Math 503 - Euclidean and Non-Euclidean Geometry

Math 503 - Euclidean and Non-Euclidean Geometry - Fall, 2006

Date: 11-28-2006

New/recent items :

(11-28-2006) Version 10.50 of Mobtran
(10-12-2006) Version 1.45 of "Complex Calculator". It now calculates hyperbolic distance.
(10-5-2006) Partial tutorial for Mobtran
(10-5-2006) Activity sheet for Circles of Apollonius
Lecture outline and technical summaries (pdf files)

Software Written materials Homework assignments Links

Instructor: Paul Glenn

Location/times: MCM 304; Monday, Wednesday, Friday from 12:10 to 1 p.m.

Text: Geometry by David Brannan, Matthew Esplen and Jeremy Gray, Cambridge University Press, 1999.

Contact:
Office: 207 MCM
Phone: 202-319-5221
Email:

Topics outline: After a careful introduction to the complex plane we will consider isometries and Mobius transformations, Klein's Erlanger Programm (transformation groups, congruence defined in terms of transformations), Mobius geometry and Poincare's disk model of hyperbolic geometry.

Software

Complex number calculator: Macintosh version and Windows version
(Version 1.45 on 10-12-2006)

The manual as a web page. The manual as a pdf file.

For the Mac version, double-click the "dmg" file and move the application to any folder you like. For the Windows version, unzip the file and put the application wherever you like.

Mobius transformation software "Mobtran" is here.

Written materials

Lecture outline (11-28-2006)
Technical summary (10-13-2006)
Brief review of tanh and tanh^-1
Hyperbolic triangle illustration
The angle sum theorem in a picture
Axioms of plane geometry

Software activities:

Homework assignments

#23: Click here
#22: Click here
#21: Click here
#20: Let L1 and L2 be ultraparallel d-lines whose centers are, respectively, q1 = (2,-.5) and q2 = (-1.5,-.5). Use the method of the proof of the "common perpendicular" theorem to find the center q3 of the d-line L3 which is perpendicular to both L1 and L2. Hint: You'll need to find the ideal points of L1 and L2. Use the exercise dealing with that task from homework #18. Also, use Mobtran to help with the numerical calculations and with plotting the results.
#19: Click here. Note: Omit question 3 on asymptotic triangles until we cover them in class, probably on Monday, October 30.
#18: Click here
#17: Click here
#16: Prove: If T is a non-euclidean transformation such that T(0)=0, then T is a composite of reflections across diameter lines of the unit circle.
#15: We proved the following theorem in class: If M(z) = (az+b)/(conj(b) z - conj(a)) is a Mobius transformation where |b| < |a| then M maps the unit disk to itself. ["conj(z)" stands for the conjugate of z]. As part of that proof, we actually found d-lines L and l such that M = r_L r_l. [where "r_l" stands for non-euclidean reflection across the d-line l.]

Let M(z) = (z - .5) / (-.5z + 1). For this transformation, determine the d-liens L and l explicitly. Optionally, you might plot these lines using "mobtran" and determine whether M is a non-euclidean rotation, non-euclidean limit rotation or non-euclidean translation.
#14: p. 324 ( sec. 6.2, exercises 2, 4 )
#13: Click here Note: Omit exercises 3 and 4 for now.
#12: Click here
#11: Click here
#10: Click here
#9: Click here
#8: Click here
#7: Click here
#6: For each exercise, sketch the result on graph paper.

(1) Let C be the circle with center (2,1) and radius 3, and let f(z) = inversion in C.
(a) Find the formula for f. (b) Use the formula to calculate f(1,2).

(2) Let C be the unit circle and let f be inversion in the unit circle. Let L be the line whose equation is x+2y+4 = 0. Find the equation of f(L).

(3) Consider two circles C_1 and C_2 both centered at (0,0) where C_1 has radius r_1 and C_2 has radius r_2. Let f_1 be inversion in C_1 and f_2 be inversion in C_2.

Calculate a formula for the composite g(z) = f_2 ( f_1(z) ) and simplify. Hint: you should get a function of the form g(z) = kz where k is a positive real number depending on r_1 and r_2.
#5: (1) Suppose a is a non-zero complex number with |a| ≠ 1. Consider the function f(z) = az. (i) Show that f is a transformation (bijection). (ii) Show that f is not an isometry. (iii) Show that f is conformal.

(2) Let H = { f : f is a conformal isometry }. Show that H satisfies the three group properties.

(3) Let J = { f : f is an anti-conformal isometry }. Show that J fails to satisfy at least some of the three group properties.
#4: (1) Suppose f and g are isometries. Show that f • g (f composed with g) and f^-1 are also isometries.

(2) Suppose f and g are conformal isometries. Show that f • g is also conformal.

(3) Suppose f is any isometry. Explain why you can calculate f(z) from knowing f(0,0), f(1,0) and f(0,1).
#3: Find formulas for the following isometries. Evaluate each at the point z = (2,3) = 2 + 3i. On graph paper, draw a diagram of each evaluation showing the relevant information: the constant vector in the case of translation, the rotation angle for the rotation and the reflection for the reflection.

(1) f(z) is rotation by the angle theta = 1.
(2) f(z) is translation by the b = (3,1).
(3) f(z) is reflection across the line containing (0,0) and (4,1).
(4) f(z) is reflection across the line containing (1,2) and (41).
#2: Click here
#1: Click here