The orth concept in 2D geometry and graphics is where, given some vector, we find a vector (line) that is at right angles to the original. We have a tiny bit of choice about how to do it – we can pick a vector to the left, or to the right – but it’s very simple (more here).

The equivalent function in 3D is one of those odd little problems where you are given some freedom with the answer, and you don’t really care about that freedom – just pick something please! – but you’re made to care about it anyway and do what feels like too much thinking. Bah!¹

I’ve exhibited at Colony of Artists several times, a vibrant art festival at Abbeyhill Colonies in Edinburgh.

One of the things I was showing and selling was my tesselation pieces: designs inspired by MC Escher’s work, particularly matamorphosing tesselations.

Some of these pieces play with negative space and the transition from 3D to 2D:

Sometimes I do some GLSL or similar: much graphics and geometry. I usually scribble down some back-of-envelope variant of the usual geometry equations, but I’ve never consistently decided what to call things or how to write them down.

Here I’m setting down the notation I’m going to use, and capturing the definitions. Reference pseudo-implementations are given too (Swift).

The equations and source given here have been unit tested; full source to come after tidying.