How to interpolate rotations?

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悲哀的现实
悲哀的现实 2021-02-14 15:29

I have two vectors describing rotations; a start rotation A and a target rotation B. How would I best go about interpolating A by a factor F to approach B?

Using a simpl

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  • 2021-02-14 16:09

    If you have decided to go with Quaternions (which will slerp very nicely), see my answer here on resources for implementing Quaternions: Rotating in OpenGL relative to the viewport

    You should find plenty of examples in the links in that post.

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  • 2021-02-14 16:15

    Well, your slerp approach would work and is probably computationally most efficient (even though it's a bit tough to understand). To get back from the quaternions to the vector, you'll need to use a set of formulas you can find here.

    There's also a bit of relevant code here, although I don't know if it corresponds to the way you have your data represented.

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  • 2021-02-14 16:17

    Since I don't seem to understand your question, here is a little SLERP implementation in python using numpy. I plotted the results using matplotlib (v.99 for Axes3D). I don't know if you can use python, but does look like your SLERP implementation? It seems to me to give fine results ...

    from numpy import *
    from numpy.linalg import norm
    
    def slerp(p0, p1, t):
            omega = arccos(dot(p0/norm(p0), p1/norm(p1)))
            so = sin(omega)
            return sin((1.0-t)*omega) / so * p0 + sin(t*omega)/so * p1
    
    
    # test code
    if __name__ == '__main__':
        pA = array([-2.0, 0.0, 2.0])
        pB = array([0.0, 2.0, -2.0])
    
        ps = array([slerp(pA, pB, t) for t in arange(0.0, 1.0, 0.01)])
    
        from pylab import *
        from mpl_toolkits.mplot3d import Axes3D
        f = figure()
        ax = Axes3D(f)
        ax.plot3D(ps[:,0], ps[:,1], ps[:,2], '.')
        show()
    
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  • 2021-02-14 16:29

    A simple LERP (and renormalizing) only works fine when the vectors are very close together, but will result in unwanted results when the vectors are further apart.

    There are two options:

    Simple cross-products:

    Determine the axis n that is orthogonal to both A and B using a cross product (take care when the vectors are aligned) and calculate the angle a between A and B using a dot product. Now you can simply approach B by letting a go from 0 to a (this will be aNew and applying the rotation of aNew about axis n on A.

    Quaternions:

    Calculate the quaternion q that moves A to B, and interpolate q with the identity quaternion I using SLERP. The resulting quaternion qNew can then be applied on A.

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