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/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://www.math.ecnu.edu.cn/~lfzhou/others/cann on.pdf http://www.geom.uiuc.edu/~crobles/hyperbolic/hy pr/modl/kb/