Hypershot Fun with Hyperbolic Geometry

- Praneet Sahgal

Modeling Hyperbolic Geometry

- Upper Half-plane Model (Poincaré half-plane

model) - Poincaré Disk Model
- Klein Model
- Hyperboloid Model (Minkowski Model)

Image Source Wikipedia

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

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 http//www.ms.uky.edu/droyster/cour

ses/spring08/math6118/Classnotes/Chapter09.pdf

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 http//www.geom.uiuc.edu/crobles/hy

perbolic/hypr/modl/kb/

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 n1 space) - Generalizes to higher dimensions

Image Source Wikipedia

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

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

Demonstration

Notable behavior

- Objects in the center take a long time to move

the space in the center is bigger (see right)

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 Cubes walls
- Requires Syzygy vector (the data structure)
- Hyperbolic viewing frustum

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 (?)

References

- http//mathworld.wolfram.com/EuclidsPostulates.htm

l - Hyperbolic Geometry by James W. Anderson
- http//mathworld.wolfram.com/EuclidsPostulates.htm

l - http//www.math.ecnu.edu.cn/lfzhou/others/cannon.

pdf - http//www.geom.uiuc.edu/crobles/hyperbolic/hypr/

modl/kb/