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.

You should save your work and your notes, because some figures and a discussion will be turned in as an assignment.

Download this Sketchpad file lab08.gsp. This file has a two sketches and 3 tools.

1. One sketch called Basic P model shows a P-point A and its inversion in h, along with the radical axis of A and h.
2. The second page is called Two Asymptotics; it will be used in Lab Activity 1, #6.
3. The tool "invpt + rad axis (interior)" is an automatching tool for inverting a P- point A in the circle h. It also constructs the radical axis of the point A and the circle h; this is needed to construct circles through A orthogonal to h.. The figure in the sketch shows the construction. You can use this as a starting point.
4. The tool "Arc from circle XA" is also automatching. It will construct an arc that is a P-line from a support circle, specifically the center of the support circle and a P-point on the circle.
5. The tool invpt C in OR will be used in Lab Activity 3. Use the other inversion tool in Lab Activity 1.

Automatch in lab08.gsp: This lab will work very well using Automatch in your tools. In the example file, the special circle of ideal points h has center P-center and passes through the point P-radius. If you use these names, the inversion tool with Automatch. To keep the names, you can duplicate pages in your file using Document Options or you can copy and paste the points with the special names.

Lab Activity 1. Parallel Line Experiments

Goal: See some examples of P-lines, intersections of lines, asymptotic parallels (also called converging or limiting parallels) and ultraparallels.

Begin with the circle h with center P-center and through the point P-radius. The points interior to h are the P-points. In the text below, the center of h will be called O and not P-center.

1. Construct the supporting circle of P-line AB. Draw two P-points A and B; then construct the supporting circle of the 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 the P-line AB using the furnished arc tool. To use the Arc tool, click on the center of the support circle and also on A. You can make the support circle dashed to make it less prominent. Notice that the tool also constructs two ideal points, since these can be useful in constructions. The ideal points are not P-points.
   
   [Note: Here is how the tool works. (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. Construct segment OK and intersect OK with the support circle to get point G; (c) construct the arc on 3 points EGF. (d) hide segment OK and point G.]
   
   
4. Experiment: Visualizing Intersections and parallels
   • Construct a second P-line CD in your figure.
   • 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 P-points of intersection can the P-lines have? How many points of intersection can the support circles have? .How are the intersection points related?
   • When the two support circles intersect at one point L, they must be tangent (why?). In this case, do the P-lines intersect in a P-point? In this case the P-lines are said to be asymptotic (or limiting or converging) parallels.
   • For a fixed P-line AB and a fixed P-point C, how many P-lines through C are asymptotically 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.
     
     
5. Experiments with critical parallels and 3 lines
• Construct a third P-line EF in your figure.
• Drag CD so that it is asymptotically parallel (approximately) to AB at one of the two ideal points (these are points at P-infinity).
• Drag EF so that it is asymptotically parallel (approximately) to AB at the other of the two ideal points . Note that this does not make CD and EF asymptotically parallel to each other. The P-line AB has two collections of asymptotic parallels, one at each direction at infinity.
• Now drag EF so that it is asymptotically parallel to both the other lines 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)?
  
  
6. 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). This shows that the relationship of being ultraparallel is not transitive!
  
• In the file lab08.gsp is a page called Two Asymptotics. This shows a P-line AB and P-lines m and n through F and through G each asymptotically parallel to AB but at different ideal points. Notice that these lines m and n are ultraparallel.
• Merge points F and G to get a single point G; now m and n are the two lines through G asymptotically parallel to AB. Now draw a P-point H and construct P-line GH. Check that this P-line GH is ultraparallel to AB so long as H is inside a pair of vertical angles defined by the two lines m and n through G.
• Trace P-line GH as you move H between m and n so that GH is ultraparallel to AB. This GH will trace a sort of "bow tie" of all the lines through G that are ultraparallel to AB.

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.)
   • Explain why the construction of such a p is impossible if m and n are 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 ultraparallel m and n belong to?
• What kind of pencil do the supporting circles of asumptotically parallel m and n belong to?
• What kind of pencil do the supporting circles of intersecting m and n belong to?
• In 3 above, 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?
• If a pencil is hyperbolic, it consists of Apollonian circles with respect to two limit points, what are the two points in 3 above?
• 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.

Congruence Defined by Transformations

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: 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.

Lab Activity 3. Mirror lines

To make this idea of congruence work, one must figure out the meaning of P-line reflection. 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'. You can use the supplied inv pt C in OR tool.
   • Drag A around the interior of h to show experimentally that for any P-point A, the point 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

The definition of the P-circle c with P-center A through B is the set of P-points C so that the P-segment AC is congruent to AB. In particular, this means that the perpendicular bisector of BC passes through A (and reflects B to C). This is very much like what was done in the D-model.

1. Constructing the P-center of an E-circle interior to h.
   • Draw an E-circle c with E-center J through K. Move the points J and K so that c is interior to h.
   • Construct two P-lines orthogonal to c. Intersect them to get the P-center of c.
   • Explain the construction in terms of pencils and orthogonal circles.
     
     
2. Construct an E-circle with P-center A
   • Let A and B be P-points. Construct any two P-lines m and n through A. Then construct the circle d through B that is orthogonal to m and n. This is the P-circle through B with P-center A.
   • Also notice that d does not intersect h; in fact d and h are both Apollonian circles of A and A'.
   • Drag B 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 (i.e., near the circle h)?
   • Make a P-circle tool.
     
     
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?
   • These 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.