Math 487 Lab 5

Exploring the Poincare Disk Model

This is an example of a model for non- Euclidean geometry.

We call the points and lines in the Poincare 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 I (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 I which is the "universe" that you are operating in.

(2) For P-lines, you can either draw the whole circle orthogonal to I and just ignore the part outside or else you can also construct the arc interior to I 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 I 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 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 I and when they are (nearly collinear with O).
  3. 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 (parallel). Note particularly the limiting parallel case 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 limiting parallel of AB, do the two circles representing these lines meet? Where? Does this contradict the fact that parallel lines have no point in common?
  4. 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 Portfolio Figure 1.1.
  5. 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 Portfolio Figure 1.2. (You may want to save this as a script.)
  6. 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 limiting parallel in the same direction (same ideal point) and if lines b and c are limiting parallel in the same direction then lines a and c are limiting parallel in the same direction, unlike the case of ultraparallels.
    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.

  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 Portfolio 2.1. 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.
    Question. What kind of family are the 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 Portfolio 2.2
    Question. The supporting circles of m and n belong to a pencil of circles. What kind? Explain why the circles, the supporting circle of p and the circle I. belong to the pencil of circles orthogonal to supports of m and n. What kind of pencil is this?

Lab Activity 3. Triangles and Angle Sums.

  1. Construct a P-triangle ABC. Measure the angles and calculate the angle sum and the defect. Save this as a script.
  2. Drag the triangle around and notice how the defect changes.
    How small can you make the defect by dragging. What is true of the triangle when the defect is small?
    How large can you make the defect by dragging. What is true of the triangle when the defect is large?
  3. Construct a P-triangle DEF calculate the defect (use your script!). Drag DEF so that it is inside triangle ABC. What is the relationship between the defects? Save and print as Portfolio 3.1
  4. Let J, K, and L be ideal points. Construct the P-lines JK, KL, and LJ and think of this an "ideal triangle." What is the defect in this case? These are sometimes called Omega triangles.

Lab Activity 4. Mirror lines.

  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.
  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 Portfolio 4.1.
  3. In a new figure, construct a P-triangle ABC and construct the three perpendicular bisectors of the sides. Are they concurrent? Save and print some interesting cases as Portfolio 4.2.

Lab Activity 5. P-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. Take a P-point A and two P-lines m and n through A. Now choose any P-point B. Reflect B across m to get B', then reflect B' across n to get B'' then reflect B'' across m to get B''', etc. This gives points so that the P-segments AB, AB', AB'', AB''', etc. are all congruent (if we believe that reflection is an isometry.
  2. All these points lie on a Euclidean circle c. (You can see that this looks correct with Sketchpad, why is it true?

  3. So construct the Euclidean circle c through B, B' and B'' to get the P- circle with P-center A through B. Save and print as Portfolio 5.1.
  4. Drag B and trace this circle to see a family of concentric circles.
  5. 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 Portfolio 5.2.

Lab Activity 6. Equal P-Steps.

  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.
  3. Also do this for B. Look for the ruler markings on p. Save and print as Portfolio 6.1.
  4. Remember the link between double reflection and translation.