Constructions in the Hyperbolic Models
Below are descriptions of hyperbolic "straightedge and compass" constructions in each of the three most well-known hyperbolic models: the Poincaré disk model, the the Poincaré half-plane model, and the Beltrami-Klein model. On other pages on this site you can find dynamic geometry tools (in GeoGebra and Geometer's Sketchpad) that automate 10 "standard" non-Euclidean constructions in the three models. On this page are the mathematical descriptions of the construction steps used to create these tools. The constructions are depicted in GeoGebra sketches and are dynamic -- they can be explored by clicking and dragging.
Please note the following regarding the constructions on this page:
- The constructions are developed in a logical order. Many of the constructions are referenced in later constructions. Moreover, there are constructions other than the 10 "standard" ones that are frequently used to support the development of the standard ones (e.g. constructing the inverse of a point in the Poincaré disk.) These constructions are provided as well. This is the reason, for example, for the constructions in 1(a) and (b) before the actual line constructions provided in 1(c).
- Although the constructions are correct, no proof is provided for their correctness.
- The constructions are described as "straightedge and compass" constructions. This means that wherever possible, the constructions rely on only a Euclidean straightedge (line-maker) and compass (circle-maker). In particular, the constructions do not rely on analytic geometry (coordinates) or on measurement. Notable exceptions include measuring lengths and angles in the models, and constructing circles in the Klein model. (Circles in the Klein model are Euclidean ellipses, which are not "straightedge and compass" constructible.)
- I do not claim that the constructions necessarily give the simplest, most intuitive, or most elegant constructions. There might well be a better way!
- The steps and sketches on this page provide the mathematical basis for the tools, but do not describe the debugging solutions to the challenges and quirks inherent to the software. One example: it is challenging, when measuring the angle between two Poincaré lines in GeoGebra, to ensure that the package always chooses the correct angle (rather than its supplement). This difficulty is not discussed in the mathematical description of the angle measurement tool below.
The 10 Constructions
1. Construct a hyperbolic line, given two points.
(a) Construct the inverse point of a point, relative to a circle (Poincaré Disk only).
(b) Construct a hyperbolic ray, given a vertex and another point on the ray.
(c) Construct a hyperbolic line, given two points on the line.
2. Construct a hyperbolic line segment.
3. Measure the length of a hyperbolic line segment.
4. Calculate the measure of a hyperbolic angle.
5. Construct the bisector of a given hyperbolic angle.
(a) Construct the pole of a hyperbolic line (Klein Disk only).
(b) Construct the bisector of a given hyperbolic angle.
6. Construct a perpendicular to a given line through a given point on the line.
7. Construct a perpendicular to a given line through a given point not on the line.
8. Construct the perpendicular bisector of a hyperbolic line segment.
(a) Construct the midpoint of a hyperbolic line segment.
(b) Construct the perpendicular bisector of a hyperbolic line segment.
9. Construct a hyperbolic circle, given its center and a point on the circle.
(a) Construct the reflection of a point about a hyperbolic line.
(b) Construct a hyperbolic circle, given its center and a point on the circle.
10. Construct a hyperbolic circle, given its center and two points determining the radius of the circle.