How to obtain axis-angle from rotation matrix?
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31-05-2021 - |
Question
I need to obtain some data from an openGL rotation matrix. I need to obtain the equivalent euler angles (already did it), the equivalent quaternion (did it, but just copying it from the Internet) and the equivalent axis-angle.
I dont know if a rotation matrix can be expresed as a single rotation of a certain angle around an certain vector. Are these equivalent? If they are, how can I obtain one from the other?
Also, i would like to understand better the meaning of a quaternion, and the insides of a rotation matrix. Where should i go to learn about this?
Solution
Yes any rotation matrix/unit quaternion is equivalent to a rotation around a single axis. If we call this axis n
and the angle theta
then the quaternion for this rotation is:
[n * sin(theta / 2) cos(theta / 2)]
To reconstruct this use acos
on the w
element of the quaternion to get theta / 2
. After you have theta
you can divide x
,y
and z
component with sin(theta / 2)
to reconstruct the axis.
OTHER TIPS
Here's a function which converts a 3x3 matrix into an axis, angle (using a quatention, so perhaps theres a more efficient way which bypasses that step).
void axis_angle_from_mat3(float r_axis[3], float *r_angle, float mat[3][3])
{
float q[4];
/* -------------------------------------------------------------------- */
/* matrix to quaternion */
double tr, s;
float tmat[3][3];
/* work on a copy */
memcpy(tmat, mat, sizeof(tmat));
/* normalize the matrix */
int i;
for (i = 0; i < 3; i++) {
float d = (tmat[i][0] * tmat[i][0] + tmat[i][1] * tmat[i][1] + tmat[i][2] * tmat[i][2]);
if (d > 1.0e-35f) {
d = sqrtf(d);
tmat[i][0] /= d;
tmat[i][1] /= d;
tmat[i][2] /= d;
}
else {
tmat[i][0] = 0.0f;
tmat[i][1] = 0.0f;
tmat[i][2] = 0.0f;
d = 0.0f;
}
}
tr = 0.25 * (double)(1.0f + tmat[0][0] + tmat[1][1] + tmat[2][2]);
if (tr > (double)1e-4f) {
s = sqrt(tr);
q[0] = (float)s;
s = 1.0 / (4.0 * s);
q[1] = (float)((double)(tmat[1][2] - tmat[2][1]) * s);
q[2] = (float)((double)(tmat[2][0] - tmat[0][2]) * s);
q[3] = (float)((double)(tmat[0][1] - tmat[1][0]) * s);
}
else {
if (tmat[0][0] > tmat[1][1] && tmat[0][0] > tmat[2][2]) {
s = 2.0f * sqrtf(1.0f + tmat[0][0] - tmat[1][1] - tmat[2][2]);
q[1] = (float)(0.25 * s);
s = 1.0 / s;
q[0] = (float)((double)(tmat[1][2] - tmat[2][1]) * s);
q[2] = (float)((double)(tmat[1][0] + tmat[0][1]) * s);
q[3] = (float)((double)(tmat[2][0] + tmat[0][2]) * s);
}
else if (tmat[1][1] > tmat[2][2]) {
s = 2.0f * sqrtf(1.0f + tmat[1][1] - tmat[0][0] - tmat[2][2]);
q[2] = (float)(0.25 * s);
s = 1.0 / s;
q[0] = (float)((double)(tmat[2][0] - tmat[0][2]) * s);
q[1] = (float)((double)(tmat[1][0] + tmat[0][1]) * s);
q[3] = (float)((double)(tmat[2][1] + tmat[1][2]) * s);
}
else {
s = 2.0f * sqrtf(1.0f + tmat[2][2] - tmat[0][0] - tmat[1][1]);
q[3] = (float)(0.25 * s);
s = 1.0 / s;
q[0] = (float)((double)(tmat[0][1] - tmat[1][0]) * s);
q[1] = (float)((double)(tmat[2][0] + tmat[0][2]) * s);
q[2] = (float)((double)(tmat[2][1] + tmat[1][2]) * s);
}
}
/* normalize the quat */
float len;
len = sqrtf(q[0] * q[0] + q[1] * q[1] + q[2] * q[2] + q[3] * q[3]);
if (len != 0.0f) {
q[0] /= len;
q[1] /= len;
q[2] /= len;
q[3] /= len;
}
else {
q[1] = 1.0f;
q[0] = q[2] = q[3] = 0.0f;
}
/* -------------------------------------------------------------------- */
/* quaternion to axis angle */
float ha, si;
ha = acosf(q[0]);
si = sinf(ha);
*r_angle = ha * 2;
if (fabsf(si) < FLT_EPSILON)
si = 1.0f;
r_axis[0] = q[1] / si;
r_axis[1] = q[2] / si;
r_axis[2] = q[3] / si;
}