/**************************************************************************/
/* transform_interpolator.cpp */
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/* Copyright (c) 2007-2014 Juan Linietsky, Ariel Manzur. */
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#include "transform_interpolator.h"
#include "core/math/transform_2d.h"
#include "core/math/transform_3d.h"
void TransformInterpolator::interpolate_transform_2d(const Transform2D &p_prev, const Transform2D &p_curr, Transform2D &r_result, real_t p_fraction) {
// Special case for physics interpolation, if flipping, don't interpolate basis.
// If the determinant polarity changes, the handedness of the coordinate system changes.
if (_sign(p_prev.determinant()) != _sign(p_curr.determinant())) {
r_result.columns[0] = p_curr.columns[0];
r_result.columns[1] = p_curr.columns[1];
r_result.set_origin(p_prev.get_origin().lerp(p_curr.get_origin(), p_fraction));
return;
}
r_result = p_prev.interpolate_with(p_curr, p_fraction);
}
void TransformInterpolator::interpolate_transform_3d(const Transform3D &p_prev, const Transform3D &p_curr, Transform3D &r_result, real_t p_fraction) {
r_result.origin = p_prev.origin + ((p_curr.origin - p_prev.origin) * p_fraction);
interpolate_basis(p_prev.basis, p_curr.basis, r_result.basis, p_fraction);
}
void TransformInterpolator::interpolate_basis(const Basis &p_prev, const Basis &p_curr, Basis &r_result, real_t p_fraction) {
Method method = find_method(p_prev, p_curr);
interpolate_basis_via_method(p_prev, p_curr, r_result, p_fraction, method);
}
void TransformInterpolator::interpolate_transform_3d_via_method(const Transform3D &p_prev, const Transform3D &p_curr, Transform3D &r_result, real_t p_fraction, Method p_method) {
r_result.origin = p_prev.origin + ((p_curr.origin - p_prev.origin) * p_fraction);
interpolate_basis_via_method(p_prev.basis, p_curr.basis, r_result.basis, p_fraction, p_method);
}
void TransformInterpolator::interpolate_basis_via_method(const Basis &p_prev, const Basis &p_curr, Basis &r_result, real_t p_fraction, Method p_method) {
switch (p_method) {
default: {
interpolate_basis_linear(p_prev, p_curr, r_result, p_fraction);
} break;
case INTERP_SLERP: {
r_result = _basis_slerp_unchecked(p_prev, p_curr, p_fraction);
} break;
case INTERP_SCALED_SLERP: {
interpolate_basis_scaled_slerp(p_prev, p_curr, r_result, p_fraction);
} break;
}
}
Quaternion TransformInterpolator::_basis_to_quat_unchecked(const Basis &p_basis) {
Basis m = p_basis;
real_t trace = m.rows[0][0] + m.rows[1][1] + m.rows[2][2];
real_t temp[4];
if (trace > 0.0) {
real_t s = Math::sqrt(trace + 1.0f);
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] + 1.0f);
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]);
}
Quaternion TransformInterpolator::_quat_slerp_unchecked(const Quaternion &p_from, const Quaternion &p_to, real_t p_fraction) {
Quaternion to1;
real_t omega, cosom, sinom, scale0, scale1;
// Calculate cosine.
cosom = p_from.dot(p_to);
// Adjust signs (if necessary)
if (cosom < 0.0f) {
cosom = -cosom;
to1.x = -p_to.x;
to1.y = -p_to.y;
to1.z = -p_to.z;
to1.w = -p_to.w;
} else {
to1.x = p_to.x;
to1.y = p_to.y;
to1.z = p_to.z;
to1.w = p_to.w;
}
// Calculate coefficients.
// This check could possibly be removed as we dealt with this
// case in the find_method() function, but is left for safety, it probably
// isn't a bottleneck.
if ((1.0f - cosom) > (real_t)CMP_EPSILON) {
// standard case (slerp)
omega = Math::acos(cosom);
sinom = Math::sin(omega);
scale0 = Math::sin((1.0f - p_fraction) * omega) / sinom;
scale1 = Math::sin(p_fraction * omega) / sinom;
} else {
// "from" and "to" quaternions are very close
// ... so we can do a linear interpolation
scale0 = 1.0f - p_fraction;
scale1 = p_fraction;
}
// Calculate final values.
return Quaternion(
scale0 * p_from.x + scale1 * to1.x,
scale0 * p_from.y + scale1 * to1.y,
scale0 * p_from.z + scale1 * to1.z,
scale0 * p_from.w + scale1 * to1.w);
}
Basis TransformInterpolator::_basis_slerp_unchecked(Basis p_from, Basis p_to, real_t p_fraction) {
Quaternion from = _basis_to_quat_unchecked(p_from);
Quaternion to = _basis_to_quat_unchecked(p_to);
Basis b(_quat_slerp_unchecked(from, to, p_fraction));
return b;
}
void TransformInterpolator::interpolate_basis_scaled_slerp(Basis p_prev, Basis p_curr, Basis &r_result, real_t p_fraction) {
// Normalize both and find lengths.
Vector3 lengths_prev = _basis_orthonormalize(p_prev);
Vector3 lengths_curr = _basis_orthonormalize(p_curr);
r_result = _basis_slerp_unchecked(p_prev, p_curr, p_fraction);
// Now the result is unit length basis, we need to scale.
Vector3 lengths_lerped = lengths_prev + ((lengths_curr - lengths_prev) * p_fraction);
// Keep a note that the column / row order of the basis is weird,
// so keep an eye for bugs with this.
r_result[0] *= lengths_lerped;
r_result[1] *= lengths_lerped;
r_result[2] *= lengths_lerped;
}
void TransformInterpolator::interpolate_basis_linear(const Basis &p_prev, const Basis &p_curr, Basis &r_result, real_t p_fraction) {
// Interpolate basis.
r_result = p_prev.lerp(p_curr, p_fraction);
// It turns out we need to guard against zero scale basis.
// This is kind of silly, as we should probably fix the bugs elsewhere in Godot that can't deal with
// zero scale, but until that time...
for (int n = 0; n < 3; n++) {
Vector3 &axis = r_result[n];
// Not ok, this could cause errors due to bugs elsewhere,
// so we will bodge set this to a small value.
const real_t smallest = 0.0001f;
const real_t smallest_squared = smallest * smallest;
if (axis.length_squared() < smallest_squared) {
// Setting a different component to the smallest
// helps prevent the situation where all the axes are pointing in the same direction,
// which could be a problem for e.g. cross products...
axis[n] = smallest;
}
}
}
// Returns length.
real_t TransformInterpolator::_vec3_normalize(Vector3 &p_vec) {
real_t lengthsq = p_vec.length_squared();
if (lengthsq == 0.0f) {
p_vec.x = p_vec.y = p_vec.z = 0.0f;
return 0.0f;
}
real_t length = Math::sqrt(lengthsq);
p_vec.x /= length;
p_vec.y /= length;
p_vec.z /= length;
return length;
}
// Returns lengths.
Vector3 TransformInterpolator::_basis_orthonormalize(Basis &r_basis) {
// Gram-Schmidt Process.
Vector3 x = r_basis.get_column(0);
Vector3 y = r_basis.get_column(1);
Vector3 z = r_basis.get_column(2);
Vector3 lengths;
lengths.x = _vec3_normalize(x);
y = (y - x * (x.dot(y)));
lengths.y = _vec3_normalize(y);
z = (z - x * (x.dot(z)) - y * (y.dot(z)));
lengths.z = _vec3_normalize(z);
r_basis.set_column(0, x);
r_basis.set_column(1, y);
r_basis.set_column(2, z);
return lengths;
}
TransformInterpolator::Method TransformInterpolator::_test_basis(Basis p_basis, bool r_needed_normalize, Quaternion &r_quat) {
// Axis lengths.
Vector3 al = Vector3(p_basis.get_column(0).length_squared(),
p_basis.get_column(1).length_squared(),
p_basis.get_column(2).length_squared());
// Non unit scale?
if (r_needed_normalize || !_vec3_is_equal_approx(al, Vector3(1.0, 1.0, 1.0), (real_t)0.001f)) {
// If the basis is not normalized (at least approximately), it will fail the checks needed for slerp.
// So we try to detect a scaled (but not sheared) basis, which we *can* slerp by normalizing first,
// and lerping the scales separately.
// If any of the axes are really small, it is unlikely to be a valid rotation, or is scaled too small to deal with float error.
const real_t sl_epsilon = 0.00001f;
if ((al.x < sl_epsilon) ||
(al.y < sl_epsilon) ||
(al.z < sl_epsilon)) {
return INTERP_LERP;
}
// Normalize the basis.
Basis norm_basis = p_basis;
al.x = Math::sqrt(al.x);
al.y = Math::sqrt(al.y);
al.z = Math::sqrt(al.z);
norm_basis.set_column(0, norm_basis.get_column(0) / al.x);
norm_basis.set_column(1, norm_basis.get_column(1) / al.y);
norm_basis.set_column(2, norm_basis.get_column(2) / al.z);
// This doesn't appear necessary, as the later checks will catch it.
// if (!_basis_is_orthogonal_any_scale(norm_basis)) {
// return INTERP_LERP;
// }
p_basis = norm_basis;
// Orthonormalize not necessary as normal normalization(!) works if the
// axes are orthonormal.
// p_basis.orthonormalize();
// If we needed to normalize one of the two bases, we will need to normalize both,
// regardless of whether the 2nd needs it, just to make sure it takes the path to return
// INTERP_SCALED_LERP on the 2nd call of _test_basis.
r_needed_normalize = true;
}
// Apply less stringent tests than the built in slerp, the standard Godot slerp
// is too susceptible to float error to be useful.
real_t det = p_basis.determinant();
if (!Math::is_equal_approx(det, 1, (real_t)0.01f)) {
return INTERP_LERP;
}
if (!_basis_is_orthogonal(p_basis)) {
return INTERP_LERP;
}
// TODO: This could possibly be less stringent too, check this.
r_quat = _basis_to_quat_unchecked(p_basis);
if (!r_quat.is_normalized()) {
return INTERP_LERP;
}
return r_needed_normalize ? INTERP_SCALED_SLERP : INTERP_SLERP;
}
// This check doesn't seem to be needed but is preserved in case of bugs.
bool TransformInterpolator::_basis_is_orthogonal_any_scale(const Basis &p_basis) {
Vector3 cross = p_basis.get_column(0).cross(p_basis.get_column(1));
real_t l = _vec3_normalize(cross);
// Too small numbers, revert to lerp.
if (l < 0.001f) {
return false;
}
const real_t epsilon = 0.9995f;
real_t dot = cross.dot(p_basis.get_column(2));
if (dot < epsilon) {
return false;
}
cross = p_basis.get_column(1).cross(p_basis.get_column(2));
l = _vec3_normalize(cross);
// Too small numbers, revert to lerp.
if (l < 0.001f) {
return false;
}
dot = cross.dot(p_basis.get_column(0));
if (dot < epsilon) {
return false;
}
return true;
}
bool TransformInterpolator::_basis_is_orthogonal(const Basis &p_basis, real_t p_epsilon) {
Basis identity;
Basis m = p_basis * p_basis.transposed();
// Less stringent tests than the standard Godot slerp.
if (!_vec3_is_equal_approx(m[0], identity[0], p_epsilon) || !_vec3_is_equal_approx(m[1], identity[1], p_epsilon) || !_vec3_is_equal_approx(m[2], identity[2], p_epsilon)) {
return false;
}
return true;
}
real_t TransformInterpolator::checksum_transform_3d(const Transform3D &p_transform) {
// just a really basic checksum, this can probably be improved
real_t sum = _vec3_sum(p_transform.origin);
sum -= _vec3_sum(p_transform.basis.rows[0]);
sum += _vec3_sum(p_transform.basis.rows[1]);
sum -= _vec3_sum(p_transform.basis.rows[2]);
return sum;
}
TransformInterpolator::Method TransformInterpolator::find_method(const Basis &p_a, const Basis &p_b) {
bool needed_normalize = false;
Quaternion q0;
Method method = _test_basis(p_a, needed_normalize, q0);
if (method == INTERP_LERP) {
return method;
}
Quaternion q1;
method = _test_basis(p_b, needed_normalize, q1);
if (method == INTERP_LERP) {
return method;
}
// Are they close together?
// Apply the same test that will revert to lerp as is present in the slerp routine.
// Calculate cosine.
real_t cosom = Math::abs(q0.dot(q1));
if ((1.0f - cosom) <= (real_t)CMP_EPSILON) {
return INTERP_LERP;
}
return method;
}