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gdnative_cpp/core/basis.cpp

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/*************************************************************************/
/* Basis.cpp */
/*************************************************************************/
/* This file is part of: */
/* PANDEMONIUM ENGINE */
/* https://pandemoniumengine.org */
/*************************************************************************/
/* Copyright (c) 2007-2022 Juan Linietsky, Ariel Manzur. */
/* Copyright (c) 2014-2022 Pandemonium 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 */
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/* 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 "defs.h"
#include "quaternion.h"
#include "vector3.h"
#include <algorithm>
const Basis Basis::IDENTITY = Basis();
const Basis Basis::FLIP_X = Basis(-1, 0, 0, 0, 1, 0, 0, 0, 1);
const Basis Basis::FLIP_Y = Basis(1, 0, 0, 0, -1, 0, 0, 0, 1);
const Basis Basis::FLIP_Z = Basis(1, 0, 0, 0, 1, 0, 0, 0, -1);
Basis::Basis(const Vector3 &row0, const Vector3 &row1, const Vector3 &row2) {
elements[0] = row0;
elements[1] = row1;
elements[2] = row2;
}
Basis::Basis(real_t xx, real_t xy, real_t xz, real_t yx, real_t yy, real_t yz, real_t zx, real_t zy, real_t zz) {
set(xx, xy, xz, yx, yy, yz, zx, zy, zz);
}
Basis::Basis() {
elements[0][0] = 1;
elements[0][1] = 0;
elements[0][2] = 0;
elements[1][0] = 0;
elements[1][1] = 1;
elements[1][2] = 0;
elements[2][0] = 0;
elements[2][1] = 0;
elements[2][2] = 1;
}
#define cofac(row1, col1, row2, col2) \
(elements[row1][col1] * elements[row2][col2] - elements[row1][col2] * elements[row2][col1])
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];
ERR_FAIL_COND(det == 0);
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);
}
#undef cofac
bool Basis::isequal_approx(const Basis &a, const Basis &b) const {
for (int i = 0; i < 3; i++) {
for (int j = 0; j < 3; j++) {
if ((::fabs(a.elements[i][j] - b.elements[i][j]) < CMP_EPSILON) == false)
return false;
}
}
return true;
}
bool Basis::is_orthogonal() const {
Basis id;
Basis m = (*this) * transposed();
return isequal_approx(id, m);
}
bool Basis::is_rotation() const {
return ::fabs(determinant() - 1) < CMP_EPSILON && is_orthogonal();
}
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::inverse() const {
Basis b = *this;
b.invert();
return b;
}
Basis Basis::transposed() const {
Basis b = *this;
b.transpose();
return b;
}
real_t Basis::determinant() const {
return elements[0][0] * (elements[1][1] * elements[2][2] - elements[2][1] * elements[1][2]) -
elements[1][0] * (elements[0][1] * elements[2][2] - elements[2][1] * elements[0][2]) +
elements[2][0] * (elements[0][1] * elements[1][2] - elements[1][1] * elements[0][2]);
}
Vector3 Basis::get_axis(int p_axis) const {
// get actual basis axis (elements is transposed for performance)
return Vector3(elements[0][p_axis], elements[1][p_axis], elements[2][p_axis]);
}
void Basis::set_axis(int p_axis, const Vector3 &p_value) {
// get actual basis axis (elements is transposed for performance)
elements[0][p_axis] = p_value.x;
elements[1][p_axis] = p_value.y;
elements[2][p_axis] = p_value.z;
}
void Basis::rotate(const Vector3 &p_axis, real_t p_phi) {
*this = rotated(p_axis, p_phi);
}
Basis Basis::rotated(const Vector3 &p_axis, real_t p_phi) const {
return Basis(p_axis, p_phi) * (*this);
}
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 b = *this;
b.scale(p_scale);
return b;
}
Vector3 Basis::get_scale() const {
// We are assuming M = R.S, and performing a polar decomposition to extract R and S.
// FIXME: We eventually need a proper polar decomposition.
// As a cheap workaround until then, to ensure that R is a proper rotation matrix with determinant +1
// (such that it can be represented by a Quaternion or Euler angles), we absorb the sign flip into the scaling matrix.
// As such, it works in conjuction with get_rotation().
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());
}
// TODO: implement this directly without using quaternions to make it more efficient
Basis Basis::slerp(Basis b, float t) const {
ERR_FAIL_COND_V(!is_rotation(), Basis());
ERR_FAIL_COND_V(!b.is_rotation(), Basis());
Quaternion from(*this);
Quaternion to(b);
return Basis(from.slerp(to, t));
}
// 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;
ERR_FAIL_COND_V(is_rotation() == false, euler);
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 = ::atan2(-elements[1][2], elements[2][2]);
euler.y = ::asin(sy);
euler.z = ::atan2(-elements[0][1], elements[0][0]);
}
} else {
euler.x = -::atan2(elements[0][1], elements[1][1]);
euler.y = -Math_PI / 2.0;
euler.z = 0.0;
}
} else {
euler.x = ::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 = ::cos(p_euler.x);
s = ::sin(p_euler.x);
Basis xmat(1.0, 0.0, 0.0, 0.0, c, -s, 0.0, s, c);
c = ::cos(p_euler.y);
s = ::sin(p_euler.y);
Basis ymat(c, 0.0, s, 0.0, 1.0, 0.0, -s, 0.0, c);
c = ::cos(p_euler.z);
s = ::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;
ERR_FAIL_COND_V(is_rotation() == false, 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 = ::cos(p_euler.x);
s = ::sin(p_euler.x);
Basis xmat(1.0, 0.0, 0.0, 0.0, c, -s, 0.0, s, c);
c = ::cos(p_euler.y);
s = ::sin(p_euler.y);
Basis ymat(c, 0.0, s, 0.0, 1.0, 0.0, -s, 0.0, c);
c = ::cos(p_euler.z);
s = ::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;
}
// transposed dot products
real_t Basis::tdotx(const Vector3 &v) const {
return elements[0][0] * v[0] + elements[1][0] * v[1] + elements[2][0] * v[2];
}
real_t Basis::tdoty(const Vector3 &v) const {
return elements[0][1] * v[0] + elements[1][1] * v[1] + elements[2][1] * v[2];
}
real_t Basis::tdotz(const Vector3 &v) const {
return elements[0][2] * v[0] + elements[1][2] * v[1] + elements[2][2] * v[2];
}
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));
}
Vector3 Basis::xform(const Vector3 &p_vector) const {
return Vector3(
elements[0].dot(p_vector),
elements[1].dot(p_vector),
elements[2].dot(p_vector));
}
Vector3 Basis::xform_inv(const Vector3 &p_vector) const {
return Vector3(
(elements[0][0] * p_vector.x) + (elements[1][0] * p_vector.y) + (elements[2][0] * p_vector.z),
(elements[0][1] * p_vector.x) + (elements[1][1] * p_vector.y) + (elements[2][1] * p_vector.z),
(elements[0][2] * p_vector.x) + (elements[1][2] * p_vector.y) + (elements[2][2] * p_vector.z));
}
void Basis::operator*=(const Basis &p_matrix) {
set(
p_matrix.tdotx(elements[0]), p_matrix.tdoty(elements[0]), p_matrix.tdotz(elements[0]),
p_matrix.tdotx(elements[1]), p_matrix.tdoty(elements[1]), p_matrix.tdotz(elements[1]),
p_matrix.tdotx(elements[2]), p_matrix.tdoty(elements[2]), p_matrix.tdotz(elements[2]));
}
Basis Basis::operator*(const Basis &p_matrix) const {
return Basis(
p_matrix.tdotx(elements[0]), p_matrix.tdoty(elements[0]), p_matrix.tdotz(elements[0]),
p_matrix.tdotx(elements[1]), p_matrix.tdoty(elements[1]), p_matrix.tdotz(elements[1]),
p_matrix.tdotx(elements[2]), p_matrix.tdoty(elements[2]), p_matrix.tdotz(elements[2]));
}
void Basis::operator+=(const Basis &p_matrix) {
elements[0] += p_matrix.elements[0];
elements[1] += p_matrix.elements[1];
elements[2] += p_matrix.elements[2];
}
Basis Basis::operator+(const Basis &p_matrix) const {
Basis ret(*this);
ret += p_matrix;
return ret;
}
void Basis::operator-=(const Basis &p_matrix) {
elements[0] -= p_matrix.elements[0];
elements[1] -= p_matrix.elements[1];
elements[2] -= p_matrix.elements[2];
}
Basis Basis::operator-(const Basis &p_matrix) const {
Basis ret(*this);
ret -= p_matrix;
return ret;
}
void Basis::operator*=(real_t p_val) {
elements[0] *= p_val;
elements[1] *= p_val;
elements[2] *= p_val;
}
Basis Basis::operator*(real_t p_val) const {
Basis ret(*this);
ret *= p_val;
return ret;
}
Basis::operator String() const {
String s;
for (int i = 0; i < 3; i++) {
for (int j = 0; j < 3; j++) {
if (i != 0 || j != 0)
s += ", ";
s += String::num(elements[i][j]);
}
}
return s;
}
/* create / set */
void Basis::set(real_t xx, real_t xy, real_t xz, real_t yx, real_t yy, real_t yz, real_t zx, real_t zy, real_t zz) {
elements[0][0] = xx;
elements[0][1] = xy;
elements[0][2] = xz;
elements[1][0] = yx;
elements[1][1] = yy;
elements[1][2] = yz;
elements[2][0] = zx;
elements[2][1] = zy;
elements[2][2] = zz;
}
Vector3 Basis::get_column(int i) const {
return Vector3(elements[0][i], elements[1][i], elements[2][i]);
}
Vector3 Basis::get_row(int i) const {
return Vector3(elements[i][0], elements[i][1], elements[i][2]);
}
Vector3 Basis::get_main_diagonal() const {
return Vector3(elements[0][0], elements[1][1], elements[2][2]);
}
void Basis::set_row(int i, const Vector3 &p_row) {
elements[i][0] = p_row.x;
elements[i][1] = p_row.y;
elements[i][2] = p_row.z;
}
Basis Basis::transpose_xform(const Basis &m) const {
return Basis(
elements[0].x * m[0].x + elements[1].x * m[1].x + elements[2].x * m[2].x,
elements[0].x * m[0].y + elements[1].x * m[1].y + elements[2].x * m[2].y,
elements[0].x * m[0].z + elements[1].x * m[1].z + elements[2].x * m[2].z,
elements[0].y * m[0].x + elements[1].y * m[1].x + elements[2].y * m[2].x,
elements[0].y * m[0].y + elements[1].y * m[1].y + elements[2].y * m[2].y,
elements[0].y * m[0].z + elements[1].y * m[1].z + elements[2].y * m[2].z,
elements[0].z * m[0].x + elements[1].z * m[1].x + elements[2].z * m[2].x,
elements[0].z * m[0].y + elements[1].z * m[1].y + elements[2].z * m[2].y,
elements[0].z * m[0].z + elements[1].z * m[1].z + elements[2].z * m[2].z);
}
void Basis::orthonormalize() {
ERR_FAIL_COND(determinant() == 0);
// 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 b = *this;
b.orthonormalize();
return b;
}
bool Basis::is_symmetric() const {
if (::fabs(elements[0][1] - elements[1][0]) > CMP_EPSILON)
return false;
if (::fabs(elements[0][2] - elements[2][0]) > CMP_EPSILON)
return false;
if (::fabs(elements[1][2] - elements[2][1]) > CMP_EPSILON)
return false;
return true;
}
Basis Basis::diagonalize() {
// I love copy paste
if (!is_symmetric())
return Basis();
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 (::fabs(elements[j][j] - elements[i][i]) < CMP_EPSILON) {
angle = Math_PI / 4;
} else {
angle = 0.5 * ::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] = ::cos(angle);
rot.elements[i][j] = -(rot.elements[j][i] = ::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;
}
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_COND(p_index >= 24);
*this = _ortho_bases[p_index];
}
Basis::Basis(const Vector3 &p_euler) {
set_euler(p_euler);
}
#include "quaternion.h"
Basis::Basis(const Quaternion &p_quaternion) {
real_t d = p_quaternion.length_squared();
real_t s = 2.0 / d;
real_t xs = p_quaternion.x * s, ys = p_quaternion.y * s, zs = p_quaternion.z * s;
real_t wx = p_quaternion.w * xs, wy = p_quaternion.w * ys, wz = p_quaternion.w * zs;
real_t xx = p_quaternion.x * xs, xy = p_quaternion.x * ys, xz = p_quaternion.x * zs;
real_t yy = p_quaternion.y * ys, yz = p_quaternion.y * zs, zz = p_quaternion.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));
}
Basis::Basis(const Vector3 &p_axis, real_t p_phi) {
// Rotation matrix from axis and angle, see https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
Vector3 axis_sq(p_axis.x * p_axis.x, p_axis.y * p_axis.y, p_axis.z * p_axis.z);
real_t cosine = ::cos(p_phi);
real_t sine = ::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);
}
Basis::operator Quaternion() const {
// commenting this check because precision issues cause it to fail when it shouldn't
// ERR_FAIL_COND_V(is_rotation() == false, Quaternion());
real_t trace = elements[0][0] + elements[1][1] + elements[2][2];
real_t temp[4];
if (trace > 0.0) {
real_t s = ::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 = ::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 Quaternion(temp[0], temp[1], temp[2], temp[3]);
}
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