Hyperbolic Geometry
Table of Contents
Introduction
Riddle
A person is standing somewhere on Earth.
They walk 10km S, 10km E, 10km N.
They are back where they started.
Where are they?
Videos
1
2
Properties
Exponential
\(\text{circumference} \propto e^\text{radius}\)
\(\text{area} \propto e^\text{radius}\)
\(\Downarrow\)
\(\text{circumference} \propto \text{area}\)
Gyrovector space
Vectors rotate when you translate them.
\(\text{NE} \neq \text{EN}\)
\(\Downarrow\)
Coordinates are very expensive relative to radius.
Special Relativity
\[\mathbf{E}^3 \mathbf{H}\]
The 3 euclidean dimensions are spatial.
The hyperbolic dimension is time.
The constant of proportionality between the two is \(c\).
Consequence
No need for “boosts”, Lorentz transforms, etc.
Adding velocity (without hyperbolic geometry:):
\[ v_\Sigma = \frac{v_1+v_2}{1+\frac{v_1v_2}{c^2}} \]
Adding rapidity (with hyperbolic geometry:):
\[ \psi_\Sigma = \psi_1 + \psi_2 \]
CS Applications
Hierarchical data
Concept spaces.
Routing
Every connected, finite graph has a greedy embedding in the hyperbolic plane. (Kleinberg 2007)