diff options
author | Bastien Montagne <montagne29@wanadoo.fr> | 2014-04-06 21:15:17 +0400 |
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committer | Bastien Montagne <montagne29@wanadoo.fr> | 2014-04-06 21:15:50 +0400 |
commit | 07f8c5c3b680c717c3bbc698cf873015f63d9798 (patch) | |
tree | 15f82b58db0dec659c804dd2506ee15b3e53034f /source | |
parent | 959ec27ac91466373d76eb92c0b508ea5932249f (diff) |
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
Diffstat (limited to 'source')
-rw-r--r-- | source/blender/blenkernel/intern/armature.c | 112 |
1 files changed, 78 insertions, 34 deletions
diff --git a/source/blender/blenkernel/intern/armature.c b/source/blender/blenkernel/intern/armature.c index 60ec4817f81..845781f9abc 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 */ |