1. Hypershot: Fun with Hyperbolic Geometry
Praneet Sahgal

2. Modeling Hyperbolic Geometry
Upper Half-plane Model (Poincaré half-plane model)
Poincaré Disk Model
Klein Model
Hyperboloid Model (Minkowski Model)
Image Source: Wikipedia

3. Upper Half Plane Model Say we have a complex plane
We define the positive portion of the complex axis as hyperbolic space
We can prove that there are infinitely many parallel lines between two points on the real axis
Image Source: Hyperbolic Geometry by James W. Anderson

4. Poincaré Disk Model
Instead of confining ourselves to the upper half plane, we use the entire unit disk on the complex plane
Lines are arcs on the disc orthogonal to the boundary of the disk
The parallel axiom also holds here
Image Source:

5. Klein Model
Similar to the Poincaré disk model, except chords are used instead of arcs
The parallel axiom holds here, there are multiple chords that do not intersect
Image Source:

6. Hyperboloid Model
Takes hyperbolic lines on the Poincaré disk (or Klein model) and maps them to a hyperboloid
This is a stereographic projection (preserves angles)
Maps a 2 dimensional disk to 3 dimensional space (maps n space to n+1 space)
Generalizes to higher dimensions
Image Source: Wikipedia

7. Motion in Hyperbolic Space
Translation in x, y, and z directions is not the same! Here are the transformation matrices:
To show things in 3D Euclidean space, we need 4D Hyperbolic space
x-direction
y-direction
z-direction

8. The Project
Create a system for firing projectiles in hyperbolic space, like a first person shooter
Provide a sandbox for understanding paths in hyperbolic space

9. Demonstration

10. Notable behavior
Objects in the center take a long time to move; the space in the center is bigger (see right)

11. Techincal challenges
Applying the transformations for hyperbolic translation
LOTS of matrix multiplication
Firing objects out of the wand
Rotational transformation of a vector
Distributing among the Cube’s walls
Requires Syzygy vector (the data structure)
Hyperbolic viewing frustum

12. Adding to the project
Multiple weapons (firing patterns that would show different behavior)
Collisions with stationary objects
Path tracing
Making sure wall distribution works…
3D models for gun and target (?)

13. References http://mathworld.wolfram.com/EuclidsPostulates. html
Hyperbolic Geometry by James W. Anderson
on.pdf
pr/modl/kb/