Quaternion from two vectors: the final version

I have published a first article about how to build a quaternion from two arbitrary direction vectors that will transform one of the directions into the other. That article was deliberately omitting the special case when the two vectors were facing away from each other, which required special treatment. So I wrote a second article explaining how to quickly pick an orthogonal vector.

Combining the results from the two articles should be straightforward. Here is the resulting, final function:

/* Build a unit quaternion representing the rotation
 * from u to v. The input vectors need not be normalised. */
quat quat::fromtwovectors(vec3 u, vec3 v)
{
    float norm_u_norm_v = sqrt(dot(u, u) * dot(v, v));
    float real_part = norm_u_norm_v + dot(u, v);
    vec3 w;

    if (real_part < 1.e-6f * norm_u_norm_v)
    {
        /* If u and v are exactly opposite, rotate 180 degrees
         * around an arbitrary orthogonal axis. Axis normalisation
         * can happen later, when we normalise the quaternion. */
        real_part = 0.0f;
        w = abs(u.x) > abs(u.z) ? vec3(-u.y, u.x, 0.f)
                                : vec3(0.f, -u.z, u.y);
    }
    else
    {
        /* Otherwise, build quaternion the standard way. */
        w = cross(u, v);
    }

    return normalize(quat(real_part, w.x, w.y, w.z));
}

If you know you are exclusively dealing with unit vectors, you can replace all occurrences of norm_u_norm_v with the value 1.0f in order to avoid a useless square root.

Comments

1. minorlogic@yahoo.com -- 2014-05-09 13:01

The complete version from nonunit vectors looks like

quat quat::fromtwovectors(vec3 u, vec3 v) {

vec3 w = cross(u, v); quat q = quat(dot(u, v), w.x, w.y, w.z); q.w += q.magnitude(); return normalize(q);

}

  1. Faster. 4 muls for quaternion magnitude instead of 6 muls for vectors magnitude.
  2. Numerical more stable (produce less roundoff error) than use of vectors length.

Derivation of original method can be found http://www.euclideanspace.com/maths/algebra/vectors/angleBetween/minorlogic.htm

2. -tHE SWINe- -- 2014-06-16 16:50

For the simple case and normalized vectors, this can be carried out with just a single dot product, a single cross product and a single square root: just taking a half vector, normalizing it, taking a dot product for cosine of the half-angle and taking a cross product to produce the complex part of the quaternion. That seems to be slightly faster than the proposed version.

I have compared to the proposed implementation (which is in fact used in the glm library), glm seems to be less numerically stable for nearly opposite vectors rotate (1, 2, 3) to (-1, -2, -3.01).

3. anonymous -- 2014-09-01 17:12

Since you blog about these topics, you might be interested to read about Grassman algebra, aka Geometric Algebra.

The math is really nice IMHO, and a lot of 'magic' of complex numbers and quaternions is explained by it.

For example, any rotation in any dimension is actually a double reflection about two vectors where the angle between the vectors is half the rotation angle. The math for a double reflection (hence rotation) in that algebra is N*M*A*M*N, See e.g. http://en.wikipedia.org/wiki/Rotor_(mathematics)

Apparently the guys from the C4 Engine gave a talk about this at GDC! http://gdcvault.com/play/1015322/Fundamentals-of-Grassmann

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