Assignment for Week 10

This assignment has 3 parts, all with the theme of learning about hyperbolic geometry. Since some on-line parts are open-ended, it affords an opportunity for extra credit for anyone who wants to earn some. Also, extra or extra-good constructions could count for extra credit.

Part 1. Web sources for Hyperbolic Geometry - first due Wednesday 3/13 with later contributions possible

Find one or more interesting sources on the web for Hyperbolic NonEuclidean Geometry and/of its models. Report on your find on the online discussion list, giving the link or and a brief description or review indicating why it is particularly interesting or enlightening. Try to make your first posting before Wednesday, but you can add others, including some later the week. Also, you should in general avoid duplication unless you have something significant to add about a site already contributed.

Post your reports at this Online Discussion of Hyperbolic Geometry Sources

Part 2. Online discussion of Questions and Constructions in the Poincare model - online discussion begins ASAP, actual constructions to turn in due Friday 3/15

You can make your first explorations on the Web, using the Java software NonEuclid. This site also has a lot of very good questions, experiments, and explanations. You can use this as well as Sketchpad to answer questions.

There is (or will be shortly) a list of constructions and experiments for the week. Lab 10 will be devoted to these constructions, but the assignment may take longer unless you prepare in advance by thinking about the subject. (See part 3.)

Post your answers, suggestions, comments, etc. on the Online Discussion of Constructions and Figures.

Part 3. Constructions and Questions in the Poincare model

This is a list of constructions and explorations that you should carry out as your construction assignment, includind Assignment 3/15 (see above). You should know how to do the constructions indicated by hand or by Sketchpad, but you may prefer to do the measuring experiments with NonEuclid instead. You can get extra credit for contributing ideas for making this constructions online in the Hyperbolic Construction List. The sooner you do this the better.

Basic Poincare model constructions

All points and lines below are P-points and P-lines unless otherwise specified!

  1. Line through 2 points. Construct the line AB through two points A and B.
  2. Perpendicular line. Given a point C not on line AB, construct the line through C orthogonal to line AB.
  3. Asymptotic parallels. Given an ideal point X and a point E, construct the line through E and X. Given a point C not on line AB, construct the two lines through C which are asymptotically parallel to line AB.
  4. Ultraparallels. Given two ultraparallel lines m = line AB and n = line CD, construct the line p that is orthogonal to both.
  5. Line Reflection. Given a point C, construct the reflection C' of C in line AB.
  6. Perpendicular Bisector as Mirror Line. Given points A and B, construct the line m which reflects A to B. Be sure to include the case when B is the center of the P-disk. (This line will be the perpendicular bisector when distance is defined. Notice that this means than any point can be moved to any other by a line reflection.)
  7. Circle given center and radius. Given a point A and a point B, construct the circle with center A through B. (Hint: This circle is a P-circle. As will be explained by an experiment below, a P-circle is actually a Euclidean circle in the interior of the disk of the P-model. As in the DWEG model and the Stereo model, the P-center is not the Euclidean center, for a P-diameter of the circle is a P-line orthogonal to the circle.0
  8. Circle given diameter. Given points A and B, construct the circle with diameter AB.

Intermediate Constructions and Experiments in the Poincare Model

All points and lines below are P-points and P-lines unless otherwise specified!

  1. P-circles are E-circles. Given a point A and point P, consider the locus of reflections P' of P in lines AB for all possible points B. Oberve that this locus is a Euclidean circle.
  2. 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.
  3. Equilateral triangles. Using circles as in Euclidean geometry, given a segment AB on line m construct an equilateral triangle ABC. Measure the angles. As AB gets longer by moving B along line m and leaving A fixed, what happens to the angles?
  4. Perpendicular bisectors and circumcircles. Construct a triangle ABC and the perpendicular bisectors of the sides. Are they always concurrent? Are they ever concurrent? Are there any special relationships that you can point out? What does this say about constructing a circle through 3 points in hyperbolic geometry?
  5. Angle bisectors and incircles. Construct a triangle ABC and the (interior) angle bisectors. Are these always concurrent? Can one always construct a circle inscribed in ABC?
  6. Perpendicular line intersecting both sides of an angle. Given an acute angle ABC, if D is a point on the ray BA, does the perpendicular to BA through D always intersect ray BC?
  7. Poincare "constant width". Given m = line AB, construct a point C on m. Then construct the line a through A perpendicular to m and line c through C perpendicular to m. 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. What does the locus look like? Is it a P-line? Is it any recognizable Euclidean figure?
  8. Poincare equal distance. 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'. 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.

Questions to answer online

  1. If lines m and n are ultraparallel and also n and p are ultraparallel, must m and p also be ultraparallel? Give an example and/or explanation.
  2. If two lines e and f are orthogonal to the same line, are they ultraparallel always?
  3. Give a clear explanation for why the locus of points P' in Experiment 1 (P Circles are E circles) is a Euclidean (or inversive) circle. Use what you know about orthogonal circles to give a clear and concise reason.
  4. Give a clear explanation for why the locus of points P' in Experiment 2 (Horocycles) is a Euclidean (or inversive) circle. Use what you know about orthogonal circles to give a clear and concise reason.
  5. The elliptic pencil of P-lines defined by a P-point A is the set of all P-lines through A. Explain why the support circles of such a pencil of P-lines form an elliptic pencil of circles.
  6. The parabolic pencil of P-lines defined by an ideal point X is the set of all P-lines whose support circle passes through X. Explain why the support circles of such a pencil of P-lines form a parabolic pencil of circles.
  7. The hyperbolic pencil of P-lines defined by a P-line m is the set of all P-lines orthogonal to m. Explain why the support circles of such a pencil of P-lines form a hyperbolic pencil of circles.
  8. If ABC is a P-triangle, define the Defect D(ABC) as 180 - (angle A + angle B + angle C). If E is a point on side BC, explain why D(ABC) = D(ABE) + D(AEC).