Lab 3: Visualizing the Hyperbolic Plane



Section 0: Starting NonEuclid

Geometers SketchPad allows us to do geometry in the Euclidean plane, but not in the hyperbolic plane. There is a nice program called NonEuclid which is available for us to use. 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 4.6 or Internet Explorer 5.0 have Java capabilities. Most of the labs on campus should have one of these two programs installed.

At the start of the lab, NonEuclid should be started and waiting for you to use it. When you are not in lab you will need to start a browser (like netscape or explorer) and go to the address: http://math.rice.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 1999.8b. 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 labelled as A. Click anywhere else on the screen. This will produce a point labelled 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.

Task: In the box on the left side of the screen the length of the segment you just drew will be displayed. Try 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, its length will be displayed in the left box. 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. Using the Draw Line Segment function, draw a line segment on the screen. Move the mouse key over the end point labelled as A. When the mouse is over A, it will change color. 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) labelled 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 is a fancy way of saying angles which appear the same in our model are actually the same in the hyperbolic plane.

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 as A. Click anywhere else on the screen. This will draw a point, label it as B, 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 lenght of line segments? Does it really? Why?

Does reflection appear to change the angle at which curves meet? Does it really? Why?

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?

Warning: Since NonEuclid doesn't run quite properly, you may have problems with the Reflect function. A simple way to cure this seems to be to quit NonEuclid and then start a fresh copy of it.

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: What About Area?

In the previous sections, we have seen that length is distorted in the hyperbolic plane while angles are preserved. The last main measurement that we can do in the plane is area. Since triangles are easiest to work with, we will concentrate on the question of whether or not the area of a triangle is distorted in the hyperbolic plane or if it is preserved.

Question: Make an initial guess as to whether or not the area of a triangle will be preserved in the hyperbolic plane. Add whatever justification you can think of.








Create a new drawing. Draw a line segment on the screen using the Draw Line Segment function. Move the mouse over the point labelled as A (A will change color when you are over it). Click the mouse key. Move the mouse somewhere else and click the mouse key. This will draw a point labelled as C and a line from A to C. Similarly, draw a line from B to C. You should now have a triangle. Go to the Measure menu and pull down to Measure Triangle. Follow the directions in the upper left box as to how to use this function. Use them to measure the triangle you have just created. In the left box, you will notice a variety of measurements that are related to the triangle. Do any strike you as odd?

Task: Draw several other triangles which appear to be the same as the original one that you drew. Measure them as you did before. Does your conjecture from before seem to be right? If so, try to explain why. If not, try to explain why the conjecture failed.








Section 5: Triangles in Hyperbolic Space

Draw a triangle on the screen. Measure the triangle as you did in the previous question.

Question 1: Is the sum of the interior angles of the triangle always 180 degrees? Always more? Always less? Move the triangle several times, 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 (drawing one triangle and just moving it around is a painless way to do construct lots of triangles). Pay attention to what is happenning to the area.

Question 2: What is the exact relationship between the sum of the interior angles of a triangle and its area? Fill in the blank at the end of this statement: If the sum of the interior angles of a triangle is S, then the area of the triangle is:_______________.
Compare you answers with Euclidian space.

Section 6: Parallel lines

Draw a line l 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 Euclidian space?