Topics in Topology and Geometry        Homework Assignment        Fall 2000, Prof. Golé
 


Assignment 5, due Monday October 23: Read through Chapter 8 (at least
Problems: # 7.10, 8.2
and

- (Problem you gave yourself on the board) Use Mathematica to compute the curvature and torsion of
(I am not quite sure about the curve you chose-any one will do!) (t^2/2,  t^3-t, sin(t^2))

- a. Give the parameterization of the torus centered at the origin, with axis of rotation the y-axis, with (Big and small) radius R and r.
  b. Find the equation of the tangent plane at the point ((R+r)/sqrt(2), r/sqrt(2),  (R+r)/sqrt(2))

-Bonus problem: Have Mathematica draw the above torus (your choice of R and r, make it easy to change) and its tangent plane at once. Better still, make it so that very minor changes are required to draw the torus and the tangent plane at any pointyou wish.

Assignment 4, due Monday October 16: Read through Chapter 7
Problems: # 7.2, 7.3, 7.4, 7.7, 7.8


Assignment 3, due Thursday October 5: # 6.3, 6.4, 6.8, 6.6, 6.10 (complement it by answering the question: what is an evolute of a curve? and by drawing some ellipses and their evolutes using Mathematica)
(4, 4, 4, 4, 6 pts)
look at 6.7: new bonus problem B3...

Assignment 2, due Thursday September 28: # 3.3, 6.1, 6.2 and:
Find a parametrization of the circle passing through the points (1, 0, 0), (0,1,0), (0,0,1). Then find a parameterization by arclength of the same circle.
(4, 4, 4, 6 points)
Read up to page 71.
Assignment 1, due  Monday, September 18:

Read Chapter 4.

Essay Assignment: write small essays on the following questions:
Q1: How is the question of existence of a measure of length treated in Chapter 2
Q2: How does the Chapter 2 adresses the measure of angles and their comparative magnitudes?
Q3: What does it mean to enter a triangle?
Q4: What are the 5 groups of axioms Hilbert proposed for a geometry, and where are they located in the book?
Q5: What is the main goal of Chapter 3? How does it prepare us for another geometry than the Euclidean one?
(each question on 4pt)

Reading, for Thursday, September 14:

Read: Chapter 3.

Reading, for Tuesday, September 12:

Read: Chapter 2.


Bonus Problems (can be handed in at any time during the semester)

B1 (20 pts) Draw a hyperbolic circle in the 1/2 half plane or the unit disk, centered at any given point (x,y). Do it rigorously, giving a parameterization of the circle. Graph it with Mathematica.

B2 (6 points) Construct a hyperbolic parallelogram with different opposite angles

B3 (10 points) #6.7

B4 (7+3 points) Have Mathematica draw the torus and its tangent plane at once. Better still, make it so very minor changes are required to draw the torus and the tangent plane at any point you wish.

Note: the point value of the extra credit problem might decrease through the semester!

Note: Your homework will be graded over a total of 200 points.