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godot/core/math/matrix3.cpp

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/*************************************************************************/
/* matrix3.cpp */
/*************************************************************************/
/* This file is part of: */
/* GODOT ENGINE */
/* http://www.godotengine.org */
/*************************************************************************/
/* Copyright (c) 2007-2017 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 "matrix3.h"
#include "math_funcs.h"
#include "os/copymem.h"
#define cofac(row1,col1, row2, col2)\
(elements[row1][col1] * elements[row2][col2] - elements[row1][col2] * elements[row2][col1])
void Basis::from_z(const Vector3& p_z) {
if (Math::abs(p_z.z) > Math_SQRT12 ) {
// choose p in y-z plane
real_t a = p_z[1]*p_z[1] + p_z[2]*p_z[2];
real_t k = 1.0/Math::sqrt(a);
elements[0]=Vector3(0,-p_z[2]*k,p_z[1]*k);
elements[1]=Vector3(a*k,-p_z[0]*elements[0][2],p_z[0]*elements[0][1]);
} else {
// choose p in x-y plane
real_t a = p_z.x*p_z.x + p_z.y*p_z.y;
real_t k = 1.0/Math::sqrt(a);
elements[0]=Vector3(-p_z.y*k,p_z.x*k,0);
elements[1]=Vector3(-p_z.z*elements[0].y,p_z.z*elements[0].x,a*k);
}
elements[2]=p_z;
}
void Basis::invert() {
real_t co[3]={
cofac(1, 1, 2, 2), cofac(1, 2, 2, 0), cofac(1, 0, 2, 1)
};
real_t det = elements[0][0] * co[0]+
elements[0][1] * co[1]+
elements[0][2] * co[2];
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 );
}
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 c = *this;
c.orthonormalize();
return c;
}
bool Basis::is_orthogonal() const {
Basis id;
Basis m = (*this)*transposed();
return isequal_approx(id,m);
}
bool Basis::is_rotation() const {
return Math::isequal_approx(determinant(), 1) && is_orthogonal();
}
bool Basis::is_symmetric() const {
if (Math::abs(elements[0][1] - elements[1][0]) > CMP_EPSILON)
return false;
if (Math::abs(elements[0][2] - elements[2][0]) > CMP_EPSILON)
return false;
if (Math::abs(elements[1][2] - elements[2][1]) > CMP_EPSILON)
return false;
return true;
}
Basis Basis::diagonalize() {
//NOTE: only implemented for symmetric matrices
//with the Jacobi iterative method method
ERR_FAIL_COND_V(!is_symmetric(), 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 (Math::abs(elements[j][j] - elements[i][i]) < CMP_EPSILON) {
angle = Math_PI / 4;
} else {
angle = 0.5 * Math::atan(2 * elements[i][j] / (elements[j][j] - elements[i][i]));
}
// Compute the rotation matrix
Basis rot;
rot.elements[i][i] = rot.elements[j][j] = Math::cos(angle);
rot.elements[i][j] = - (rot.elements[j][i] = Math::sin(angle));
// Update the off matrix norm
off_matrix_norm_2 -= elements[i][j] * elements[i][j];
// Apply the rotation
*this = rot * *this * rot.transposed();
acc_rot = rot * acc_rot;
}
return acc_rot;
}
Basis Basis::inverse() const {
Basis inv=*this;
inv.invert();
return inv;
}
void Basis::transpose() {
SWAP(elements[0][1],elements[1][0]);
SWAP(elements[0][2],elements[2][0]);
SWAP(elements[1][2],elements[2][1]);
}
Basis Basis::transposed() const {
Basis tr=*this;
tr.transpose();
return tr;
}
// Multiplies the matrix from left by the scaling matrix: M -> S.M
// See the comment for Basis::rotated for further explanation.
void Basis::scale(const Vector3& p_scale) {
elements[0][0]*=p_scale.x;
elements[0][1]*=p_scale.x;
elements[0][2]*=p_scale.x;
elements[1][0]*=p_scale.y;
elements[1][1]*=p_scale.y;
elements[1][2]*=p_scale.y;
elements[2][0]*=p_scale.z;
elements[2][1]*=p_scale.z;
elements[2][2]*=p_scale.z;
}
Basis Basis::scaled( const Vector3& p_scale ) const {
Basis m = *this;
m.scale(p_scale);
return m;
}
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 Quat 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()
);
}
// 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);
}
Basis Basis::rotated(const Vector3& p_euler) const {
return Basis(p_euler) * (*this);
}
void Basis::rotate(const Vector3& p_euler) {
*this = rotated(p_euler);
}
Vector3 Basis::get_rotation() 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();
}
// get_euler 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() 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);
euler.y = Math::asin(elements[0][2]);
if ( euler.y < Math_PI*0.5) {
if ( euler.y > -Math_PI*0.5) {
euler.x = Math::atan2(-elements[1][2],elements[2][2]);
euler.z = Math::atan2(-elements[0][1],elements[0][0]);
} else {
real_t r = Math::atan2(elements[1][0],elements[1][1]);
euler.z = 0.0;
euler.x = euler.z - r;
}
} else {
real_t r = Math::atan2(elements[0][1],elements[1][1]);
euler.z = 0;
euler.x = r - euler.z;
}
return euler;
}
// set_euler expects a vector containing the Euler angles in the format
// (c,b,a), where a is the angle of the first rotation, and c is the last.
// The current implementation uses XYZ convention (Z is the first rotation).
void Basis::set_euler(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);
}
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 (Math::isequal_approx(a.elements[i][j],b.elements[i][j]) == false)
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;
}
Basis::operator Quat() const {
ERR_FAIL_COND_V(is_rotation() == false, Quat());
real_t trace = elements[0][0] + elements[1][1] + 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]=((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 = Math::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 Quat(temp[0],temp[1],temp[2],temp[3]);
}
static const Basis _ortho_bases[24]={
Basis(1, 0, 0, 0, 1, 0, 0, 0, 1),
Basis(0, -1, 0, 1, 0, 0, 0, 0, 1),
Basis(-1, 0, 0, 0, -1, 0, 0, 0, 1),
Basis(0, 1, 0, -1, 0, 0, 0, 0, 1),
Basis(1, 0, 0, 0, 0, -1, 0, 1, 0),
Basis(0, 0, 1, 1, 0, 0, 0, 1, 0),
Basis(-1, 0, 0, 0, 0, 1, 0, 1, 0),
Basis(0, 0, -1, -1, 0, 0, 0, 1, 0),
Basis(1, 0, 0, 0, -1, 0, 0, 0, -1),
Basis(0, 1, 0, 1, 0, 0, 0, 0, -1),
Basis(-1, 0, 0, 0, 1, 0, 0, 0, -1),
Basis(0, -1, 0, -1, 0, 0, 0, 0, -1),
Basis(1, 0, 0, 0, 0, 1, 0, -1, 0),
Basis(0, 0, -1, 1, 0, 0, 0, -1, 0),
Basis(-1, 0, 0, 0, 0, -1, 0, -1, 0),
Basis(0, 0, 1, -1, 0, 0, 0, -1, 0),
Basis(0, 0, 1, 0, 1, 0, -1, 0, 0),
Basis(0, -1, 0, 0, 0, 1, -1, 0, 0),
Basis(0, 0, -1, 0, -1, 0, -1, 0, 0),
Basis(0, 1, 0, 0, 0, -1, -1, 0, 0),
Basis(0, 0, 1, 0, -1, 0, 1, 0, 0),
Basis(0, 1, 0, 0, 0, 1, 1, 0, 0),
Basis(0, 0, -1, 0, 1, 0, 1, 0, 0),
Basis(0, -1, 0, 0, 0, -1, 1, 0, 0)
};
int Basis::get_orthogonal_index() const {
//could be sped up if i come up with a way
Basis orth=*this;
for(int i=0;i<3;i++) {
for(int j=0;j<3;j++) {
real_t v = orth[i][j];
if (v>0.5)
v=1.0;
else if (v<-0.5)
v=-1.0;
else
v=0;
orth[i][j]=v;
}
}
for(int i=0;i<24;i++) {
if (_ortho_bases[i]==orth)
return i;
}
return 0;
}
void Basis::set_orthogonal_index(int p_index){
//there only exist 24 orthogonal bases in r3
ERR_FAIL_INDEX(p_index,24);
*this=_ortho_bases[p_index];
}
void Basis::get_axis_and_angle(Vector3 &r_axis,real_t& r_angle) const {
ERR_FAIL_COND(is_rotation() == false);
real_t angle,x,y,z; // variables for result
real_t epsilon = 0.01; // margin to allow for rounding errors
real_t epsilon2 = 0.1; // margin to distinguish between 0 and 180 degrees
if ( (Math::abs(elements[1][0]-elements[0][1])< epsilon)
&& (Math::abs(elements[2][0]-elements[0][2])< epsilon)
&& (Math::abs(elements[2][1]-elements[1][2])< epsilon)) {
// singularity found
// first check for identity matrix which must have +1 for all terms
// in leading diagonaland zero in other terms
if ((Math::abs(elements[1][0]+elements[0][1]) < epsilon2)
&& (Math::abs(elements[2][0]+elements[0][2]) < epsilon2)
&& (Math::abs(elements[2][1]+elements[1][2]) < epsilon2)
&& (Math::abs(elements[0][0]+elements[1][1]+elements[2][2]-3) < epsilon2)) {
// this singularity is identity matrix so angle = 0
r_axis=Vector3(0,1,0);
r_angle=0;
return;
}
// otherwise this singularity is angle = 180
angle = Math_PI;
real_t xx = (elements[0][0]+1)/2;
real_t yy = (elements[1][1]+1)/2;
real_t zz = (elements[2][2]+1)/2;
real_t xy = (elements[1][0]+elements[0][1])/4;
real_t xz = (elements[2][0]+elements[0][2])/4;
real_t yz = (elements[2][1]+elements[1][2])/4;
if ((xx > yy) && (xx > zz)) { // elements[0][0] is the largest diagonal term
if (xx< epsilon) {
x = 0;
y = 0.7071;
z = 0.7071;
} else {
x = Math::sqrt(xx);
y = xy/x;
z = xz/x;
}
} else if (yy > zz) { // elements[1][1] is the largest diagonal term
if (yy< epsilon) {
x = 0.7071;
y = 0;
z = 0.7071;
} else {
y = Math::sqrt(yy);
x = xy/y;
z = yz/y;
}
} else { // elements[2][2] is the largest diagonal term so base result on this
if (zz< epsilon) {
x = 0.7071;
y = 0.7071;
z = 0;
} else {
z = Math::sqrt(zz);
x = xz/z;
y = yz/z;
}
}
r_axis=Vector3(x,y,z);
r_angle=angle;
return;
}
// as we have reached here there are no singularities so we can handle normally
real_t s = Math::sqrt((elements[1][2] - elements[2][1])*(elements[1][2] - elements[2][1])
+(elements[2][0] - elements[0][2])*(elements[2][0] - elements[0][2])
+(elements[0][1] - elements[1][0])*(elements[0][1] - elements[1][0])); // s=|axis||sin(angle)|, used to normalise
angle = Math::acos(( elements[0][0] + elements[1][1] + elements[2][2] - 1)/2);
if (angle < 0) s = -s;
x = (elements[2][1] - elements[1][2])/s;
y = (elements[0][2] - elements[2][0])/s;
z = (elements[1][0] - elements[0][1])/s;
r_axis=Vector3(x,y,z);
r_angle=angle;
}
Basis::Basis(const Vector3& p_euler) {
set_euler( p_euler );
}
Basis::Basis(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)) ;
}
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= Math::cos(p_phi);
real_t sine= Math::sin(p_phi);
elements[0][0] = axis_sq.x + cosine * ( 1.0 - axis_sq.x );
elements[0][1] = p_axis.x * p_axis.y * ( 1.0 - cosine ) - p_axis.z * sine;
elements[0][2] = p_axis.z * p_axis.x * ( 1.0 - cosine ) + p_axis.y * sine;
elements[1][0] = p_axis.x * p_axis.y * ( 1.0 - cosine ) + p_axis.z * sine;
elements[1][1] = axis_sq.y + cosine * ( 1.0 - axis_sq.y );
elements[1][2] = p_axis.y * p_axis.z * ( 1.0 - cosine ) - p_axis.x * sine;
elements[2][0] = p_axis.z * p_axis.x * ( 1.0 - cosine ) - p_axis.y * sine;
elements[2][1] = p_axis.y * p_axis.z * ( 1.0 - cosine ) + p_axis.x * sine;
elements[2][2] = axis_sq.z + cosine * ( 1.0 - axis_sq.z );
}
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