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 lab may take more than one lab session to finish. You should keep your work and your notes, because the figures and a discussion will be turned in as an assignment.

Lab Activity 1. Parallel Line Experiments

Goal: See some examples of P-lines, intersections of lines, convering parallels and ultraparallels.

Download this Sketchpad file lab08.gsp. It has an automatching tool for inverting a point in a circle and also constructing the radical axis of the point and the circle. The figure in the sketch shows the construction. You can use this as a starting point.

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 continue to use these names and create new tools with automatching..

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

1. Construct the supporting circle of a P-line AB. Given two P-points A and B construct a P-line through A and B. Translating this into a Euclidean plane statement, A and B are two points inside the disk. Construct a circle through A and B orthogonal to the circle h. This is the supporting circle of the line. The P-line itself is the arc inside the circle.
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 tool for the P-line AB. The actual P-line can be constructed as an arc sitting on the support circle, which can be made a dashed circle so indicate its "ghostly" nature (its points are not points of the model). Here is a good way to do this after you make the support circle dashed: (a) Intersect the support circle of the P-line with the circle h to get points E and F; (b) Let K be the (Euclidean) center of the support circle of the P-line AB. Connect K to O, the center of h by a segment OKand intersect OK with the support circle to get point G; (c) construct the arc on 3 points EGF. Make the arc either thin or thick, not dashed. (d) hide segment OK and point G. (e) Make a tool. (f) Set up automatching.
4. Construct a second P-line CD in your figure.
5. Experiment: Visualizing Intersections and parallels
   • Drag D with A, B and C fixed. Look for cases when the lines intersect and when they do not meet.
   • Suppose the two P-lines intersect at a P-point. How many points of intersection can the P-lines have? How many points of intersection can the support circles have. Explain.
   • When the two supporting circles intersect at one point L, where must the circles intersect? How are the support circles related? In this case, do the P-lines intersect in a P-point? In this case the P-lines are said to be a (critical or converging) parallel. For a fixed P-line AB and a fixed P-point C, how many P-lines through C are critically parallel to P-line AB?
   • If the supporting circles of the P-lines do not meet at all, we say the P-lines are ultra-parallel.
6. Construct a third P-line EF in your figure.
7. 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?
8. 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 drag 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 orthogonal 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. With the built-in radical axis tool, you should be able to make if work for G on or off of P-line AB.
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 to be 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 use them to define congruence. Then we can ask later whether it is possible to define a distance so that these transformations preserve this distance. But we ccn go ahead and test for congruence using these special transformations.

We can define congruence; we say two figures are congruent (with respect to these special transformations) if one is the image of the other by one of the special transformations. The special transformations are called "congruence transformations" or "isometries". The actual definition of how to measure distance with a number will come later.

In the P-model, we will first define reflection in a P-line. Then we will say that the isometries are the transformations of the P-model that are compositions of reflections of P-lines. Thus two figures will be P-congruent if there is a sequence of P-line reflections that will take one to the other. A distance measure is not needed at this point, but we think of two P-congruent segments as usual as segments that have the same length in hyperbolic geometry.

To make this idea work, one must figure out the meaning of P-line reflection. The obvious candidate is to define reflection in a P-line as inversion in the supporting circle of the P-line. But there is an important point to check to see that this definition is valid (see below). Is this really a transformation of the P-model. More precisely, if A is a P-point is the inversion A' also a P-point or can it be inverted to a point outside the P-disk to a "non-point" for the P-model?

Once this point is checked below, then we can use the definition to check that figures are congruent by reflecting them.

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).
   • Think of a reason that A' must be a P-point (i.e., is inside the P-disk).
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 (as P-triangles, not as Euclidean figures).
   • 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. Explain why m is perpendicular to the P-line AB. Let M be the intersection of the two P-lines; then MA and MB are congruent (in the P-model) since m reflects one segment to the other. Thus by definition, M is the P-midpoint of the P-segment AB and m is the perpendicular bisector. (Thus the P-line of symmetry of AB, which is the perpendicular bisector can be defined using congruence of segments and angles without needing a distance measure. 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 any 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 tool.

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

This activity begins to explore measurement by finding "strips of constant width" and "equal steps" along a P-line. Since this is all in the P-model the widths and equal steps are given by P-congruent P-segments. While we are inside the P-model it is not necessary to write "P-" every single time, since congruence means P-congruence if we are working inside the P-model.

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 (in the P-model) from P-line m. Why? You should review the concept of distance from a point to a line and think how this transfers to the P-model using congruence but not numerical measurement.
   • 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?