Rotations in N dimensions
Jeffrey P. Freeman
Feb 9, 2020
1 min read
TL;DR Rotations in 4D+ space aren't just 3D rotations with extra steps — they require a fundamentally different approach using rotation planes and Householder reflections. This post walks through the math and provides a complete PDF reference. Several years ago I was writing a Machine Learning paper that required me to do rotations in an arbitrary number of dimensions. As such I had an entire section of the paper devoted to explaining how that was done before moving on to the actual algorithm.
Rotations in 4D+ space aren't just 3D rotations with extra steps — they require a fundamentally different approach using rotation planes and Householder reflections. This post walks through the math and provides a complete PDF reference.
Several years ago I was writing a Machine Learning paper that required me to do rotations in an arbitrary number of dimensions. As such I had an entire section of the paper devoted to explaining how that was done before moving on to the actual algorithm. Here I extracted the portion where I explain N-dimensional rotations, basically rotations in 4-dimensional space or higher, and created its own PDF out of it. I hope this will be of use to some of you to help explain the process. Its a bit math-heavy but I am, as always, happy to answer any questions.
The PDF covers rotation planes, the N-dimensional rotation matrix, Householder reflections, and a worked example in 4D. It's written for someone who knows linear algebra but has never had to rotate a vector in more than three dimensions.
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