Math 487 Lab 10: Exploring the Poincaré Disk Model

The work of this lab is closely related to Assignment 3/15. Be sure to compare with the assignment. The first part of lab will the collected by observation and by checking off the Construction Checklist handed out in lab.

Background and definitions

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 o with center O and radius r.
• The lines are "arcs" m consisting of the intersection of a "circle" m_ with the interior of o. (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 o (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 o which is the "universe" that you are operating in.

(2) For P-lines, you can either draw the whole circle orthogonal to o and just ignore the part outside or else you can also construct the arc interior to o 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.

Begin by drawing the circle o 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). You will find it very convenient if you make a P-line script with givens O, R, A, B.
   • Drag point B around., leaving A fixed. Trace the supporting circle of P-line AB. This is a pencil. What pencil of circles is this?
     
   • 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 o and when they are (nearly) collinear with O.
2. Add a second line CD to your figure. Keep A, B and C fixed and drag D and observe the appearance of the figure when line CD intersects line AB and when it is disjoint.
   • If the two supporting circles of the P-lines meet at a common (ideal) point X on o, we say the lines are asymptotically parallel. If the supporting circles of the P-lines do not meet at all, we say they are ultra-parallel. So to summarize, if the lines do not meet; they are either asymptotically parallel or ultra-parallel. Note the asymptotically parallel case occurs when you pass from having a point of intersection of two P-lines to the ultraparallel case. In other words the supporting circles are passing from two intersection points to zero intersection points.
   • Question. In the P-model, when the P-line CD is an asymptotic parallel of AB, must the two supporting circles be tangent? Does this contradict the idea that parallel lines should have no point in common?
     
   • Sketch 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.
3. Given a P-point A and an ideal point J, construct a P-line a whose supporting circle passes through A and J. (You may also want to save this as a script.). Terminology: we will say that the line a passes through A and J even though J is an ideal point. 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 asymptotically parallel in the same direction (same ideal point) and if lines b and c are asymptotically parallel in the same direction then lines a and c are asymptotically parallel in the same direction, unlike the case of ultra-parallels.
• Question. Drag A and trace P-line a. What kind of pencil is the family of supporting circles of the moving P-linea look like?
• If I and J are both ideal points, how can you construct a P-line through I and J? Is there exactly one line always?

Lab Activity 2. Perpendicular Lines.

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. 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?).
   • Drag A back and forth and trace P-line n. What kind of pencil is the set of supporting circles of the P-lines n? Is this a family of Apollonian circles with respect to two points P and Q (i.e., a hyperbolic pencil of circles)? What are points P and Q in this case?
2. Given two ultraparallel P-lines m and n, construct the P-line p which is orthogonal to both m and n.
   • 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 o belong to the pencil of circles orthogonal to the supporting circles of m and n. What kind of pencil is do o and p belong to?

Lab Activity 3. Mirror lines

1. Construct 3 P-lines to form a P-triangle ABC. Reflect the triangle 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.
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 P-line of symmetry of AB, that is the mirror line of A and B or also the perpendicular bisector.) Save as a script.

Lab Activity 4. P-Circles are E-circles

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. Constructing circle points by reflecting in a moving mirror: Take a P-point A and P-line m = P-line AB. Now choose any P-point Q. Reflect Q across m to get Q'. Now trace Q' as you drag B (and thus rotate the P-line AB around A). Note that Q' appears to trace a Euclidean circle.
2. Construct an E-circle which is a P-circle: Construct the Euclidean circle d through Q which is orthogonal to the supporting circles m_ and n_ of two P-lines m and n through A. Make a script for this or use an old one.
   • Explain why for any P-line through A, the P-reflection of Q 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 o; in fact d and o are both Apollonian circles of A and A'.
   • Drag Q and trace this circle to see a family of concentric P-circles.
3. Circumcircle 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.
4. Horocycles: Given an ideal point X and point P, consider the locus of reflections P' of P in lines XB for all possible points B.
   • Observe that this locus is a Euclidean circle. But this locus is not a P-circle because one point of the Euclidean circle is not a P-point.

Lab Activity 5. Compass constructions with P-circles

This works best if you have made a P-circle script. Remember that the P-circle is just a special Apollonian circle, so you may be able to adapt an old script.

1. Given two P-points A and B, construct the circle with P-center A through B and the circle with P-center B through A. Then if C and D are the points of intersection of the two circles, observe that you have constructed two equilateral triangle ABC and ABD.
2. Use the Euclidean centers of the supporting circles of AB and BC to measure the angle ABC (this is the angle between the circles, not the Euclidean angle ABC).
   • Move the triangle around and see what happens to the size of the angle as the triangle gets bigger. What happens if A and B get very near the circle o?
   • If you measure the other angles of the triangle they will be the same. You can eyeball this or you can measure to make sure.
3. Also, construct the P-line CD and note that it is the P-perpendicular bisector of AB.

Lab Activity 6. Equal width, equal P-Steps and P-translations

1. Poincare "constant width". Given m = line AB, construct a point C on m.
2. Then construct the line a through A perpendicular to m and line c through C perpendicular to m.
3. Construct a point P on a. Then reflect line a and points A and P in c to get a', A' and P'.
   • Then trace P' as you drag point C. Notice that the segment A'P' is congruent to AP, so the lengths of the segments perpendicular to m are the same in hyperbolic geometry.
   • What does the locus of P' look like? Is it a P-line?
   • Is the locus any recognizable Euclidean figure? Can you prove or explain what you see?
4. Poincare equal distance and ruler. In the same setup as above, with m = line AB and lines a and b perpendicular to m through A and B. Reflect A in b to get A'.
5. Then reflect A'' in a to get A'''. Then reflect A''' in b to get A''''; then reflect the result in a again, then reflect the result in b again, and on and on.
   • You should get a set of points A, A', A''', A''''', etc., stretching on one direction one line m and A, A'', A'''', A'''''', etc. in the other direction.
   • If you assume that line reflection are isometries, these sequences of points are equally spaced and give a ruler on line m.
6. Recall the link between double reflection in parallel lines and translations in Euclidean geometry. Describe the hyperbolic transformation which is this double reflections in ultraparallels in the P-model.