Hypershot: Fun with Hyperbolic Geometry

1
Hypershot Fun with Hyperbolic Geometry

• Praneet Sahgal

2
Motivation for Hyperbolic Geometry

• Euclids 5 Axioms
• 1. A straight line segment can be drawn joining
  any two points.
• 2. Any straight line segment can be extended
  indefinitely in a straight line.
• 3. Given any straight line segment, a circle can
  be drawn having the segment as radius and one
  endpoint as center.
• 4. All right angles are congruent.
• 5. If two lines are drawn which intersect a third
  in such a way that the sum of the inner angles on
  one side is less than two right angles, then the
  two lines inevitably must intersect each other on
  that side if extended far enough. This postulate
  is equivalent to what is known as the parallel
  postulate.

Source http//mathworld.wolfram.com/EuclidsPostul
ates.html
3
Motivation for Hyperbolic Geometry

• What if we tweak that last axiom, the Parallel
  Postulate?
• Say there arent ANY parallel lines (spherical
  geometry)
• Say theres MORE THAN ONE parallel line
  (hyperbolic geometry)
• Note Theres actually infinite parallel lines in
  hyperbolic space

4
Modeling Hyperbolic Geometry

• Upper Half-plane Model (Poincaré half-plane
  model)
• Poincaré Disk Model
• Klein Model
• Hyperboloid Model (Minkowski Model)

Image Source Wikipedia
5
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
6
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
7
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/
8
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
9
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
10
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

11
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/