Math 487 Lab 8: Exploring the Poincaré Disk Model

Background and Definitions

The definition of the Poincaré disk model (with some construction tips) is at this link. This is listed as Lab 8, but some of the work will be done in 445 class and some will be on a homework assignment.

Lab Activity 1. Parallel Line Experiments

This first Activity is already done in the Example file pdisk.gsp that you can download here.

Example file and Automatch: This lab will work very well using Automatch for O and for R in your tools. In the example file, the points are given the distinctive names P-center instead of O and P-radius instead of R. If you add pages to your file using Document Options, you can keep these names intact.

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

1. Construct P-line AB. Given two P-points A and B construct a P-line through A and B. (For this dynamic construction, we ignore the special case when the P-line is a Euclidean line). Make a P-line tool with givens O, R, A, B. [You can automatch.]
2. Experiment - visualizing lines through A
   • Trace the supporting circle of P-line AB as B moves and A is fixed. This is the trace of a pencil of circles. Why is this so? What pencil is it, and what kind of pencil?
     
   • Turn off 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 h and when they are (nearly) collinear with O.
3. Construct a second P-line CD in your figure.
4. Experiment: Visualizing Intersections and parallels
   • Drag D with A, B and C fixed.
   • Observe the appearance of the figure when P-line CD intersects P-line AB at some P-point. In this case, at how many points do the supporting circles intersect?
   • When the two supporting circles intersect at one point, how are the circles related? In this case, do the P-lines intersect? In this case the P-lines are said to be (critically) parallel.
   • If the supporting circles of the P-lines do not meet at all, we say the P-lines are ultra-parallel.
5. Construct a third P-line EF in your figure.
6. Experiments with critical parallels and 3 lines
• Drag CD so that it is critically parallel (approximately) to AB at one of the two points at infinity and drag EF so that it is critically parallel (approximately) to AB at the other of the two points at infinity. Note that this does not make CD and EF critically parallel to each other. The P-line AB has two collections of critical parallels, one at each direction at infinity.
• Drag EF so that it is critically parallel to both the other lines and so that the 3 lines form a "triangle" with vertices at infinity. What would you say are the angles at infinity of this triangle (i.e, what are the angles between the supporting circles)?
• Drag the lines so that AB and CD are critically parallel. Is is possible to drag P-line EF so that it is orthogonal to both these lines? Explain why this is or is not possible, reasoning using the support circles.
• Finally, drag the 3 lines so that they are approximately critical parallels in the same direction at infinity. If any other line is critically parallel to AB in this direction is is also critically parallel to the other 2 lines?
7. Experiments with ultraparallels and 3 lines
• Drag the 3 P-lines in your figure so that line AB is ultraparallel to line CD and line CD is ultraparallel to line EF, but line AB and line EF are not ultraparallel (or even parallel).
  
• Drag the 3 P-lines in your figure so that line AB is ultraparallel to line CD and drage line EF to be critically parallel to line AB and also to line CD.
• Drag the 3 P-lines in your figure so that line AB is ultraparallel to line CD and line EF is orhtogonal to line AB and also to line CD. Is this possible? Are there many such lines?

Lab Activity 2. Perpendicular Line Constructions

Add a new sketch page to your file by copying Page 1 using Document Options.

1. Construct the perpendicular to a P-line through a point. Given the P-line m through A and B and a P-point g, construct a P-line n through g which is perpendicular to m. (Hint: Translate this into a construction problem in Euclidean geometry of circles.)
   • Make a Perpendicular P-line tool, with givens O, R, A, B, and G.
2. Experiment with the perpendicular
• Drag G back and forth and trace P-line n.
• What kind of pencil is the set of supporting circles of the P-lines n?
• This is a family of Apollonian circles with respect to two limit points P and Q (i.e., a hyperbolic pencil of circles)? What are points P and Q in this case?
3. Construct the perpendicular to two P-lines. Given two ultraparallel P-lines m and n, construct a P-line p which is orthogonal to both m and n. (Hint: Translate this into a construction problem in Euclidean geometry of circles.)
   • What happens to p if m and n are dragged to be almost critically parallel?
4. Explanations and Connections
• Recall that any two circles belong to a unique pencil of circles.
• What kind of pencil do the supporting circles of m and n belong to?
• If they are Apollonian circles with respect to two limit points, what are the two points?
• Explain why the supporting circle of p and the circle h belong to the pencil of circles orthogonal to the supporting circles of m and n. What kind of pencil do h and p belong to?
• Just using orthogonal circles, explain why there can be only one circle orthogonal to the supporting circles of m, n and also orthogonal to circle h.

Lab Activity 3. Mirror lines

In our study of Euclidean geometry, we began with a known distance measure and concept of congruence and defined isometries, which were the transformations that preserve distance (and hence congruence). It is possible to reverse this process. Start with a collection of transformations that seem geometrically natural, and see whether it is possible to define a distance so that these are the isometries. To begin with, we can say two figures are congruent (with respect to these transformations) if one is the image of the other by one of the special transformation.

A good place to start for this process is to figure out the meaning of line reflection. Then let the isometry candidates be the compositions of line reflections. In the P-model, line reflection is defined as inversion in the supporting circle of the line.

1. Reflect a point
   • Construct a p-line m and a point A. Invert the point A in the support circle of m to get A'.
   • Check by experiment that if A is any P-point (i.e., inside h) then A' is also a P-point on the opposite side of m (but also inside h).
2. Reflect a triangle
   • 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.
3. Construct the perpendicular bisector (important construction)
   • 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 tool.

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. The idea is that if AQ and AQ' are two radii of a circle, then AQ' is the reflection of AQ in the perpendicular bisector p of QQ'. The line p will pass through A because A is equidistant from Q and Q'. Conversely, for any line p through A, the reflection of AQ is a segment AQ' congruent to AQ, so Q' is on the circle through Q with center A. The set of all these reflections is exactly the set of points on the circle.

1. Constructing P-circle points by reflecting in a moving mirror
   • Take a P-point A and construct the P-line m = P-line AB.
   • Take 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.
   • This circle is the P-circle with P-center A. Is the E-center of the circle also A?
2. Construct an E-circle which is a P-circle:
   • Construct the Euclidean circle d through Q which is orthogonal to the supporting circle m_ of a P-line m through A. Make a tool 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 P-line through A will reflect d to itself.
   • Also notice that d does not intersect h; in fact d and h are both Apollonian circles of A and A'.
   • Drag Q and trace this circle to see a family of concentric P-circles. What do the circles look like when the center is very near infinity (the circle h)?
3. Circumcircle Question
   • Construct a triangle ABC and the P-perpendicular bisectors of the sides. Are they always concurrent?
   • In the P-model, do three non-collinear points A, B, C always lie on a circle?
   • Do the supporting circles of the P-perpendicular bisectors always belong to a pencil? Explain the cases.
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. This gives a locus that can be considered as a "circle with center at infinity".
   • 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. Which point?
   • This figures, which have no analog in Euclidean geometry, are called horocycles.

Lab Activity 5. Compass constructions with P-circles

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

1. Euclid's First Construction - Equilateral triangle
   • 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.
   • 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. Measure the angles of an equilateral triangle
   • Use the Euclidean centers of the supporting circles of AB and BC to measure the angle ABC (this is the angle between the tangents to 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 h?
• 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. Perpendicular Bisector - with P-straightedge and P-compass
   • 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" strips
   • Given m = line AB, construct a point C on m.
   • Then construct P-line c through C perpendicular to m.
   • Take any a P-point P. Reflect P in m to get P'. Reflect P and P' in c to get Q and Q'. P and P' are the same distance from line m. Why?
   • Then trace Q and Q' as you drag point C. Notice that the segment PP' is P-congruent to QQ', so the lengths of the segments perpendicular to m are the same in hyperbolic geometry.
   • What do the loci of Q and Q' look like? Are they parts of E-circles? Are they P-lines?
   • Explain what you see.
2. Poincare equal distance and ruler
   • In the same setup as above, with m = line AB, construct P-lines a and b perpendicular to m through A and B. Reflect A in b to get A'.
   • 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 line reflections are isometries, these sequences of points are equally spaced and give a ruler on line m.
3. Poincare "translations"
   • Recall the link between double reflection in parallel lines and translations and parallel strips in Euclidean geometry. Describe the hyperbolic transformation which is this double reflections in ultraparallels in the P-model.
   • Explain why the loci of Q and Q' are transformed into themselves by this double reflection.
   • Are lines mapped to parallel lines by this transformation?