Math 141 Computer Lab: Visualizing the Hyperbolic Plane

Objectives: At the completion of this lab, students will be able to:

Use the program NonEuclid.
Visualize the distortion of length and the preservation of angles in the hyperbolic plane.
Explain similarities between and difference between E² and H².

Instructions: You may complete this assignment in groups.
Complete each step of the lab indicated below. When you come to a question, think about the answer and talk about it, then write down an answer. If you experience difficulties or get stuck, please ask each other for help, email Professor Klodginski, or visit office hours. Make sure that each person in your group has an opportunity to use NonEuclid.
At then end of the lab, you will also need to answer a few questions about the sphere.
To do this, you may use your model, or another online program described later.

Section 0: Starting NonEuclid

Geometers SketchPad is a program for doing geometry in the Euclidean plane, but not in the hyperbolic plane or the sphere. There is a nice program called NonEuclid which is available for us to use. (Spherical Easel is for the sphere.) The advantage of this program is that is can be run from any computer which is connected to the internet and has a Java-enabled browser. For example, Netscape Navigator 6.2 or Internet Explorer 6.0 have Java capabilities. You may also download the program onto your computer.

You will need to start a browser (like netscape or explorer) and go to the address: http://www.cs.unm.edu/~joel/NonEuclid/

Scroll down the page a bit. If the browser that you are using is Java enabled, you will see a banner for NonEuclid 2002.05a. Click on this and the program will begin. Click okay in the information box that appears.

Note: If you scroll towards the bottom of the web page listed above, you will find other activities and information about the hyperbolic plane and non-Euclidean geometry.

Section 1: Lines and Distortion of Length in the Hyperbolic Plane

There are two models for the hyperbolic plane available in NonEuclid: The Poincare disk model and the upper half plane model. When NonEuclid starts, the Poincare disk model is the default. We want to use the upper half plane model. To change to the upper half plane model, click on the View menu, pull down to Hyperbolic Model, and then select Upper Half-Plane. Click on the Constructions menu and pull down to Draw Line Segment.

Note: The current function that is being used is explained in a box in the upper left corner of the screen. In this box, you should see a message telling you that Draw Line Segment is the current function selected and an explanation of how to use the function.

Click anywhere on the drawing screen. This will produce a point which is labeled as A. Click anywhere else on the screen. This will produce a point labeled as B and a line segment between them. The reason that the computer can always do this is that there is a unique line segment between every two points. Why?

This line may appear to be curved. Why? What are the possible curves that you will get?

Note: If you need to delete something, go to the Edit menu and pull down to Delete. Then click on the object that you want deleted. Notice that an object becomes brighter when the cursor is over it.

Task: To measure the (hyperbolic) length of the segment you just drew, click on the Measurements menu, then on Measure Distance. Now click on the point A, then on the point B. In the box on the left side of the screen the length of the segment AB will be displayed. Your task is to draw another line segment which appears to be the same length as the first one you drew. This is an eyeballing procedure, so don't worry about being too exact. Once you draw a line segment which appears to have the same length, measure its length as before. Repeat this process several times. Is there a place on the screen where there is more distortion of length than at another? Less?

Section 2: Preservation of Angles in the Hyperbolic Plane

Create a new drawing by going to the File menu and pulling down to New. Note: you will not need to turn in any printouts from the computer for this lab, so feel free to start a new drawing whenever yours gets too cluttered.

Using the Draw Line Segment function, draw a line segment on the screen. Move the mouse key over the end point labeled as A. When the mouse is over A, it will become brighter. Click the mouse key. Move the mouse somewhere else and click the mouse key. This will draw a line segment from A to another point (where you clicked) labeled as C. You should have something that looks like a hinge.

Task: Click on the Measure menu and pull down to Measure Angle. In the upper left hand box, instructions for using the Measure Angle function will appear. Follow them to measure the angle BAC. Try to draw another hinge somewhere else on the screen which appears to have the same angle as the first one you drew. Measure its angle. Repeat the process several times. Is angle distorted in the same way that length is?

If you have carried out the above task, you should notice that, unlike length, angle is not distorted in the hyperbolic plane. This is a feature of the model that we are using: It is a conformal model of the hyperbolic plane in the Euclidean plane. This means that the hyperbolic angle measure is the same as the Euclidean angle measure. For example, if you measure an angle which appears to be a right angle, the measurement will be close to 90 degrees.

Section 3: Reflection

Go to the Construction menu and pull down to the Draw Infinite Line function. Click anywhere on the screen. This will draw a point and label it. Click anywhere else on the screen. This will draw a point, label it, and draw an infinite line between them.

Go to the Constructions menu and pull down to Reflect. The upper left hand box will give you instructions as to how to reflect objects about lines. Does reflection appear to change the length of line segments? Does it really? Why? Experiment by reflecting several line segments and measuring the segments and their reflections.

Does reflection appear to change the angle at which curves meet? Does it really? Why? Again, experiment with several angles and measure.

Create a new drawing screen. Draw another infinite line.

Task:Does the infinite line really appear to be infinitely long? Repeat the process several times until you get a line which definitely does not appear to be infinitely long (it should look like a semicircle). Where on the screen do you have to put the two points? Why does it appear to be finite?

Recall that reflections preserve lengths. For us, this means that if you refect an infinitely long line about another line, you get another infinitely long line. Draw another infinitely long line on the screen. Reflect your first line about the one that you just drew. What do you get?

Task: Try to find an infinite line so that when you reflect your original line about it, you get a line which really looks infinitely long (infinite looking lines appear vertical). Use the Move Point function, which is in the Edit menu, to move your lines around. You may have to delete objects several times. See above for how to do this. Since reflections preserve distances, this shows that lines which appear to be of finite length can actually be infinitely long.

Section 4: Triangles in Hyperbolic Space

Draw a triangle on the screen, using Draw Line Segment from the Constructions menu. Click on Measure Triangle from the Measurements menu and follow the directions to measure the side lengths and angles of the triangle. Do this with several triangles.

Is the sum of the interior angles of the triangle always 180 degrees? Always more? Always less? Try drawing similarly shaped triangles in different locations, noting what happens to the sum of the angles.

Try to construct triangles whose sum of angles is as small as possible and triangles whose sum of angles is as large as possible.

What is the largest possible angle sum for a triangle in hyperbolic space? What is the smallest possible angle sum for a triangle in hyperbolic space? Compare your answers with Euclidian space.

Section 5: Parallel lines
Draw a line l (using Draw Infinite Line from the Constructions menu) and a point P not on l. Try to draw a line m through P that does not intersect l. Can you? Can you draw more than one such line?

How does this compare with Euclidean space?

Section 6: What about the sphere?
The program Spherical Easel works similarly to Non-Euclid, and can be found at http://merganser.math.gvsu.edu/easel/.
To answer these questions, you may try using Spherical Easel, or you can use your model with tape and string.

Make a triangle on a sphere.

Measure (you can estimate) the sum of the interior angles, and record your measurements. Based on your work above, is the angle sum of the triangle less than 180, greater than 180, or equal to 180? Approximately what is the angle sum?

Repeat this for several other triangles. Make some large and some small, with various types of angles.

Do you notice any pattern? Is the angle sum related to the size of the triangle?