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Mathsteroids on hyperbolic surfaces
2023-03-14
A circle of radius \(r\) on a piece of paper can be thought of as a line through all the points on the paper that are a distance \(r\) from the centre of the circle.
The length of this line (ie the circumference) is \(2\pi r\).
Due to the curvature of the Earth, the line through all the points that are a distance \(r\) from where you are currently standing will be less than \(2\pi r\) (although you might
not notice the difference unless \(r\) is very big). The type of curvature that a sphere has is often called positive curvature.
It is also possible to create surfaces that have negative curvature; or surfaces where the length of the line through all the points that are a distance \(r\) from a point is more than \(2\pi r\).
These surfaces are called hyperbolic surfaces.
Hyperbolic surfaces
The most readily available example of a hyperbolic surface is a Pringle: Pringles are curved upwards in one direction, and curved downwards in the other direction.
![](javascript:showlimage("pringles.jpg"))
Pringles
It's not immediately obvious what a larger hyperbolic surface would look like, but you can easily make one if you know how to crochet: simply increase the number of stitches at a
constant rate.
![](javascript:showlimage("crochet-hyper.jpg"))
A hyperolic surface that I crocheted
If it was possible to continue crocheting forever, you'd get an unbounded hyperbolic surface: the hyperbolic plane.
Over the last few weeks, I've been working on adding hyperbolic levels to Mathsteroids, the asteroids-inspired game that I started making levels for in 2018.
There are quite a few different ways to represent the hyperbolic plane in 2D. In this blog post we'll take a look at some of these; I encourage you to play the Mathsteroids level for each one as you
read so that you can get a feeling for their behaviour.
Poincaré disk
The Poincaré disk represents the hyperbolic plane as the interior of a circle. As objects move away from the centre of the circle, they appear smaller and smaller; and the circumference of
the circle is infinitely far from the centre of the circle. Straight lines on the hyperbolic plane appear on the Poincaré disk as either circles that meet the edge of the disk at right angles,
or straight lines through the centre of the circle.
![](javascript:showlimage("hyper-poincare-lines.png"))
Some examples of straight lines represented on the Poincaré disk
One of the nicest properties of the Poincaré disk is that it correctly represents angles: If you measure the angle between to intersecting straight lines represented on the disk, then the value
you measure will be the same as the angles between the two lines in the actual hyperbolic plane.
This representation can be used to demonstrate some interesting properties of the hyperbolic plane.
In normal two dimensional space, if you are given a line and a point then there is exactly one line that goes through the point and is parallel to the first line.
But in hyperbolic space, there are many such parallel lines.
![](javascript:showlimage("hyper-poincare-parallel.png"))
Parallel lines on the Poincaré disk: all of the red lines are parallel to the blue line
In Mathsteroids, there are two levels based on the Poincaré disk. The first level is bounded: you can only fly so far from the centre of the circle before
you hit the edge of the level and bounce back. (This prevents you from getting lost in the part of the disk where you would appear very very small.)
The second level is unbounded. In this level, you can fly as far as you like in any direction, but the point which is at the centre of the
disk will change when you go too far to prevent the ship from getting to small.
Beltrami–Klein disk
Simlar to the Poincaré disk, the Beltrami–Klein disk represents the hyperbolic plane using the interior of a circle, with the edge of the circle infinitely far away. Straight lines
in hyperbolic space will appear as straight lines in the Beltrami–Klein disk. The Beltrami–Klein disk is closely related to the Poincaré disk: the same line represented on both disks
will have its endpoints at the same points on the circle (although these endpoints are arguably not part of the line as they are on the circle that is infinitely far away).
![](javascript:showlimage("hyper-bk-lines.png"))
Some examples of straight lines represented on the Beltrami–Klein disk
Unlike in the Poincaré disk, angles are not correctly represented in the Beltrami–Klein disk. It can however be a very useful representation as straight lines appearing as straight lines
is a helpful feature.
Again, there are two Mathsteroids levels based on the Beltrami–Klein disk: a bounded level and an unbounded level.
Poincaré half-plane
The Poincaré half-plane represents the hyperbolic plane using the half-plane above the \(x\)-axis. Straight lines in hyperbolic space are represented by either vertical
straight lines or semicircles with their endpoints on the \(x\)-axis.
![](javascript:showlimage("hyper-php-lines.png"))
Some examples of straight lines represented on the Poincaré half-plane
Like the Poincaré disk, the Poincaré half-plane correctly represents the angles between lines.
There is one Mathsteroids level based on the Poincaré half-plane: this level is bounded so you can only travel so far away from where you start.
Band
The band representation represents the hyperbolic plane as the strip between two horizontal lines. The strip extends forever to the left and right.
Straight lines on the band come in a variety of shapes.
![](javascript:showlimage("hyper-band-lines.png"))
Some examples of straight lines represented on the band
There is one Mathsteroids level based on the band representation: the bounded level.
Hyperboloid and Gans plane
The hyperbolic plane can be represented on the surface of a hyperboloid. A hyperboloid is the shape you get if you rotate and hyperbola (a curve shaped like \(y=1/x\).)
![](javascript:showlimage("hyperbola.png"))
A hyperbola
![](javascript:showlimage("hyperboloid.png"))
A hyperboloid
Straight lines on this representation are the intersections of the hyperboloid with flat planes that go through the origin.
You can see what these straight lines look like by playing the (bounded) Mathsteroids level based on the hyperboloid.
The Gans plane is simply the hyperboloid viewed from above. There is a (bounded) Mathsteroids level based on the Gans plane.
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