Constructions in hyperbolic geometry

Constructions in Hyperbolic Geometry

Hyperbolic Geometry is a non-Euclidean geometry where the first four axioms of Euclidean geometry are kept but the the fifth axiom is changed. The fifth axiom of hyperbolic geometry says that when given a line and a point not on that line, there are at least two lines going through that point that is parallel to the given line.

Tools

In hyperbolic geometry, you can use the standard ruler and compass that is often used in Euclidean planar geometry. However, there are a variety of compasses and rulers developed for hyperbolic constructions.

A hypercompass can be used to construct a hypercycle if you have the central line and radius. A horocompass can be used to construct a horocycle through a specific point if the diameter and direction are also provided. Both of these also require a straight edge, like the standard ruler. When doing constructions in hyperbolic geometry, as long as you are using the proper ruler for the construction, the three compasses (meaning the horocompass, hypercompass, and the standard compass) can all perform the same constructions.

A parallel ruler can be used to draw a line through a given point A and parallel to a given ray a. If you are given any 2 unique lines, a hyperbolic ruler can be used to construct a line that is parallel to the first line and perpendicular to the second.

A few notes on the uses of rulers are:

• A parallel ruler can be used to construct anything that a standard ruler and the three rulers can also construct
• A parallel ruler can act as a ruler in Euclidean geometry
• A hyperbolic ruler cannot perform Euclidean geometry constructions
• In hyperbolic geometry, constructions that can be done using any one of the three compasses listed above and the parallel ruler can also be done using the hyperbolic ruler

Definitions for Complex Constructions

When giving angles a positive or negative sign, angle XYZ will be positive if the direction of the shortest path from XY to YZ is counterclockwise

For the purposes of the following definitions, the following assumptions will be made, which usually cannot be made in hyperbolic geometry

• Three distinct points create a unique circle
• Given any two lines, they meet at a unique point (normally, this would contradict the parallel axiom of hyperbolic geometry, since there can be many different lines parallel to the same line)
• Angle measures have signs. Here, they will be defined in the following way: Consider a triangle XYZ. The sign of angle XYZ is positive if and only if the direction of the path along the shortest arc from side XY to side YZ is counterclockwise. The picture of the triangle on the right describes this. To make a comparison, when working with the unit circle, the angle measure is positive when going counterclockwise and negative when going clockwise.

Cyclic Quadrilaterals

A quadrilateral is cyclic if the two opposite vertices add up to pi radians or 180 degrees. Also, if a quadrilateral is inscribed in a circle in a way that all of its vertexes lie on the circle, it is cyclic.

Pseudoaltitudes

Consider triangle ABC where the points are labeled in a clockwise manner so all angles are positive. Let X be a point moving along BC from B to C. As X moves closer to C, angle AXB will decrease and angle AXC will increase. When X is close enough to B, angle AXB>angle AXC. When X is close enough to C, angle AXB<angle AXC. This means that at some point, X will be in a position where angle AXB= angle AXC. When X is in this position, it is defined as the foot of the pseudoaltitude from vertex A. The pseudoaltitude would then be the line segment AX.

Here, examples of pseudoaltitudes would be A₁H₁, A₂H₂, and A₃H₃.

Pseudolengths

Let d_E(A,B) denote the pseudolength for a given hyperbolic line segment AB. Let a transformation move A to the center of a Poincaré disk with a radius equalling 1. The pseudolength d_E(A,B) is the length of this segment in Euclidean geometry.

Applying to Spherical Geometry and Models of Hyperbolic Geometry

Algebraically, hyperbolic and spherical geometry have the same structure. This allows us to apply concepts and theorems to one geometry to the other. Applying hyperbolic geometry to spherical geometry can make it easier to understand because spheres are much more concrete, which then makes spherical geometry easier to conceptualize.

In addition to this, portions of the hyperbolic plane can be placed onto a pseudosphere and maintain angles and hyperbolic distances, as well as be bent around the pseudosphere and still keep its properties. However, not the entire hyperbolic plane can be placed onto the pseudosphere as a model, only a portion of the hyperbolic plane. The entire hyperbolic plane can also be placed on a Poincaré disk and maintain its angles. However, the lines will turn into circular arcs, which warps them.