[Bf-blender-cvs] [07f8c5c] master: Better code for (bone axis + roll) to mat
Bastien Montagne
noreply at git.blender.org
Sun Apr 6 19:21:25 CEST 2014
Commit: 07f8c5c3b680c717c3bbc698cf873015f63d9798
Author: Bastien Montagne
Date: Sun Apr 6 19:15:17 2014 +0200
https://developer.blender.org/rB07f8c5c3b680c717c3bbc698cf873015f63d9798
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
===================================================================
M source/blender/blenkernel/intern/armature.c
===================================================================
diff --git a/source/blender/blenkernel/intern/armature.c b/source/blender/blenkernel/intern/armature.c
index 60ec481..845781f 100644
--- a/source/blender/blenkernel/intern/armature.c
+++ b/source/blender/blenkernel/intern/armature.c
@@ -1414,6 +1414,7 @@ void BKE_rotMode_change_values(float quat[4], float eul[3], float axis[3], float
* pose_mat(b)= arm_mat(b) * chan_mat(b)
*
* *************************************************************************** */
+
/* Computes vector and roll based on a rotation.
* "mat" must contain only a rotation, and no scaling. */
void mat3_to_vec_roll(float mat[3][3], float r_vec[3], float *r_roll)
@@ -1433,52 +1434,95 @@ void mat3_to_vec_roll(float mat[3][3], float r_vec[3], float *r_roll)
}
}
-/* Calculates the rest matrix of a bone based
- * On its vector and a roll around that vector */
+/* Calculates the rest matrix of a bone based on its vector and a roll around that vector. */
+/* Given v = (v.x, v.y, v.z) our (normalized) bone vector, we want the rotation matrix M
+ * from the Y axis (so that M * (0, 1, 0) = v).
+ * -> The rotation axis a lays on XZ plane, and it is orthonormal to v, hence to the projection of v onto XZ plane.
+ * -> a = (v.z, 0, -v.x)
+ * We know a is eigenvector of M (so M * a = a).
+ * Finally, we have w, such that M * w = (0, 1, 0) (i.e. the vector that will be aligned with Y axis once transformed).
+ * We know w is symmetric to v by the Y axis.
+ * -> w = (-v.x, v.y, -v.z)
+ *
+ * Solving this, we get (x, y and z being the components of v):
+ * ┌ (x^2 * y + z^2) / (x^2 + z^2), x, x * z * (y - 1) / (x^2 + z^2) ┐
+ * M = │ x * (y^2 - 1) / (x^2 + z^2), y, z * (y^2 - 1) / (x^2 + z^2) │
+ * └ x * z * (y - 1) / (x^2 + z^2), z, (x^2 + z^2 * y) / (x^2 + z^2) ┘
+ *
+ * This is stable as long as v (the bone) is not too much aligned with +/-Y (i.e. x and z components
+ * are not too close to 0).
+ *
+ * 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).
+ * This allows to simplifies M like this:
+ * ┌ 1 - x^2 / (1 + y), x, -x * z / (1 + y) ┐
+ * M = │ -x, y, -z │
+ * └ -x * z / (1 + y), z, 1 - z^2 / (1 + y) ┘
+ *
+ * Written this way, we see the case v = +Y is no more a singularity. The only one remaining is the bone being
+ * aligned with -Y.
+ *
+ * Let's handle the asymptotic behavior when bone vector is reaching the limit of y = -1. Each of the four corner
+ * elements can vary from -1 to 1, depending on the axis a chosen for doing the rotation. And the "rotation" here
+ * is in fact established by mirroring XZ plane by that given axis, then inversing the Y-axis.
+ * For sufficiently small x and z, and with y approaching -1, all elements but the four corner ones of M
+ * will degenerate. So let's now focus on these corner elements.
+ *
+ * We rewrite M so that it only contains its four corner elements, and combine the 1 / (1 + y) factor:
+ * ┌ 1 + y - x^2, -x * z ┐
+ * M* = 1 / (1 + y) * │ │
+ * └ -x * z, 1 + y - z^2 ┘
+ *
+ * When y is close to -1, computing 1 / (1 + y) will cause severe numerical instability, so we ignore it and
+ * normalize M instead. We know y^2 = 1 - (x^2 + z^2), and y < 0, hence y = -sqrt(1 - (x^2 + z^2)).
+ * Since x and z are both close to 0, we apply the binomial expansion to the first order:
+ * y = -sqrt(1 - (x^2 + z^2)) = -1 + (x^2 + z^2) / 2. Which gives:
+ * ┌ z^2 - x^2, -2 * x * z ┐
+ * M* = 1 / (x^2 + z^2) * │ │
+ * └ -2 * x * z, x^2 - z^2 ┘
+ */
void vec_roll_to_mat3(const float vec[3], const float roll, float mat[3][3])
{
- float nor[3], axis[3], target[3] = {0, 1, 0};
+ float nor[3];
float theta;
float rMatrix[3][3], bMatrix[3][3];
normalize_v3_v3(nor, vec);
- /* Find Axis & Amount for bone matrix */
- cross_v3_v3v3(axis, target, nor);
+ theta = 1 + nor[1];
- /* was 0.0000000000001, caused bug [#23954], smaller values give unstable
- * roll when toggling editmode.
- *
- * was 0.00001, causes bug [#27675], with 0.00000495,
- * so a value inbetween these is needed.
+ /* With old algo, 1.0e-13f caused T23954 and T31333, 1.0e-6f caused T27675 and T30438,
+ * so using 1.0e-9f as best compromise.
*
- * was 0.000001, causes bug [#30438] (which is same as [#27675, imho).
- * Resetting it to org value seems to cause no more [#23954]...
- *
- * was 0.0000000000001, caused bug [#31333], smaller values give unstable
- * roll when toggling editmode again...
- * No good value here, trying 0.000000001 as best compromise. :/
+ * New algo is supposed much more precise, since less complex computations are performed,
+ * but it uses two different threshold values...
*/
- if (len_squared_v3(axis) > 1.0e-9f) {
- /* if nor is *not* a multiple of target ... */
- normalize_v3(axis);
-
- theta = angle_normalized_v3v3(target, nor);
-
- /* Make Bone matrix*/
- axis_angle_normalized_to_mat3(bMatrix, axis, theta);
+ if (theta > 1.0e-9f) {
+ /* nor is *not* -Y.
+ * We got these values for free... so be happy with it... ;)
+ */
+ bMatrix[0][1] = -nor[0];
+ bMatrix[1][0] = nor[0];
+ bMatrix[1][1] = nor[1];
+ bMatrix[1][2] = nor[2];
+ bMatrix[2][1] = -nor[2];
+ if (theta > 1.0e-5f) {
+ /* If nor is far enough from -Y, apply the general case. */
+ bMatrix[0][0] = 1 - nor[0] * nor[0] / theta;
+ bMatrix[2][2] = 1 - nor[2] * nor[2] / theta;
+ bMatrix[2][0] = bMatrix[0][2] = -nor[0] * nor[2] / theta;
+ }
+ else {
+ /* If nor is too close to -Y, apply the special case. */
+ theta = nor[0] * nor[0] + nor[2] * nor[2];
+ bMatrix[0][0] = (nor[0] + nor[2]) * (nor[0] - nor[2]) / theta;
+ bMatrix[2][2] = -bMatrix[0][0];
+ bMatrix[2][0] = bMatrix[0][2] = 2.0f * nor[0] * nor[2] / theta;
+ }
}
else {
- /* if nor is a multiple of target ... */
- float updown;
-
- /* point same direction, or opposite? */
- updown = (dot_v3v3(target, nor) > 0) ? 1.0f : -1.0f;
-
- /* I think this should work... */
- bMatrix[0][0] = updown; bMatrix[0][1] = 0.0; bMatrix[0][2] = 0.0;
- bMatrix[1][0] = 0.0; bMatrix[1][1] = updown; bMatrix[1][2] = 0.0;
- bMatrix[2][0] = 0.0; bMatrix[2][1] = 0.0; bMatrix[2][2] = 1.0;
+ /* If nor is -Y, simple symmetry by Z axis. */
+ unit_m3(bMatrix);
+ bMatrix[0][0] = bMatrix[1][1] = -1.0;
}
/* Make Roll matrix */
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