A collection of experiments with 3D conformal inversions. The goal is to offer unique perspectives on various aspects of what it means to be a creature living in living in space and to learn about math in the process.
The horse is inverse stereographically projected onto a hypersphere, rotated, and projected back again. Dragging moves the camera. Pressing the left & right arrow keys decrements and increments the rotation speed.
Daniel Piker’s blog posts give a great overview of this stuff.
Another big help is Moebius Transformations
Revealed, which shows the 2 dimensional Moebius transformations.
It really gives an idea of what's going on in a more familiar space.
And of course Wikipedia: Stereographic projection, Möbius
transformation, Isoclinic decomposition.
This GLSL code does the actual geometry work of the stereographic projection and 4 dimensional rotation. You can see that it takes two four-dimensional vectors as input. Those are the parameters for the isoclinic rotation matrices, which should each have determinants of 1 since they are rotations. So you have to be careful what you put in them! I usually generate these from a normalized quaternion representing the rotation of some dummy vector.
This basic one just shines red, green and blue lights along x, y and z respectively.
There's an amazing video series online called Dimensions Math, check it out as soon as possible.
