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Differential D436 Diff 1429 source/blender/blenkernel/intern/armature.c

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source/blender/blenkernel/intern/armature.c

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* *
* pose_mat(b)= pose_mat(b-1) * yoffs(b-1) * d_root(b) * bone_mat(b) * chan_mat(b) * pose_mat(b)= pose_mat(b-1) * yoffs(b-1) * d_root(b) * bone_mat(b) * chan_mat(b)
* *
* we then - in init deform - store the deform in chan_mat, such that: * we then - in init deform - store the deform in chan_mat, such that:
* *
* pose_mat(b)= arm_mat(b) * chan_mat(b) * pose_mat(b)= arm_mat(b) * chan_mat(b)
* *
* *************************************************************************** */ * *************************************************************************** */
/* Computes vector and roll based on a rotation. /* Computes vector and roll based on a rotation.
* "mat" must contain only a rotation, and no scaling. */ * "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) void mat3_to_vec_roll(float mat[3][3], float r_vec[3], float *r_roll)
{ {
if (r_vec) { if (r_vec) {
copy_v3_v3(r_vec, mat[1]); copy_v3_v3(r_vec, mat[1]);
} }
if (r_roll) { if (r_roll) {
float vecmat[3][3], vecmatinv[3][3], rollmat[3][3]; float vecmat[3][3], vecmatinv[3][3], rollmat[3][3];
vec_roll_to_mat3(mat[1], 0.0f, vecmat); vec_roll_to_mat3(mat[1], 0.0f, vecmat);
invert_m3_m3(vecmatinv, vecmat); invert_m3_m3(vecmatinv, vecmat);
mul_m3_m3m3(rollmat, vecmatinv, mat); mul_m3_m3m3(rollmat, vecmatinv, mat);
*r_roll = atan2f(rollmat[2][0], rollmat[2][2]); *r_roll = atan2f(rollmat[2][0], rollmat[2][2]);
} }
} }
/* Calculates the rest matrix of a bone based /* Calculates the rest matrix of a bone based on its vector and a roll around that vector. */
* 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]) 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 theta;
float rMatrix[3][3], bMatrix[3][3]; float rMatrix[3][3], bMatrix[3][3];
normalize_v3_v3(nor, vec); normalize_v3_v3(nor, vec);
/* Find Axis & Amount for bone matrix */ theta = 1 + nor[1];
cross_v3_v3v3(axis, target, nor);
/* was 0.0000000000001, caused bug [#23954], smaller values give unstable /* With old algo, 1.0e-13f caused T23954 and T31333, 1.0e-6f caused T27675 and T30438,
* roll when toggling editmode. * so using 1.0e-9f as best compromise.
*
* was 0.00001, causes bug [#27675], with 0.00000495,
* so a value inbetween these is needed.
* *
* was 0.000001, causes bug [#30438] (which is same as [#27675, imho). * New algo is supposed much more precise, since less complex computations are performed,
* Resetting it to org value seems to cause no more [#23954]... * but it uses two different threshold values...
*
* 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. :/
*/ */
if (len_squared_v3(axis) > 1.0e-9f) { if (theta > 1.0e-9f) {
/* if nor is *not* a multiple of target ... */ /* nor is *not* -Y.
normalize_v3(axis); * We got these values for free... so be happy with it... ;)
*/
theta = angle_normalized_v3v3(target, nor); bMatrix[0][1] = -nor[0];
bMatrix[1][0] = nor[0];
/* Make Bone matrix*/ bMatrix[1][1] = nor[1];
axis_angle_normalized_to_mat3(bMatrix, axis, theta); 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 { else {
/* if nor is a multiple of target ... */ /* If nor is -Y, simple symmetry by Z axis. */
float updown; unit_m3(bMatrix);
bMatrix[0][0] = bMatrix[1][1] = -1.0;
/* 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;
} }
/* Make Roll matrix */ /* Make Roll matrix */
axis_angle_normalized_to_mat3(rMatrix, nor, roll); axis_angle_normalized_to_mat3(rMatrix, nor, roll);
/* Combine and output result */ /* Combine and output result */
mul_m3_m3m3(mat, rMatrix, bMatrix); mul_m3_m3m3(mat, rMatrix, bMatrix);
} }
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