The following will constantly be updated during the quarter.

Homework1

1. 1) Let a circle of radius r roll on a fixed circle of radius R from the outside. Write down the equation of a marked point on the rolling cirlce.
2) How many rounds of self-rotations does the rolling circle complete when it rolls one round?

2. Do the above problem, but let the circle of radius r roll on the inside of the circle of radius R. Here we assume that r is no bigger than R.

3. (optional) What if we consider ellipses instead of circles in the above problems?

4. (optional) Write down the equation for a "cycloid", where we roll an ellipse instead of a circle on a line.

5. Problem 1 on page 8.

Homework2

1. Problem 3 and 5 on Page 8.

2. Problem 1 and 2 on Page 17 and 18.

3. Problem 3 on Page 18.

4. Problem 8 on Page 18.

5. Problem 9, Part a, on Page 18.

Homework3

1. Recall that a generalized helix is defined to be a constant speed space curve whose curvature is nonzero everywhere, and
whose tangent vectors make a constant angle with a fixed vector. Prove that a constant speed space curve p (t) is a generalized helix if and only if in a suitable orthogonal coordinate system the following holds
p(t)=q(t)+ct e'_3,
where q(t) is a constant speed curve in the x'y'-plane with curvature being nonzero everywhere, and c is a constant. Here e'_3 denotes the unit vector in the z'-direction.
Note: planar generalized helices are not so interesting. To avoid them, we can require the above c to be nonzero. Equivalently, we can require the constant angle to be smaller than 90 degrees. (We can assume that the angle is between 0 and 90 degrees.)

2. Problem 11 on Page 18 of the text.

3. Problem 12 on Page 19 of the text.

4. Problem 18 on Page 20 of the text.

Homework4

1. Problem 11 on page 32.
2. Problem 1 on Page 41.
3. Problem 2 on Page 41.
4. Problem 3 on Page 41.
5. Problem 4 on Page 41. Hint: consider curves on the surface and take derivative in a suitable equation along the curves.

Homework5

Good news: The first two homework sets have been graded!

1. Show that a round cylinder with one generating line removed is
isometric to a plane domain. (The special case of cross radius 1 was done in the lectures.)
2. Show that the forward part of a circular cone (with one generating
line removed) is isometric to a plane domain. The basic idea was explained in the lectures. You need to work out the explicit formulas. In particular, you need to find the relations
between the various involved quantities, such as the angles.
3. Determine the image of the Gauss map of a rotation torus. (Make a choice of its orientation.) Explain.
4. Determine the image of the Gauss map of the catenoid. (Make a choice of its orientation.) Explain.
5. Determine the image of the Gauss map of the graph of z=x^2+y^2, using the downward orientation.

Homework6

1. problem 3 a, b, c, d on page 53.
2. problem 4 on page 53. (only for problem 3 a, b, c, d).
3. problem 7 on page 53.
4. problem 13 on page 54.
5. problem 14 on page 54.
6. problem 15 on page 54.