Complex Sphere

A single fragment shader (available here), rendered using CoolLab.

What you see here is a simple sphere, plus two rendering tricks. The main one is inspired by Conformal Geometric Algebra, an alternative way of doing geometry.

Geometric Algebra is a theory that generalizes many concepts such as complex numbers, quaternions, dual numbers and many more. It offers a very abstract and powerful way of working with geometric primitives.

In particular, 3D Conformal Geometric Algebra is a 32-dimensional vector space where both points, pairs of points, directions, lines, planes, circles and spheres can be natively represented as (32-dimensional) vectors. Using the many products defined on this algebra we can express usual operations very easily.

For example we can compute the two intersection points between a ray and a sphere using the so-called outer product between the dual representations of our objects:

intersectionPtPair = dual(outer(dualSphere, dual(rayLine)));

And note that the two points are represented as a single vector! From there on we can extract the two points from the pair and proceed with our rendering as usual.

But now comes the very interesting part: what happens if there is no actual intersection?

Well, we then obtain an imaginary point pair (the "dot product" of the vector with itself is negative). But it still represents a pair of points nonetheless! We could check the sign of the "dot product" and discard the negative ones to get a traditional sphere rendering. Or we could consider those imaginary point pairs as valid ones and still render them as if they were part of the sphere!

And there you are. This is what I did to get this interesting result outside of the regular sphere, and this is why I called it a complex sphere.

The second rendering trick is more mundane. What you see rendered is the normals of the sphere. But to make sure that negative values still appear, I take an absolute value. And finally to get brighter colors I apply a square root, and that's it!