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godot-cpp/src/core/Basis.cpp

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#include "Basis.hpp"
#include "Defs.hpp"
#include "Vector3.hpp"
#include "Quat.hpp"
#include <algorithm>
namespace godot {
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;
}
const Vector3& Basis::operator[](int axis) const {
return elements[axis];
}
Vector3&Basis:: operator[](int axis) {
return elements[axis];
}
#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()
{
std::swap(elements[0][1],elements[1][0]);
std::swap(elements[0][2],elements[2][0]);
std::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 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()
);
}
// 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 "Quat.hpp"
namespace godot {
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= ::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 Quat() const {
//commenting this check because precision issues cause it to fail when it shouldn't
//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 = ::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 Quat(temp[0],temp[1],temp[2],temp[3]);
}
}
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