1024 lines
32 KiB
C++
1024 lines
32 KiB
C++
/*************************************************************************/
|
|
/* basis.cpp */
|
|
/*************************************************************************/
|
|
/* This file is part of: */
|
|
/* GODOT ENGINE */
|
|
/* https://godotengine.org */
|
|
/*************************************************************************/
|
|
/* Copyright (c) 2007-2022 Juan Linietsky, Ariel Manzur. */
|
|
/* Copyright (c) 2014-2022 Godot Engine contributors (cf. AUTHORS.md). */
|
|
/* */
|
|
/* Permission is hereby granted, free of charge, to any person obtaining */
|
|
/* a copy of this software and associated documentation files (the */
|
|
/* "Software"), to deal in the Software without restriction, including */
|
|
/* without limitation the rights to use, copy, modify, merge, publish, */
|
|
/* distribute, sublicense, and/or sell copies of the Software, and to */
|
|
/* permit persons to whom the Software is furnished to do so, subject to */
|
|
/* the following conditions: */
|
|
/* */
|
|
/* The above copyright notice and this permission notice shall be */
|
|
/* included in all copies or substantial portions of the Software. */
|
|
/* */
|
|
/* THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, */
|
|
/* EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF */
|
|
/* MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.*/
|
|
/* IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY */
|
|
/* CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, */
|
|
/* TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE */
|
|
/* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. */
|
|
/*************************************************************************/
|
|
|
|
#include "basis.h"
|
|
|
|
#include "core/math/math_funcs.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) > (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);
|
|
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 / 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 / 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_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 identity;
|
|
Basis m = (*this) * transposed();
|
|
|
|
return m.is_equal_approx(identity);
|
|
}
|
|
|
|
bool Basis::is_diagonal() const {
|
|
return (
|
|
Math::is_zero_approx(elements[0][1]) && Math::is_zero_approx(elements[0][2]) &&
|
|
Math::is_zero_approx(elements[1][0]) && Math::is_zero_approx(elements[1][2]) &&
|
|
Math::is_zero_approx(elements[2][0]) && Math::is_zero_approx(elements[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(elements[0][1], elements[1][0], (real_t)UNIT_EPSILON)) {
|
|
return false;
|
|
}
|
|
if (!Math::is_equal_approx_ratio(elements[0][2], elements[2][0], (real_t)UNIT_EPSILON)) {
|
|
return false;
|
|
}
|
|
if (!Math::is_equal_approx_ratio(elements[1][2], elements[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 = 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 > (real_t)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 = 0.5f * 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 * 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_angle) const {
|
|
return Basis(p_axis, p_angle) * (*this);
|
|
}
|
|
|
|
void Basis::rotate(const Vector3 &p_axis, real_t p_angle) {
|
|
*this = rotated(p_axis, p_angle);
|
|
}
|
|
|
|
void Basis::rotate_local(const Vector3 &p_axis, real_t p_angle) {
|
|
// performs a rotation in object-local coordinate system:
|
|
// M -> (M.R.Minv).M = M.R.
|
|
*this = rotated_local(p_axis, p_angle);
|
|
}
|
|
Basis Basis::rotated_local(const Vector3 &p_axis, real_t p_angle) const {
|
|
return (*this) * Basis(p_axis, p_angle);
|
|
}
|
|
|
|
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;
|
|
real_t sy = elements[0][2];
|
|
if (sy < (1 - (real_t)CMP_EPSILON)) {
|
|
if (sy > -(1 - (real_t)CMP_EPSILON)) {
|
|
// is this a pure Y rotation?
|
|
if (elements[1][0] == 0 && elements[0][1] == 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[2][1], elements[1][1]);
|
|
euler.y = -Math_PI / 2.0;
|
|
euler.z = 0.0;
|
|
}
|
|
} else {
|
|
euler.x = Math::atan2(elements[2][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, 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 = elements[0][1];
|
|
if (sz < (1 - (real_t)CMP_EPSILON)) {
|
|
if (sz > -(1 - (real_t)CMP_EPSILON)) {
|
|
euler.x = Math::atan2(elements[2][1], elements[1][1]);
|
|
euler.y = Math::atan2(elements[0][2], elements[0][0]);
|
|
euler.z = Math::asin(-sz);
|
|
} else {
|
|
// It's -1
|
|
euler.x = -Math::atan2(elements[1][2], elements[2][2]);
|
|
euler.y = 0.0;
|
|
euler.z = Math_PI / 2.0;
|
|
}
|
|
} else {
|
|
// It's 1
|
|
euler.x = -Math::atan2(elements[1][2], elements[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 = elements[1][0];
|
|
if (sz < (1 - (real_t)CMP_EPSILON)) {
|
|
if (sz > -(1 - (real_t)CMP_EPSILON)) {
|
|
euler.x = Math::atan2(-elements[1][2], elements[1][1]);
|
|
euler.y = Math::atan2(-elements[2][0], elements[0][0]);
|
|
euler.z = Math::asin(sz);
|
|
} else {
|
|
// It's -1
|
|
euler.x = Math::atan2(elements[2][1], elements[2][2]);
|
|
euler.y = 0.0;
|
|
euler.z = -Math_PI / 2.0;
|
|
}
|
|
} else {
|
|
// It's 1
|
|
euler.x = Math::atan2(elements[2][1], elements[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 = elements[1][2];
|
|
|
|
if (m12 < (1 - (real_t)CMP_EPSILON)) {
|
|
if (m12 > -(1 - (real_t)CMP_EPSILON)) {
|
|
// 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, 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 = elements[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(-elements[2][0], elements[2][2]);
|
|
euler.z = Math::atan2(-elements[0][1], elements[1][1]);
|
|
} else {
|
|
// It's -1
|
|
euler.x = -Math_PI / 2.0;
|
|
euler.y = Math::atan2(elements[0][2], elements[0][0]);
|
|
euler.z = 0;
|
|
}
|
|
} else {
|
|
// It's 1
|
|
euler.x = Math_PI / 2.0;
|
|
euler.y = Math::atan2(elements[0][2], elements[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 = elements[2][0];
|
|
if (sy < (1 - (real_t)CMP_EPSILON)) {
|
|
if (sy > -(1 - (real_t)CMP_EPSILON)) {
|
|
euler.x = Math::atan2(elements[2][1], elements[2][2]);
|
|
euler.y = Math::asin(-sy);
|
|
euler.z = Math::atan2(elements[1][0], elements[0][0]);
|
|
} else {
|
|
// It's -1
|
|
euler.x = 0;
|
|
euler.y = Math_PI / 2.0;
|
|
euler.z = -Math::atan2(elements[0][1], elements[1][1]);
|
|
}
|
|
} else {
|
|
// It's 1
|
|
euler.x = 0;
|
|
euler.y = -Math_PI / 2.0;
|
|
euler.z = -Math::atan2(elements[0][1], elements[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 elements[0].is_equal_approx(p_basis.elements[0]) && elements[1].is_equal_approx(p_basis.elements[1]) && elements[2].is_equal_approx(p_basis.elements[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.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
|
|
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() if the Basis contains linearly independent vectors.");
|
|
#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) {
|
|
real_t s = Math::sqrt(trace + 1);
|
|
temp[3] = (s * 0.5f);
|
|
s = 0.5f / 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);
|
|
temp[i] = s * 0.5f;
|
|
s = 0.5f / 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 > 0.5f) {
|
|
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
|
|
*/
|
|
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 = Math_SQRT12;
|
|
z = Math_SQRT12;
|
|
} 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 = Math_SQRT12;
|
|
y = 0;
|
|
z = Math_SQRT12;
|
|
} 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 = 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 = 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 / 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_angle) {
|
|
// 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_angle);
|
|
elements[0][0] = axis_sq.x + cosine * (1 - axis_sq.x);
|
|
elements[1][1] = axis_sq.y + cosine * (1 - axis_sq.y);
|
|
elements[2][2] = axis_sq.z + cosine * (1 - axis_sq.z);
|
|
|
|
real_t sine = Math::sin(p_angle);
|
|
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;
|
|
}
|
|
|
|
void Basis::set_axis_angle_scale(const Vector3 &p_axis, real_t p_angle, const Vector3 &p_scale) {
|
|
set_diagonal(p_scale);
|
|
rotate(p_axis, p_angle);
|
|
}
|
|
|
|
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 &p_to, const real_t &p_weight) const {
|
|
//consider scale
|
|
Quat from(*this);
|
|
Quat to(p_to);
|
|
|
|
Basis b(from.slerp(to, p_weight));
|
|
b.elements[0] *= Math::lerp(elements[0].length(), p_to.elements[0].length(), p_weight);
|
|
b.elements[1] *= Math::lerp(elements[1].length(), p_to.elements[1].length(), p_weight);
|
|
b.elements[2] *= Math::lerp(elements[2].length(), p_to.elements[2].length(), p_weight);
|
|
|
|
return b;
|
|
}
|