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Asked: 2026-06-01T06:10:05+00:00

Given an arbitrary 4×4 transformation matrix, how do I find out the center of

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Given an arbitrary 4×4 transformation matrix, how do I find out the center of rotation?

m = [m11 m12 m13 m14; 
     m21 m22 m23 m24;
     m31 m32 m33 m34; 
     m41 m42 m43 m44]
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1 Answer

  1. Editorial Team

    If you know that m is purely a rotation matrix, and not the aggregation of multiple transformations of different types, you can find the axis of rotation (vector v) by solving the following equation:

    mv = v

    This works because rotating a vector about itself does not change the vector. (Note there are multiple solutions to this equation, but they all differ only by a scalar factor.)

    Unfortunately, if you cannot be sure that m does not include other transformations, I don’t know if you can find the axis of the rotation, or even if there is a unique axis of rotation to be found.

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