# Hypershot: Fun with Hyperbolic Geometry - PowerPoint PPT Presentation

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## Hypershot: Fun with Hyperbolic Geometry

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### HYPERSHOT: FUN WITH HYPERBOLIC GEOMETRY Praneet Sahgal MODELING HYPERBOLIC GEOMETRY Upper Half-plane Model (Poincar half-plane model) Poincar Disk Model Klein ... – PowerPoint PPT presentation

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Title: Hypershot: Fun with Hyperbolic Geometry

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 http//www.ms.uky.edu/droyster/cour
ses/spring08/math6118/Classnotes/Chapter09.pdf
5
Klein Model
• Similar to the Poincaré disk model, except chords
• 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/
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 n1 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 Cubes walls
• Requires Syzygy vector (the data structure)
• Hyperbolic viewing frustum

12
• 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.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/