Combine Rotation Axis Vectors

Question

I'm experimenting with using axis-angle vectors for rotations in my hobby game engine. This is a 3-component vector along the axis of rotation with a length of the rotation in radians. I like them because:

• Unlike quats or rotation matrices, I can actually see the numbers and visualize the rotation in my mind
• They're a little less memory than quaternions or matrices.
• I can represent values outside the range of -Pi to Pi (This is important if I store an angular velocity)

However, I have a tight loop that updates the rotation of all of my objects (tens of thousands) based on their angular velocity. Currently, the only way I know to combine two rotation axis vectors is to convert them to quaternions, multiply them, and then convert the result back to an axis/angle. Through profiling, I've identified this as a bottleneck. Does anyone know a more straightforward approach?

Answer 1

You representation is equivalent to quaternion rotation, provided your rotation vectors are unit length. If you don't want to use some canned quaternion data structure you should simply ensure your rotation vectors are of unit length, and then work out the equivalent quaternion multiplications / reciprocal computation to determine the aggregate rotation. You might be able to reduce the number of multiplications or additions.

If your angle is the only thing that is changing (i.e. the axis of rotation is constant), then you can simply use a linear scaling of the angle, and, if you'd like, mod it to be in the range [0, 2π). So, if you have a rotation rate of α raidans per second, starting from an initial angle of θ₀ at time t₀, then the final rotation angle at time t is given by:

θ(t) = θ₀+α(t-t₀) mod 2π

You then just apply that rotation to your collection of vectors.

If none of this improves your performance, you should consider using a canned quaternion library as such things are already optimized for the kinds of application you're disucssing.

Answer 2

You can keep them as angle axis values.

Build a cross-product (anti-symmetric) matrix using the angle axis values (x,y,z) and weight the elements of this matrix by multiplying them by the angle value. Now sum up all of these cross-product matrices (one for each angle axis value) and find the final rotation matrix by using the matrix exponential.

If matrix A represents this cross-product matrix (built from Angle Axis value) then,

exp(A) is equivalent to the rotation matrix R (i.e., equivalent to your quaternion in matrix form).

Therefore,

exp (A1 + A2) = R1 * R2

probably a more expensive calucation in the end...