Poincaré Hyperbolic Half Plane

1. viz 1
2. viz 2
3. viz 3
4. viz 4

Axioms of Geometry

Over 2200 years ago, the mathematician Euclid made a deep study of geometry, and created a list of five intuitive axioms from which all known facts of geometry could be derived:

1. A straight line can be drawn between any two points
2. Any finite straight line can be continued indefinitely in a straight line
3. A circle can be drawn with a given center and radius
4. All right angles are equal to one another
5. In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

The fifth axiom, known as the parallel postulate, seemed to be more awkward than the others, and many wondered whether it could be derived from the other four axioms. However, in the 19th century, mathematicians discovered geometric systems that satisfied the first four axioms while violating the parallel postulate.

These non-Euclidean geometries have been a rich area of exploration for mathematicians and have found diverse applications in other fields.

In this post, we'll explore the hyperbolic half plane. Given a line and a point not on it, we can draw infinitely many parallel lines through the point that is parallel to the line, in violation of the parallel postulate. The hyperbolic half plane is thus an example of a non-Euclidean geometry. We'll use interactive visualizations to give a concrete understanding of the properties of this interesting geometric space.

Drawing lines between points

In the visualizations below, we draw lines between points, and extend those lines infinitely in both directions. We have one visualization for the Euclidean plane and one for the hyperbolic half plane.

The Euclidean visualization is utterly unsurprising. Given two points on the plane, the line connecting them is just what you would expect. This is the geometry of our day-to-day experience.

On the other hand, the hyperbolic situation is strange and unusual, if you have not encountered it before. The geometry consists of an infinite half plane that has a boundary along the bottom edge.

In the hyperbolic half plane, distances are not measured in the standard way, with a normal ruler. Instead, the ruler shrinks as one moves closer and closer to the bottom of the half plane. More precisely, the ruler shrinks as \(1/y\). Which means that if you divide a point's \(y\) coordinate in half, your ruler for measuring distances shrinks by a factor of two. An ant walking downward would appear to shrink and walk slower as it got closer to the bottom of the half plane.

Straight lines, which are the shortest path between two points, take the form of circular arcs. When these circular arcs are extended in both directions, they become half circles that terminate on the \(x\)-axis. Note that as one approaches the bottom of the half-plane, the measuring ruler becomes infinitely small, and so these half circles are actually infinitely long lines, even though they are rendered as finite entities.

The reason why straight lines in the hyperbolic half plane take this circular form is that the optimal path between two points needs to pass through a region where the \(y\)-coordinate is higher, because distances are shorter there. In the visualization, the distance between the two points is shown, and you can confirm that points with a higher \(y\) coordinate are generally closer together.

Drawing circles around points

In this group of visualizations, we draw circles around points, and we draw straight lines emanating from the center out to the circle boundary.

Once again, in the Euclidean case, there are no surprises. Circles look the same everywhere. Radial lines go straight out from the center to the boundary of the circle.

In the hyperbolic case, we see that a circle centered around a point still looks like a circle, even with the fact that our rulers shrink in size as we go toward the bottom of the half plane, which may be somewhat surprising. However, the center of the circle is situated closer to the bottom of the circle to compensate for the fact that distances are longer toward the bottom than at the top. We also see that the radial lines take on the same circular form that we saw in the previous set.

As you move the circle toward the bottom of the half-plane, it appears to shrink. As a matter of fact, the radius of the circle stays constant, what your are actually seeing, is that the space itself is expanding!