Math 487 Lab 8

Exploring the Poincaré Disk Model

This lab is also Assignment 8. (due Monday, 2/26) You may not be able to finish the lab during lab period for this reason. Be sure to do the reading in Chapters 4 and 5 of Sved.

This is one model for hyperbolic non-Euclidean geometry. We will write "circle" when we mean a circle in the sense of inversive geometry (it is either a Euclidean circle or a Euclidean line).

• The points are the points inside a circle C with center O and radius r.
• The lines are "arcs" m consisting of the intersection of a "circle" m_ with the interior of C. (As a special case the "circle" m_ may be a straight line and then the "arc" will be a segment which is a diameter of C.

We call the points and lines in the Poincaré model (when it is not clear from the context) P-points and P-lines. If a P-line m is an arc of a circle m_, then m_ is called the support or the supporting circle of m.

The points on the circle C (i.e., on the circle itself, not the interior) are called ideal points. They are not true points of the model but we will see that they represent directions at infinity. They are useful in making some constructions in the model.

Line reflection of a P-point A in a P-line m is the P-point A', where A' is the inversion (or reflection if m is a line) of point A in "circle" m.

The angle between P-lines is measured as the usual angle measure between Euclidean circles.

GENERAL CONSTRUCTION NOTES:

(1) For each construction in non-Euclidean geometry, interpret the statement as a construction in the P-model using "circles" and then carry out the construction. Begin by drawing the circle C which is the "universe" that you are operating in.

(2) For P-lines, you can either draw the whole circle orthogonal to C and just ignore the part outside or else you can also construct the arc interior to C on top of the circle. The latter is a bit more complicated keeping straight but looks better; you can decide for yourself (maybe on a case by case basis) whether to work with the arc or the circle.

Lab Activity 1. Parallel Lines. (for Assignment 8.1)

Begin by drawing the circle C with center O through radius point R.

1. Given two P-points A and B construct a P-line through A and B. (For this dynamic construction, you can set aside the special case when the P-line is a Euclidean line). Make a P-line script with givens O, R, A, B.
2. Drag point B around., leaving A fixed. Trace the supporting circle of P-line AB. This is a pencil. What pencil is it?
   
3. Stop tracing. Drag point A or B around to get a feel for what the P-line AB looks like when A and B are close together, far apart, near the ideal circle C and when they are (nearly collinear with O).
4. Add a second line CD to your figure. Keep A, B and C fixed and drag D and observe when line CD intersects line AB and when it is disjoint. If the two arcs of the P-lines have a common endpoint on C (so the P-lines are disjoint, but just barely) we say in the Sved terminology that the lines are critically parallel. In BEG, we just say they are parallel. If the supporting circles of the P-lines do not meet at all, we say in Sved terminology that the lines are parallel and in the BEG terminology that they are ultra-parallel. In the couse, we will try to use this terminology: The lines are disjoint if they do not meet; they are either critical (or limiting) parallel if the supporting circles meet on C, and they are ultra-parallel if the supporting circles do not meet. Note the critical parallel case is when you pass from a point of intersection to no point of intersection.
   Question. In the P-model, when the P-line CD is a critical parallel of AB, are the two supporting circles tangent? Does this contradict the idea that parallel lines should have no point in common?
   
5. Give an example of three P-lines m, n, p so that m is ultra-parallel to n and n is ultra-parallel to p but p is not parallel to m. Save and print this figure as Assignment Figure 8.1A.
6. Given a P-point A and an ideal point J, construct a P-line a through A which also passes through J. Save and print this figure as Assignment Figure 8.1B. (You may also want to save this as a script.)
7. Given additional P-points B and C, construct P-lines b through B and J and c through C and J. Notice that if lines a and b are critical parallel in the same direction (same ideal point) and if lines b and c are critical parallel in the same direction then lines a and c are critical parallel in the same direction, unlike the case of ultra-parallels.
   Question. Drag A and trace P-line a. What does the family of circles supporting the P-lines a look like? Does it have a name?

Lab Activity 2. Perpendicular Lines. (for Assignment 8.2)

1. Given a P-line m through E and F and a P-point A, construct a P-line n through A which is perpendicular to m. Save and print as Assignment Figure 8.2A. Also make a Perpendicular P-line script, with givens O, R, E, F, and A. (Note two possible cases: A is on m and A is not on m. Can you handle both with one script?).
2. Drag A back and forth and trace P-line n.
   Assignment Question 8.2B. What kind of pencil is the set of supporting circles of the P-lines n? Is this a family of circles of Apollonian circles with respect to two points P and Q (i.e., a hyperbolic pencil of circles)? What are points P and Q?
3. Given two ultraparallel P-lines m and n, construct the P-line p which is orthogonal to both m and n. Save and print as Assignment Figure 8.2C.
   Assignment Question 8.2D. The supporting circles of m and n belong to a pencil of circles. What kind? If they are Apollonian circles with respect to two points, what are the two points? Explain why the supporting circle of p and the circle C. belong to the pencil of circles orthogonal to supports of m and n. What kind of pencil is this?

Lab Activity 3. Mirror lines. (for Assignment 8.3)

1. Construct a P-triangle ABC. Reflect it across a P-line m to form a P-triangle A'B'C'. These triangles are congruent. Drag the triangle ABC and also m around to observe what congruent triangles look like in the model. Assignment Figure 8.3A.
2. Given two P-points A and B, construct a P-line m so that the P-reflection of A in m is B. (This is the line of symmetry of AB, or the mirror line of A and B or also the perpendicular bisector.) Save as a script. Save and print as Assignment Figure 8.3B.
3. In a new figure, construct a P-triangle ABC and construct the three perpendicular bisectors of the sides. Are they concurrent? If not, are the 3 supporting circles of the 3 lines always in a pencil of circles? Save and print some interesting cases as Assignment Figure 8.3C and 8.3D.

Lab Activity 4. P-Circles. (for Assignment 8.4)

We don't yet know how to measure distance in the P-model, but we do know how to reflect across a line so we can find what a circle looks like.

1. Reflection tracing method of visualization. Take a P-point A and a P-lines m = P-line AB. Now choose any P-point X. Reflect X across m to get X'. Now trace X' as you drag B (and thus rotate the P-line AB around A). Note that X' appears to trace a Euclidean circle.
2. Construct the Euclidean circle d through X which is orthogonal to the supporting circles m_ and n_ of two P-lines m and n through A. Explain why for any P-line through A, the P-reflection of X in the P-line will be on this Euclidean circle d and why the P-reflection in any AB will reflect d to itself. Also notice that d does not intersect C; in fact d and C are both Apollonian circles of A and A'. Save and print as Assignment Figure 8.4A.
3. Drag X and trace this circle to see a family of concentric P-circles.
4. Question: In the P-model, do three non-collinear points A, B, C always lie on a circle? Make a figure that sheds some light on this. Save and print as Assignment Figure 8.4B.

Lab Activity 5. Equal P-Steps and P-translations. (for Assignment 8.5)

1. Let A and B be P-points on a P-line p. Construct P-lines a and b through A and B which are perpendicular to p.
2. Now mark equal steps on p in this way: reflect A across b to get A' and then reflect A' across a and then continue reflecting across b and then a over and over to get a set of evenly-spaced points on p.
3. Also do this for B. Think of these as ruler markings on p. Save and print as Assignment Figure 8.5A.
4. Now let Q be a P-point not on p. Reflect Q in a, then reflect its image in b, then reflect the new image in a, then b, then a, then b, etc. Then do the reverse by reflecting in b, then a, then b, then a, then b, .... Do these points lie on a P-line? Do they seem to lie on any simple Euclidean object? Save and print as Assignment Figure 8.5B. (1) Explain that what you observe is correct, using the theory of orthogonal circles and inversions. Also (2) explain what would happen if you carried out the same construction and transformations in Euclidean plane geometry. Write down both (1) and (2) as Assignment Question 8.5C.
5. Recall the link between double reflection in parallel lines and translations in Euclidean geometry. What do the corresponding double reflections in parallels do in the P-model?

Lab Activity 6. Triangles and Angle Sums. (for Assignment 8.6)

1. Given a P-point A, construct 3 lines through A which make angles of 120 degrees. Intersect the lines with a P-circle centered at A to form an equilateral triangle XYZ.
2. Measure the angles of the triangle. Are they equal? [They should be.] Save and print as Assignment Figure 8.6A.
3. Observe what happens to the angles at the circle becomes larger (and so the triangles become larger). How small can you make the angle? Save and print as Assignment Figure 8.6B.