Better code for (bone axis + roll) to mat
See T39470 and D436. Code by @tippisum, with some minor edits by @mont29. Tested with various rigs, including Rigify, CGcookie flex rig, and gooseberry/pataz caterpillar. Riggers, please test it, no change expected in behaviour. Reviewers: aligorith CC: tippisum Differential Revision: https://developer.blender.org/D436
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2023-02-14 10:53:20 +01:00
Referenced by issue #39470, New algorithm on computing bone matrix with improved accuracy
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@ -1414,6 +1414,7 @@ void BKE_rotMode_change_values(float quat[4], float eul[3], float axis[3], float
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* pose_mat(b)= arm_mat(b) * chan_mat(b)
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*
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* *************************************************************************** */
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/* Computes vector and roll based on a rotation.
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* "mat" must contain only a rotation, and no scaling. */
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void mat3_to_vec_roll(float mat[3][3], float r_vec[3], float *r_roll)
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@ -1433,52 +1434,95 @@ void mat3_to_vec_roll(float mat[3][3], float r_vec[3], float *r_roll)
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}
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}
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/* Calculates the rest matrix of a bone based
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* On its vector and a roll around that vector */
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/* Calculates the rest matrix of a bone based on its vector and a roll around that vector. */
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/* Given v = (v.x, v.y, v.z) our (normalized) bone vector, we want the rotation matrix M
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* from the Y axis (so that M * (0, 1, 0) = v).
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* -> The rotation axis a lays on XZ plane, and it is orthonormal to v, hence to the projection of v onto XZ plane.
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* -> a = (v.z, 0, -v.x)
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* We know a is eigenvector of M (so M * a = a).
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* Finally, we have w, such that M * w = (0, 1, 0) (i.e. the vector that will be aligned with Y axis once transformed).
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* We know w is symmetric to v by the Y axis.
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* -> w = (-v.x, v.y, -v.z)
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*
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* Solving this, we get (x, y and z being the components of v):
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* ┌ (x^2 * y + z^2) / (x^2 + z^2), x, x * z * (y - 1) / (x^2 + z^2) ┐
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* M = │ x * (y^2 - 1) / (x^2 + z^2), y, z * (y^2 - 1) / (x^2 + z^2) │
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* └ x * z * (y - 1) / (x^2 + z^2), z, (x^2 + z^2 * y) / (x^2 + z^2) ┘
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*
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* This is stable as long as v (the bone) is not too much aligned with +/-Y (i.e. x and z components
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* are not too close to 0).
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*
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* Since v is normalized, we have x^2 + y^2 + z^2 = 1, hence x^2 + z^2 = 1 - y^2 = (1 - y)(1 + y).
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* This allows to simplifies M like this:
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* ┌ 1 - x^2 / (1 + y), x, -x * z / (1 + y) ┐
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* M = │ -x, y, -z │
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* └ -x * z / (1 + y), z, 1 - z^2 / (1 + y) ┘
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*
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* Written this way, we see the case v = +Y is no more a singularity. The only one remaining is the bone being
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* aligned with -Y.
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*
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* Let's handle the asymptotic behavior when bone vector is reaching the limit of y = -1. Each of the four corner
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* elements can vary from -1 to 1, depending on the axis a chosen for doing the rotation. And the "rotation" here
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* is in fact established by mirroring XZ plane by that given axis, then inversing the Y-axis.
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* For sufficiently small x and z, and with y approaching -1, all elements but the four corner ones of M
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* will degenerate. So let's now focus on these corner elements.
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*
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* We rewrite M so that it only contains its four corner elements, and combine the 1 / (1 + y) factor:
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* ┌ 1 + y - x^2, -x * z ┐
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* M* = 1 / (1 + y) * │ │
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* └ -x * z, 1 + y - z^2 ┘
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*
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* When y is close to -1, computing 1 / (1 + y) will cause severe numerical instability, so we ignore it and
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* normalize M instead. We know y^2 = 1 - (x^2 + z^2), and y < 0, hence y = -sqrt(1 - (x^2 + z^2)).
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* Since x and z are both close to 0, we apply the binomial expansion to the first order:
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* y = -sqrt(1 - (x^2 + z^2)) = -1 + (x^2 + z^2) / 2. Which gives:
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* ┌ z^2 - x^2, -2 * x * z ┐
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* M* = 1 / (x^2 + z^2) * │ │
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* └ -2 * x * z, x^2 - z^2 ┘
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*/
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void vec_roll_to_mat3(const float vec[3], const float roll, float mat[3][3])
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{
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float nor[3], axis[3], target[3] = {0, 1, 0};
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float nor[3];
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float theta;
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float rMatrix[3][3], bMatrix[3][3];
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normalize_v3_v3(nor, vec);
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/* Find Axis & Amount for bone matrix */
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cross_v3_v3v3(axis, target, nor);
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theta = 1 + nor[1];
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/* was 0.0000000000001, caused bug [#23954], smaller values give unstable
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* roll when toggling editmode.
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/* With old algo, 1.0e-13f caused T23954 and T31333, 1.0e-6f caused T27675 and T30438,
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* so using 1.0e-9f as best compromise.
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*
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* was 0.00001, causes bug [#27675], with 0.00000495,
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* so a value inbetween these is needed.
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*
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* was 0.000001, causes bug [#30438] (which is same as [#27675, imho).
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* Resetting it to org value seems to cause no more [#23954]...
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*
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* was 0.0000000000001, caused bug [#31333], smaller values give unstable
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* roll when toggling editmode again...
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* No good value here, trying 0.000000001 as best compromise. :/
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* New algo is supposed much more precise, since less complex computations are performed,
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* but it uses two different threshold values...
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*/
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if (len_squared_v3(axis) > 1.0e-9f) {
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/* if nor is *not* a multiple of target ... */
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normalize_v3(axis);
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theta = angle_normalized_v3v3(target, nor);
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/* Make Bone matrix*/
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axis_angle_normalized_to_mat3(bMatrix, axis, theta);
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if (theta > 1.0e-9f) {
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/* nor is *not* -Y.
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* We got these values for free... so be happy with it... ;)
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*/
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bMatrix[0][1] = -nor[0];
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bMatrix[1][0] = nor[0];
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bMatrix[1][1] = nor[1];
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bMatrix[1][2] = nor[2];
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bMatrix[2][1] = -nor[2];
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if (theta > 1.0e-5f) {
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/* If nor is far enough from -Y, apply the general case. */
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bMatrix[0][0] = 1 - nor[0] * nor[0] / theta;
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bMatrix[2][2] = 1 - nor[2] * nor[2] / theta;
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bMatrix[2][0] = bMatrix[0][2] = -nor[0] * nor[2] / theta;
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}
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else {
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/* If nor is too close to -Y, apply the special case. */
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theta = nor[0] * nor[0] + nor[2] * nor[2];
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bMatrix[0][0] = (nor[0] + nor[2]) * (nor[0] - nor[2]) / theta;
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bMatrix[2][2] = -bMatrix[0][0];
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bMatrix[2][0] = bMatrix[0][2] = 2.0f * nor[0] * nor[2] / theta;
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}
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}
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else {
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/* if nor is a multiple of target ... */
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float updown;
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/* point same direction, or opposite? */
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updown = (dot_v3v3(target, nor) > 0) ? 1.0f : -1.0f;
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/* I think this should work... */
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bMatrix[0][0] = updown; bMatrix[0][1] = 0.0; bMatrix[0][2] = 0.0;
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bMatrix[1][0] = 0.0; bMatrix[1][1] = updown; bMatrix[1][2] = 0.0;
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bMatrix[2][0] = 0.0; bMatrix[2][1] = 0.0; bMatrix[2][2] = 1.0;
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/* If nor is -Y, simple symmetry by Z axis. */
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unit_m3(bMatrix);
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bMatrix[0][0] = bMatrix[1][1] = -1.0;
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}
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/* Make Roll matrix */
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