This is going to be a collection of my experiments related to moebius transformations and interesting mappings of 3d geometry. 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.
What's going on here?
The horse is inverse stereographically projected onto the hypersphere, rotated, and projected back again. Dragging the mouse moves the camera. Pressing the left & right arrow keys decrements and increments the rotation speed.
How does it work?
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 articles: 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.