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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 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).

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch

   (b) Construct a hyperbolic ray, given a vertex and another point on the ray.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch

   (c) Construct a hyperbolic line, given two points on the line.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


2. Construct a hyperbolic line segment.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


3. Measure the length of a hyperbolic  line segment.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


4. Calculate the measure of a hyperbolic angle.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


5. Construct the bisector of a given hyperbolic angle.

   (a) Construct the pole of a hyperbolic line (Klein Disk only).

          Show Klein Disk Construction Sketch                           Download Construction Sketch

   (b) Construct the bisector of a given hyperbolic angle.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


6. Construct a perpendicular to a given line through a given point on the line.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


7. Construct a perpendicular to a given line through a given point not on the line.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


8. Construct the perpendicular bisector of a hyperbolic line segment.

   (a) Construct the midpoint of a hyperbolic line segment.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch

   (b) Construct the perpendicular bisector of a hyperbolic line segment.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


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.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch

   (b) Construct a hyperbolic circle, given its center and a point on the circle.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch


10. Construct a hyperbolic circle, given its center and two points determining the radius of the circle.

          Show Poincaré Disk Construction Sketch                     Download Construction Sketch


          Show Klein Disk Construction Sketch                           Download Construction Sketch