Transcript Hypershot: Fun with Hyperbolic Geometry

HYPERSHOT:
FUN WITH HYPERBOLIC
GEOMETRY
Praneet Sahgal
MOTIVATION FOR HYPERBOLIC GEOMETRY
Euclid’s 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/EuclidsPostulates.html
MOTIVATION FOR HYPERBOLIC GEOMETRY
What if we tweak that last axiom, the Parallel
Postulate?
1.
Say there aren’t ANY parallel lines (spherical
geometry)
2.
Say there’s MORE THAN ONE parallel line
(hyperbolic geometry)
Note: There’s actually infinite parallel lines in
hyperbolic space
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/sp
ring08/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:
x-direction

y-direction
z-direction
To show things in 3D Euclidean space, we need
4D Hyperbolic space
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

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/cann
on.pdf
 http://www.geom.uiuc.edu/~crobles/hyperbolic/hy
pr/modl/kb/
