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pandemonium_engine/core/math/basis.cpp

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/*************************************************************************/
/* basis.cpp */
/*************************************************************************/
/* This file is part of: */
/* PANDEMONIUM ENGINE */
/* https://github.com/Relintai/pandemonium_engine */
/*************************************************************************/
/* Copyright (c) 2022-present Péter Magyar. */
/* Copyright (c) 2014-2022 Godot Engine contributors (cf. AUTHORS.md). */
/* Copyright (c) 2007-2022 Juan Linietsky, Ariel Manzur. */
/* */
/* Permission is hereby granted, free of charge, to any person obtaining */
/* a copy of this software and associated documentation files (the */
/* "Software"), to deal in the Software without restriction, including */
/* without limitation the rights to use, copy, modify, merge, publish, */
/* distribute, sublicense, and/or sell copies of the Software, and to */
/* permit persons to whom the Software is furnished to do so, subject to */
/* the following conditions: */
/* */
/* The above copyright notice and this permission notice shall be */
/* included in all copies or substantial portions of the Software. */
/* */
/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, */
/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF */
/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/
/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY */
/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, */
/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE */
/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */
/*************************************************************************/
#include "basis.h"
#include "core/math/math_funcs.h"
#include "core/string/print_string.h"
#define cofac(row1, col1, row2, col2) \
(rows[row1][col1] * rows[row2][col2] - rows[row1][col2] * rows[row2][col1])
void Basis::from_z(const Vector3 &p_z) {
if (Math::abs(p_z.z) > (real_t)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 / Math::sqrt(a);
rows[0] = Vector3(0, -p_z[2] * k, p_z[1] * k);
rows[1] = Vector3(a * k, -p_z[0] * rows[0][2], p_z[0] * rows[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 / Math::sqrt(a);
rows[0] = Vector3(-p_z.y * k, p_z.x * k, 0);
rows[1] = Vector3(-p_z.z * rows[0].y, p_z.z * rows[0].x, a * k);
}
rows[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 = rows[0][0] * co[0] +
rows[0][1] * co[1] +
rows[0][2] * co[2];
#ifdef MATH_CHECKS
ERR_FAIL_COND(det == 0);
#endif
real_t s = 1 / 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() {
// Gram-Schmidt Process
Vector3 x = get_column(0);
Vector3 y = get_column(1);
Vector3 z = get_column(2);
x.normalize();
y = (y - x * (x.dot(y)));
y.normalize();
z = (z - x * (x.dot(z)) - y * (y.dot(z)));
z.normalize();
set_column(0, x);
set_column(1, y);
set_column(2, z);
}
Basis Basis::orthonormalized() const {
Basis c = *this;
c.orthonormalize();
return c;
}
bool Basis::is_orthogonal() const {
Basis identity;
Basis m = (*this) * transposed();
return m.is_equal_approx(identity);
}
void Basis::orthogonalize() {
Vector3 scl = get_scale();
orthonormalize();
scale_local(scl);
}
Basis Basis::orthogonalized() const {
Basis c = *this;
c.orthogonalize();
return c;
}
bool Basis::is_diagonal() const {
return (
Math::is_zero_approx(rows[0][1]) && Math::is_zero_approx(rows[0][2]) &&
Math::is_zero_approx(rows[1][0]) && Math::is_zero_approx(rows[1][2]) &&
Math::is_zero_approx(rows[2][0]) && Math::is_zero_approx(rows[2][1]));
}
bool Basis::is_rotation() const {
return Math::is_equal_approx(determinant(), 1, (real_t)UNIT_EPSILON) && is_orthogonal();
}
bool Basis::is_symmetric() const {
if (!Math::is_equal_approx_ratio(rows[0][1], rows[1][0], (real_t)UNIT_EPSILON)) {
return false;
}
if (!Math::is_equal_approx_ratio(rows[0][2], rows[2][0], (real_t)UNIT_EPSILON)) {
return false;
}
if (!Math::is_equal_approx_ratio(rows[1][2], rows[2][1], (real_t)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 = rows[0][1] * rows[0][1] + rows[0][2] * rows[0][2] + rows[1][2] * rows[1][2];
int ite = 0;
Basis acc_rot;
while (off_matrix_norm_2 > (real_t)CMP_EPSILON2 && ite++ < ite_max) {
real_t el01_2 = rows[0][1] * rows[0][1];
real_t el02_2 = rows[0][2] * rows[0][2];
real_t el12_2 = rows[1][2] * rows[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(rows[j][j], rows[i][i])) {
angle = Math_PI / 4;
} else {
angle = 0.5f * Math::atan(2 * rows[i][j] / (rows[j][j] - rows[i][i]));
}
// Compute the rotation matrix
Basis rot;
rot.rows[i][i] = rot.rows[j][j] = Math::cos(angle);
rot.rows[i][j] = -(rot.rows[j][i] = Math::sin(angle));
// Update the off matrix norm
off_matrix_norm_2 -= rows[i][j] * rows[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(rows[0][1], rows[1][0]);
SWAP(rows[0][2], rows[2][0]);
SWAP(rows[1][2], rows[2][1]);
}
Basis Basis::transposed() const {
Basis tr = *this;
tr.transpose();
return tr;
}
Basis Basis::from_scale(const Vector3 &p_scale) {
return Basis(p_scale.x, 0, 0, 0, p_scale.y, 0, 0, 0, p_scale.z);
}
// 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) {
rows[0][0] *= p_scale.x;
rows[0][1] *= p_scale.x;
rows[0][2] *= p_scale.x;
rows[1][0] *= p_scale.y;
rows[1][1] *= p_scale.y;
rows[1][2] *= p_scale.y;
rows[2][0] *= p_scale.z;
rows[2][1] *= p_scale.z;
rows[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;
}
void Basis::scale_orthogonal(const Vector3 &p_scale) {
*this = scaled_orthogonal(p_scale);
}
Basis Basis::scaled_orthogonal(const Vector3 &p_scale) const {
Basis m = *this;
Vector3 s = Vector3(-1, -1, -1) + p_scale;
Vector3 dots;
Basis b;
for (int i = 0; i < 3; i++) {
for (int j = 0; j < 3; j++) {
dots[j] += s[i] * Math::abs(m.get_column(i).normalized().dot(b.get_column(j)));
}
}
m.scale_local(Vector3(1, 1, 1) + dots);
return m;
}
real_t Basis::get_uniform_scale() const {
return (rows[0].length() + rows[1].length() + rows[2].length()) / 3.0f;
}
void Basis::make_scale_uniform() {
float l = (rows[0].length() + rows[1].length() + rows[2].length()) / 3.0f;
for (int i = 0; i < 3; i++) {
rows[i].normalize();
rows[i] *= l;
}
}
Vector3 Basis::get_scale_abs() const {
return Vector3(
Vector3(rows[0][0], rows[1][0], rows[2][0]).length(),
Vector3(rows[0][1], rows[1][1], rows[2][1]).length(),
Vector3(rows[0][2], rows[1][2], rows[2][2]).length());
}
Vector3 Basis::get_scale_local() const {
real_t det_sign = SGN(determinant());
return det_sign * Vector3(rows[0].length(), rows[1].length(), rows[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 rows.
//
// The rotation part of this decomposition is returned by get_rotation* functions.
real_t det_sign = SGN(determinant());
return det_sign * get_scale_abs();
}
// 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 Quaternion &p_quat) const {
return Basis(p_quat) * (*this);
}
void Basis::rotate(const Quaternion &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();
}
Quaternion Basis::get_rotation_quaternion() 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_quaternion();
}
void Basis::rotate_to_align(const Vector3 &p_start_direction, const Vector3 &p_end_direction) {
// Takes two vectors and rotates the basis from the first vector to the second vector.
// Adopted from: https://gist.github.com/kevinmoran/b45980723e53edeb8a5a43c49f134724
const Vector3 axis = p_start_direction.cross(p_end_direction).normalized();
if (axis.length_squared() != 0) {
real_t dot = p_start_direction.dot(p_end_direction);
dot = CLAMP(dot, -1.0, 1.0);
const real_t angle_rads = Math::acos(dot);
*this = Basis(axis, angle_rads) * (*this);
}
}
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;
real_t sy = rows[0][2];
if (sy < (1 - (real_t)CMP_EPSILON)) {
if (sy > -(1 - (real_t)CMP_EPSILON)) {
// is this a pure Y rotation?
if (rows[1][0] == 0 && rows[0][1] == 0 && rows[1][2] == 0 && rows[2][1] == 0 && rows[1][1] == 1) {
// return the simplest form (human friendlier in editor and scripts)
euler.x = 0;
euler.y = atan2(rows[0][2], rows[0][0]);
euler.z = 0;
} else {
euler.x = Math::atan2(-rows[1][2], rows[2][2]);
euler.y = Math::asin(sy);
euler.z = Math::atan2(-rows[0][1], rows[0][0]);
}
} else {
euler.x = Math::atan2(rows[2][1], rows[1][1]);
euler.y = -Math_PI / 2.0;
euler.z = 0.0;
}
} else {
euler.x = Math::atan2(rows[2][1], rows[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, c, -s, 0, s, c);
c = Math::cos(p_euler.y);
s = Math::sin(p_euler.y);
Basis ymat(c, 0, s, 0, 1, 0, -s, 0, c);
c = Math::cos(p_euler.z);
s = Math::sin(p_euler.z);
Basis zmat(c, -s, 0, s, c, 0, 0, 0, 1);
//optimizer will optimize away all this anyway
*this = xmat * (ymat * zmat);
}
Vector3 Basis::get_euler_xzy() const {
// Euler angles in XZY convention.
// See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
//
// rot = cz*cy -sz cz*sy
// sx*sy+cx*cy*sz cx*cz cx*sz*sy-cy*sx
// cy*sx*sz cz*sx cx*cy+sx*sz*sy
Vector3 euler;
real_t sz = rows[0][1];
if (sz < (1 - (real_t)CMP_EPSILON)) {
if (sz > -(1 - (real_t)CMP_EPSILON)) {
euler.x = Math::atan2(rows[2][1], rows[1][1]);
euler.y = Math::atan2(rows[0][2], rows[0][0]);
euler.z = Math::asin(-sz);
} else {
// It's -1
euler.x = -Math::atan2(rows[1][2], rows[2][2]);
euler.y = 0.0;
euler.z = Math_PI / 2.0;
}
} else {
// It's 1
euler.x = -Math::atan2(rows[1][2], rows[2][2]);
euler.y = 0.0;
euler.z = -Math_PI / 2.0;
}
return euler;
}
void Basis::set_euler_xzy(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, c, -s, 0, s, c);
c = Math::cos(p_euler.y);
s = Math::sin(p_euler.y);
Basis ymat(c, 0, s, 0, 1, 0, -s, 0, c);
c = Math::cos(p_euler.z);
s = Math::sin(p_euler.z);
Basis zmat(c, -s, 0, s, c, 0, 0, 0, 1);
*this = xmat * zmat * ymat;
}
Vector3 Basis::get_euler_yzx() const {
// Euler angles in YZX convention.
// See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
//
// rot = cy*cz sy*sx-cy*cx*sz cx*sy+cy*sz*sx
// sz cz*cx -cz*sx
// -cz*sy cy*sx+cx*sy*sz cy*cx-sy*sz*sx
Vector3 euler;
real_t sz = rows[1][0];
if (sz < (1 - (real_t)CMP_EPSILON)) {
if (sz > -(1 - (real_t)CMP_EPSILON)) {
euler.x = Math::atan2(-rows[1][2], rows[1][1]);
euler.y = Math::atan2(-rows[2][0], rows[0][0]);
euler.z = Math::asin(sz);
} else {
// It's -1
euler.x = Math::atan2(rows[2][1], rows[2][2]);
euler.y = 0.0;
euler.z = -Math_PI / 2.0;
}
} else {
// It's 1
euler.x = Math::atan2(rows[2][1], rows[2][2]);
euler.y = 0.0;
euler.z = Math_PI / 2.0;
}
return euler;
}
void Basis::set_euler_yzx(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, c, -s, 0, s, c);
c = Math::cos(p_euler.y);
s = Math::sin(p_euler.y);
Basis ymat(c, 0, s, 0, 1, 0, -s, 0, c);
c = Math::cos(p_euler.z);
s = Math::sin(p_euler.z);
Basis zmat(c, -s, 0, s, c, 0, 0, 0, 1);
*this = ymat * zmat * xmat;
}
// 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;
real_t m12 = rows[1][2];
if (m12 < (1 - (real_t)CMP_EPSILON)) {
if (m12 > -(1 - (real_t)CMP_EPSILON)) {
// is this a pure X rotation?
if (rows[1][0] == 0 && rows[0][1] == 0 && rows[0][2] == 0 && rows[2][0] == 0 && rows[0][0] == 1) {
// return the simplest form (human friendlier in editor and scripts)
euler.x = atan2(-m12, rows[1][1]);
euler.y = 0;
euler.z = 0;
} else {
euler.x = asin(-m12);
euler.y = atan2(rows[0][2], rows[2][2]);
euler.z = atan2(rows[1][0], rows[1][1]);
}
} else { // m12 == -1
euler.x = Math_PI * 0.5;
euler.y = atan2(rows[0][1], rows[0][0]);
euler.z = 0;
}
} else { // m12 == 1
euler.x = -Math_PI * 0.5;
euler.y = -atan2(rows[0][1], rows[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, c, -s, 0, s, c);
c = Math::cos(p_euler.y);
s = Math::sin(p_euler.y);
Basis ymat(c, 0, s, 0, 1, 0, -s, 0, c);
c = Math::cos(p_euler.z);
s = Math::sin(p_euler.z);
Basis zmat(c, -s, 0, s, c, 0, 0, 0, 1);
//optimizer will optimize away all this anyway
*this = ymat * xmat * zmat;
}
Vector3 Basis::get_euler_zxy() const {
// Euler angles in ZXY convention.
// See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
//
// rot = cz*cy-sz*sx*sy -cx*sz cz*sy+cy*sz*sx
// cy*sz+cz*sx*sy cz*cx sz*sy-cz*cy*sx
// -cx*sy sx cx*cy
Vector3 euler;
real_t sx = rows[2][1];
if (sx < (1 - (real_t)CMP_EPSILON)) {
if (sx > -(1 - (real_t)CMP_EPSILON)) {
euler.x = Math::asin(sx);
euler.y = Math::atan2(-rows[2][0], rows[2][2]);
euler.z = Math::atan2(-rows[0][1], rows[1][1]);
} else {
// It's -1
euler.x = -Math_PI / 2.0;
euler.y = Math::atan2(rows[0][2], rows[0][0]);
euler.z = 0;
}
} else {
// It's 1
euler.x = Math_PI / 2.0;
euler.y = Math::atan2(rows[0][2], rows[0][0]);
euler.z = 0;
}
return euler;
}
void Basis::set_euler_zxy(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, c, -s, 0, s, c);
c = Math::cos(p_euler.y);
s = Math::sin(p_euler.y);
Basis ymat(c, 0, s, 0, 1, 0, -s, 0, c);
c = Math::cos(p_euler.z);
s = Math::sin(p_euler.z);
Basis zmat(c, -s, 0, s, c, 0, 0, 0, 1);
*this = zmat * xmat * ymat;
}
Vector3 Basis::get_euler_zyx() const {
// Euler angles in ZYX convention.
// See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
//
// rot = cz*cy cz*sy*sx-cx*sz sz*sx+cz*cx*cy
// cy*sz cz*cx+sz*sy*sx cx*sz*sy-cz*sx
// -sy cy*sx cy*cx
Vector3 euler;
real_t sy = rows[2][0];
if (sy < (1 - (real_t)CMP_EPSILON)) {
if (sy > -(1 - (real_t)CMP_EPSILON)) {
euler.x = Math::atan2(rows[2][1], rows[2][2]);
euler.y = Math::asin(-sy);
euler.z = Math::atan2(rows[1][0], rows[0][0]);
} else {
// It's -1
euler.x = 0;
euler.y = Math_PI / 2.0;
euler.z = -Math::atan2(rows[0][1], rows[1][1]);
}
} else {
// It's 1
euler.x = 0;
euler.y = -Math_PI / 2.0;
euler.z = -Math::atan2(rows[0][1], rows[1][1]);
}
return euler;
}
void Basis::set_euler_zyx(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, c, -s, 0, s, c);
c = Math::cos(p_euler.y);
s = Math::sin(p_euler.y);
Basis ymat(c, 0, s, 0, 1, 0, -s, 0, c);
c = Math::cos(p_euler.z);
s = Math::sin(p_euler.z);
Basis zmat(c, -s, 0, s, c, 0, 0, 0, 1);
*this = zmat * ymat * xmat;
}
bool Basis::is_equal_approx(const Basis &p_basis) const {
return rows[0].is_equal_approx(p_basis.rows[0]) && rows[1].is_equal_approx(p_basis.rows[1]) && rows[2].is_equal_approx(p_basis.rows[2]);
}
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.rows[i][j], b.rows[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 (rows[i][j] != p_matrix.rows[i][j]) {
return false;
}
}
}
return true;
}
bool Basis::operator!=(const Basis &p_matrix) const {
return (!(*this == p_matrix));
}
Basis::operator String() const {
return "[X: " + get_axis(0).operator String() +
", Y: " + get_axis(1).operator String() +
", Z: " + get_axis(2).operator String() + "]";
}
Quaternion Basis::get_quaternion() const {
#ifdef MATH_CHECKS
ERR_FAIL_COND_V_MSG(!is_rotation(), Quaternion(), "Basis must be normalized in order to be casted to a Quaternion. Use get_rotation_quaternion() or call orthonormalized() if the Basis contains linearly independent vectors.");
#endif
/* Allow getting a quaternion from an unnormalized transform */
Basis m = *this;
real_t trace = m.rows[0][0] + m.rows[1][1] + m.rows[2][2];
real_t temp[4];
if (trace > 0) {
real_t s = Math::sqrt(trace + 1);
temp[3] = (s * 0.5f);
s = 0.5f / s;
temp[0] = ((m.rows[2][1] - m.rows[1][2]) * s);
temp[1] = ((m.rows[0][2] - m.rows[2][0]) * s);
temp[2] = ((m.rows[1][0] - m.rows[0][1]) * s);
} else {
int i = m.rows[0][0] < m.rows[1][1]
? (m.rows[1][1] < m.rows[2][2] ? 2 : 1)
: (m.rows[0][0] < m.rows[2][2] ? 2 : 0);
int j = (i + 1) % 3;
int k = (i + 2) % 3;
real_t s = Math::sqrt(m.rows[i][i] - m.rows[j][j] - m.rows[k][k] + 1);
temp[i] = s * 0.5f;
s = 0.5f / s;
temp[3] = (m.rows[k][j] - m.rows[j][k]) * s;
temp[j] = (m.rows[j][i] + m.rows[i][j]) * s;
temp[k] = (m.rows[k][i] + m.rows[i][k]) * s;
}
return Quaternion(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 > 0.5f) {
v = 1;
} else if (v < -0.5f) {
v = -1;
} 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
*/
// https://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToAngle/index.htm
real_t x, y, z; // Variables for result.
if (Math::is_zero_approx(rows[0][1] - rows[1][0]) && Math::is_zero_approx(rows[0][2] - rows[2][0]) && Math::is_zero_approx(rows[1][2] - rows[2][1])) {
// Singularity found.
// First check for identity matrix which must have +1 for all terms in leading diagonal and zero in other terms.
if (is_diagonal() && (Math::abs(rows[0][0] + rows[1][1] + rows[2][2] - 3) < 3 * CMP_EPSILON)) {
// This singularity is identity matrix so angle = 0.
r_axis = Vector3(0, 1, 0);
r_angle = 0;
return;
}
// Otherwise this singularity is angle = 180
real_t xx = (rows[0][0] + 1) / 2;
real_t yy = (rows[1][1] + 1) / 2;
real_t zz = (rows[2][2] + 1) / 2;
real_t xy = (rows[0][1] + rows[1][0]) / 4;
real_t xz = (rows[0][2] + rows[2][0]) / 4;
real_t yz = (rows[1][2] + rows[2][1]) / 4;
if ((xx > yy) && (xx > zz)) { // rows[0][0] is the largest diagonal term.
if (xx < CMP_EPSILON) {
x = 0;
y = Math_SQRT12;
z = Math_SQRT12;
} else {
x = Math::sqrt(xx);
y = xy / x;
z = xz / x;
}
} else if (yy > zz) { // rows[1][1] is the largest diagonal term.
if (yy < CMP_EPSILON) {
x = Math_SQRT12;
y = 0;
z = Math_SQRT12;
} else {
y = Math::sqrt(yy);
x = xy / y;
z = yz / y;
}
} else { // rows[2][2] is the largest diagonal term so base result on this.
if (zz < CMP_EPSILON) {
x = Math_SQRT12;
y = Math_SQRT12;
z = 0;
} else {
z = Math::sqrt(zz);
x = xz / z;
y = yz / z;
}
}
r_axis = Vector3(x, y, z);
r_angle = Math_PI;
return;
}
// As we have reached here there are no singularities so we can handle normally
real_t s = Math::sqrt((rows[2][1] - rows[1][2]) * (rows[2][1] - rows[1][2]) + (rows[0][2] - rows[2][0]) * (rows[0][2] - rows[2][0]) + (rows[1][0] - rows[0][1]) * (rows[1][0] - rows[0][1])); // Used to normalise.
if (Math::abs(s) < CMP_EPSILON) {
// Prevent divide by zero, should not happen if matrix is orthogonal and should be caught by singularity test above.
s = 1;
}
x = (rows[2][1] - rows[1][2]) / s;
y = (rows[0][2] - rows[2][0]) / s;
z = (rows[1][0] - rows[0][1]) / s;
r_axis = Vector3(x, y, z);
// acos does clamping.
r_angle = Math::acos((rows[0][0] + rows[1][1] + rows[2][2] - 1) / 2);
}
void Basis::set_quaternion(const Quaternion &p_quat) {
real_t d = p_quat.length_squared();
real_t s = 2 / 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 - (yy + zz), xy - wz, xz + wy,
xy + wz, 1 - (xx + zz), yz - wx,
xz - wy, yz + wx, 1 - (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_MSG(!p_axis.is_normalized(), "The axis Vector3 must be 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);
rows[0][0] = axis_sq.x + cosine * (1 - axis_sq.x);
rows[1][1] = axis_sq.y + cosine * (1 - axis_sq.y);
rows[2][2] = axis_sq.z + cosine * (1 - 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;
rows[0][1] = xyzt - zyxs;
rows[1][0] = xyzt + zyxs;
xyzt = p_axis.x * p_axis.z * t;
zyxs = p_axis.y * sine;
rows[0][2] = xyzt + zyxs;
rows[2][0] = xyzt - zyxs;
xyzt = p_axis.y * p_axis.z * t;
zyxs = p_axis.x * sine;
rows[1][2] = xyzt - zyxs;
rows[2][1] = xyzt + zyxs;
}
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_quaternion_scale(const Quaternion &p_quat, const Vector3 &p_scale) {
set_diagonal(p_scale);
rotate(p_quat);
}
void Basis::set_diagonal(const Vector3 &p_diag) {
rows[0][0] = p_diag.x;
rows[0][1] = 0;
rows[0][2] = 0;
rows[1][0] = 0;
rows[1][1] = p_diag.y;
rows[1][2] = 0;
rows[2][0] = 0;
rows[2][1] = 0;
rows[2][2] = p_diag.z;
}
Basis Basis::slerp(const Basis &p_to, const real_t p_weight) const {
//consider scale
Quaternion from(*this);
Quaternion to(p_to);
Basis b(from.slerp(to, p_weight));
b.rows[0] *= Math::lerp(rows[0].length(), p_to.rows[0].length(), p_weight);
b.rows[1] *= Math::lerp(rows[1].length(), p_to.rows[1].length(), p_weight);
b.rows[2] *= Math::lerp(rows[2].length(), p_to.rows[2].length(), p_weight);
return b;
}
void Basis::rotate_sh(real_t *p_values) {
// code by John Hable
// http://filmicworlds.com/blog/simple-and-fast-spherical-harmonic-rotation/
// this code is Public Domain
const static real_t s_c3 = 0.94617469575; // (3*sqrt(5))/(4*sqrt(pi))
const static real_t s_c4 = -0.31539156525; // (-sqrt(5))/(4*sqrt(pi))
const static real_t s_c5 = 0.54627421529; // (sqrt(15))/(4*sqrt(pi))
const static real_t s_c_scale = 1.0 / 0.91529123286551084;
const static real_t s_c_scale_inv = 0.91529123286551084;
const static real_t s_rc2 = 1.5853309190550713 * s_c_scale;
const static real_t s_c4_div_c3 = s_c4 / s_c3;
const static real_t s_c4_div_c3_x2 = (s_c4 / s_c3) * 2.0;
const static real_t s_scale_dst2 = s_c3 * s_c_scale_inv;
const static real_t s_scale_dst4 = s_c5 * s_c_scale_inv;
const real_t src[9] = { p_values[0], p_values[1], p_values[2], p_values[3], p_values[4], p_values[5], p_values[6], p_values[7], p_values[8] };
real_t m00 = rows[0][0];
real_t m01 = rows[0][1];
real_t m02 = rows[0][2];
real_t m10 = rows[1][0];
real_t m11 = rows[1][1];
real_t m12 = rows[1][2];
real_t m20 = rows[2][0];
real_t m21 = rows[2][1];
real_t m22 = rows[2][2];
p_values[0] = src[0];
p_values[1] = m11 * src[1] - m12 * src[2] + m10 * src[3];
p_values[2] = -m21 * src[1] + m22 * src[2] - m20 * src[3];
p_values[3] = m01 * src[1] - m02 * src[2] + m00 * src[3];
real_t sh0 = src[7] + src[8] + src[8] - src[5];
real_t sh1 = src[4] + s_rc2 * src[6] + src[7] + src[8];
real_t sh2 = src[4];
real_t sh3 = -src[7];
real_t sh4 = -src[5];
// Rotations. R0 and R1 just use the raw matrix columns
real_t r2x = m00 + m01;
real_t r2y = m10 + m11;
real_t r2z = m20 + m21;
real_t r3x = m00 + m02;
real_t r3y = m10 + m12;
real_t r3z = m20 + m22;
real_t r4x = m01 + m02;
real_t r4y = m11 + m12;
real_t r4z = m21 + m22;
// dense matrix multiplication one column at a time
// column 0
real_t sh0_x = sh0 * m00;
real_t sh0_y = sh0 * m10;
real_t d0 = sh0_x * m10;
real_t d1 = sh0_y * m20;
real_t d2 = sh0 * (m20 * m20 + s_c4_div_c3);
real_t d3 = sh0_x * m20;
real_t d4 = sh0_x * m00 - sh0_y * m10;
// column 1
real_t sh1_x = sh1 * m02;
real_t sh1_y = sh1 * m12;
d0 += sh1_x * m12;
d1 += sh1_y * m22;
d2 += sh1 * (m22 * m22 + s_c4_div_c3);
d3 += sh1_x * m22;
d4 += sh1_x * m02 - sh1_y * m12;
// column 2
real_t sh2_x = sh2 * r2x;
real_t sh2_y = sh2 * r2y;
d0 += sh2_x * r2y;
d1 += sh2_y * r2z;
d2 += sh2 * (r2z * r2z + s_c4_div_c3_x2);
d3 += sh2_x * r2z;
d4 += sh2_x * r2x - sh2_y * r2y;
// column 3
real_t sh3_x = sh3 * r3x;
real_t sh3_y = sh3 * r3y;
d0 += sh3_x * r3y;
d1 += sh3_y * r3z;
d2 += sh3 * (r3z * r3z + s_c4_div_c3_x2);
d3 += sh3_x * r3z;
d4 += sh3_x * r3x - sh3_y * r3y;
// column 4
real_t sh4_x = sh4 * r4x;
real_t sh4_y = sh4 * r4y;
d0 += sh4_x * r4y;
d1 += sh4_y * r4z;
d2 += sh4 * (r4z * r4z + s_c4_div_c3_x2);
d3 += sh4_x * r4z;
d4 += sh4_x * r4x - sh4_y * r4y;
// extra multipliers
p_values[4] = d0;
p_values[5] = -d1;
p_values[6] = d2 * s_scale_dst2;
p_values[7] = -d3;
p_values[8] = d4 * s_scale_dst4;
}
Basis Basis::looking_at(const Vector3 &p_target, const Vector3 &p_up) {
#ifdef MATH_CHECKS
ERR_FAIL_COND_V_MSG(p_target.is_equal_approx(Vector3()), Basis(), "The target vector can't be zero.");
ERR_FAIL_COND_V_MSG(p_up.is_equal_approx(Vector3()), Basis(), "The up vector can't be zero.");
#endif
Vector3 v_z = -p_target.normalized();
Vector3 v_x = p_up.cross(v_z);
#ifdef MATH_CHECKS
ERR_FAIL_COND_V_MSG(v_x.is_equal_approx(Vector3()), Basis(), "The target vector and up vector can't be parallel to each other.");
#endif
v_x.normalize();
Vector3 v_y = v_z.cross(v_x);
Basis basis;
basis.set_columns(v_x, v_y, v_z);
return basis;
}
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