Fixed Timestep Interpolation (3D)

Adds 3D fixed timestep interpolation to the rendering server.
This does not yet include support for multimeshes or particles.

Co-authored-by: lawnjelly <lawnjelly@gmail.com>

core/math/transform_interpolator.cpp

@@ -31,6 +31,7 @@
 #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) {
 	// Extract parameters.
@@ -74,3 +75,340 @@ void TransformInterpolator::interpolate_transform_2d(const Transform2D &p_prev,
 	r_result = Transform2D(Math::atan2(v.y, v.x), p1.lerp(p2, p_fraction));
 	r_result.scale_basis(s1.lerp(s2, 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, TransformInterpolator::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, TransformInterpolator::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;
+}