HYPERSHOT: FUN WITH HYPERBOLIC GEOMETRYPraneet 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 hereImage Source:

http://www.ms.uky.edu/~droyster/courses/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/hyperbolic/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 n+1 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 Cube’s 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.html

Hyperbolic Geometry by James W. Anderson http://mathworld.wolfram.com/

EuclidsPostulates.html http://www.math.ecnu.edu.cn/~lfzhou/

others/cannon.pdf http://www.geom.uiuc.edu/~crobles/

hyperbolic/hypr/modl/kb/