Hyperbolic Geometry
Table of Contents
- Introduction
- Purpose of the Book
- Introduction to Hyperbolic Geometry
- Kline Model of Hyperbolic Geometry- using tools
- How do we create a tool for a Kline line?
- A Model of the Kline Disk
- How are perpendiculars determined in the Kline Model?
- Poincaré Model of Hyperbolic Geometry Worksheet
- What is the Poincaré Model?
- Poincare Model Worksheet
- Comparing Kline Model and Poincaré Model
- Comparing Kline Model and Poincaré Model
Purpose of the Book
The purpose of this book is to introduce hyperbolic geometry to students who are already familiar with Euclidean geometry. We will introduce the Kline model of hyperbolic geometry and have students learn how to create a tool in Geogebra. Students will also use a pre-made Kline model to explore perpendiculars and rectangles in hyperbolic geometry. After students have spent some time with the Kline model, we will introduce the Poincaré model. Students will explore what happens when the midpoints of a line are connected diagonally in the Poincaré model. Then students will compare the two models.
How do we create a tool for a Kline line?
The first thing we need to know is how a Kline line relates to a line in Euclidean Geometry. In the Kline model of hyperbolic geometry, lines are chords of a Euclidean circle.
Step 1:
Construct a circle
Step 2:
Make your circle an object by increasing the opacity of the inside of the circle. This can be done by right clicking on the circle, selecting settings, selecting color, and increasing opacity.
Step 3:
Construct two points C and D on the circle, and construct a line through those two points.
Step 4:
Construct the intersection points E and F between the line and the circle. Construct a segment g, between those two points. This segment is your Kline line.
Step 5:
In order to create a tool, click the three bars in the top right corner, then click tools, then create new tool. A box pops up and asks for an output object. The output object you want is your Kline line, which is segment g. Choose segment g from the drop down menu. Then choose input objects. This is what we want the user to select in order to create the line. Choose Point C, Point D and circle c from the drop down menu, and remove Point A and B. Finally click on Name and Icon and rename your tool as "Kline Line."
Try Creating this tool on your own.
What is the Poincaré Model?
The Poincaré hyperbolic disk is a two-dimensional space having hyperbolic geometry defined as the disk [math]\left\{x\in\mathbb{R}:|x|<1\right\}[/math]with hyperbolic metric [math]ds=\frac{dx^2+dy^2}{\left(1-x^2-y^2\right)^2}[/math]. The Poincaré disk is a model for hyperbolic geometry in which a line is represented as an arc of a circle whose ends are perpendicular to the disk’s boundary (and diameters are also permitted). Two arcs which do not meet correspond to parallel rays, arcs which meet orthogonally correspond to perpendicular lines, and arcs which meet on the boundary are a pair of limits rays. [br]
A bit more information can be found at the below link:[br][br]http://mathworld.wolfram.com/PoincareHyperbolicDisk.html
Comparing Kline Model and Poincaré Model
Since you have now had the opportunity to take a look at the Kline Model and the Poincaré Model, let's compare the two models a bit and see what we have noticed.
What do you notice about line segments in the Kline Model? in the Poincaré Model? How are they different how are they the same?
What do you notice about angles in the Kline Model? in the Poincaré Model? How are they different how are they the same?