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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
    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/
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
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.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/
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