Binary Trees and Hyper Spherical Coordinates

Tim Anderton — Sun, 07 Apr 2019 06:00:00 GMT

When dealing with vectors it is often very helpful to decouple the direction of the vector from its magnitude. This is a trick I was first taught in high school physics, and which I have never stopped finding extremely useful. In high school physics problems were mostly in 2D and so a direction was uniquely specified by just one rotation angle. As an undergraduate I was properly introduced to spherical polar coordinates, which let you express directions in 3D as a function of 2 angles. But it wasn't until close to the end of my graduate education in physics that one day I stumbled over the idea that polar coordinates in 2D, spherical polar in 3D can be thought of as special cases of a more general formalism that allows you to turn any binary tree with D leaves into expressions for the D components of a unit normal direction vector as a function of a set of standard angles.

I recently saw the outline of this idea scribbled down in one of my notebooks and I was again struck by the beauty of it, and I would like to share.

At the cost of getting a little ahead of myself here is the tree that corresponds to the usual spherical polar coordinate expansion.

To read off the expression for each coordinate you simply run up the tree from the appropriate labeled node and collect terms and then remember to put the overall vector magnitude $r$ out front of each term. I think even just this diagram alone might possibly be worth a blog post since it is much easier to remember for me than the pile of trigonometric terms that it represents. I certainly would have liked to have seen something like this as a mnemonic for spherical polar as an undergrad (or for that matter as a grad student).

Asymptotic Labs (Posts about higher dimensions)

Binary Trees and Hyper Spherical Coordinates