The following exploration activities are to be used with the Java applet łNonEuclid 2002˛ for the Poincare disc found at http://cs.unm.edu/~joel/NonEuclid/. To use this you must have a Java-enabled browser. Go to the page and click on the button to start the applet. Maximizing the window is nice.
In this exploration, you will learn to draw line segments, move points, and measure distances. You will be asked to describe appearance of lines in the Poincare disc model of hyperbolic geometry.
1. On the Constructions menu, choose Draw Line Segment.
2. Click on two points. They will be automatically labeled A and B, and the line segement will be drawn.
3. On the Edit menu, choose Move Point.
4. Click on point A and move it around the disc.
5. Try to make the line segment look straight. Write down how A and B can be positioned to make the line segment look straight (in the Euclidean sense).
6. Try to make the line segment look very curved. Write down how A and B can be positioned to make the line segment look straight (in the Euclidean sense).
7. One the Measurements menu, choose Measure Distance
8. Click on points A and B.
9. Move A close to the center of the disc. Move B so that the segment AB has length 1.
10. Move A close to the boundary of the disc. Move B so that the segment AB has length 1. Compare your results from (9) and (10) and write a statement describing the appearance of a segment of length 1
Exploration 2. Triangles
In this exploration you will construct an equilateral triangle as in Euclid Proposition 1. You will learn to draw circles, find points of intersection, hide objects, and measure triangles.
1. If necessary, clear any old constructions. On the File menu, choose New.
2. Construct a line segment AB in the central part of the disc.
3. On the Constructions menu, choose Draw Circle.
4. Click on point A and then point B to construct a circle with center A and radius AB.
5. In a similar fashion, construct a circle with center B and radius AB. (You do not have to reselect the Draw Circle option.)
6. On the Constructions menu, choose Plot Intersection Point.
7. Click on each circle. You should see two points, C and D at the intersection of the circles.
8. Draw line segments AC and BC.
9. Be careful. If you see the label E where the point C used to be, it means that you did not have C selected when you drew the line segment. If you have a new point, get rid of the point and any line segments.
a. Move point E away from point C,
b. On the Edit menu, choose Delete.
c. Click on each line or point to be deleted.
d. If you make a mistake, you can also Undo one step.
10. Now hide the circles and point D.
a. On the View menu, choose Hide Objects
b. Click on each circle and point D.
11. On the Measurements menu, choose Measure Triangle
12. Click on points A, B, and C in turn.
13. Verify that ABC is really an equilateral triangle. Now move the points A and B to different locations. (You will not be able to move C independently, since it is constructed.)
14. Write: As you move A and B, what is true of the lengths of the line segments? What is true of the angles? What is true of the angle sum? How can you mave A and B so that the angle sum is close to two right angles?