs
	elements[2][1] = xyzt + zyxs


def 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)


def Basis::set_euler_scale(const Vector3 &p_euler, const Vector3 &p_scale):
	set_diagonal(p_scale)
	rotate(p_euler)


def Basis::set_quat_scale(const Quat &p_quat, const Vector3 &p_scale):
	set_diagonal(p_scale)
	rotate(p_quat)


def 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

@const
def Basis::slerp(const Basis &target, const real_t &t) ->Basis:
	//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