Investigate a Surface

Explore the following via researching our text and elsewhere (keep track of your references) and our Maple files. Write it up in your own words but you may use pictures from elsewhere (with proper reference). You will turn in all of the following, and share some portions with your classmates.

1. Print a picture of the surface. Be prepared to project this to the class from the document camera and provide the reference.
2. In bullet point format, summarize the physically interesting features of your surface
3. In bullet point format, for your surface, summarize the historical significance and/or significance in current research and/or significance in applications
   [try to find as many as you can here]
4. Write down formulas for the following entities as a review, using letters, words, etc... Assume that you have a surface parametrized as X(u,v) to start with. Do NOT do any calculations for your surface here, but do explain how to calculate each from X(u,v) (and your answers may build upon one another, ie using part a in another part).
   1. Normal to a surface X(u,v)
   2. Curvature of a curve gamma(t) on X(u,v)
   3. Normal Curvature of gamma(t) on X(u,v)
   4. Geodesic Curvature of gamma(t) on X(u,v)
   5. A curve is a geodesic if the geodesic curvature = __________.
   6. E (of a surface X(u,v))
   7. F
   8. G
5. Provide parametrization(s) for your surface that can be used for the Maple worksheets
6. Use the Maple file on geodesic and normal curvatures in order to provide your new values for each of the following (these are the commands I used for a sphere or radius 1) for a curve that NOT is a geodesic:
   g := (x,y) -> [cos(x)*sin(y), sin(x)*sin(y), cos(y)]:
   a1:=0: a2:=Pi: b1:=0: b2:=Pi:
   c1 := 1: c2 := 3:
   Point := 2:
   f1:= (t) -> t:
   f2:= (t) -> 1:
   and print a graph that shows part of the surface, part of a NOT-a-geodesic curve, and the curvature vectors (label which is which).
7. Use the Maple file on geodesic and normal curvatures in order to provide your new values for each of the following (these are the commands I used for a sphere or radius 1) for a curve that is a geodesic, or as close to a geodesic as you can get:
   g := (x,y) -> [cos(x)*sin(y), sin(x)*sin(y), cos(y)]:
   a1:=0: a2:=Pi: b1:=0: b2:=Pi:
   c1 := 1: c2 := 3:
   Point := 2:
   f1:= (t) -> t:
   f2:= (t) -> 1:
   and print a graph that shows part of the surface, part of a geodesic, and the curvatures.
8. Write down formulas for the following entities as a review, using letters, words, etc... Assume that you have a surface parametrized as X(u,v) to start with. Do NOT do any calculations for your surface here, but do explain how to calculate each from X(u,v) (and your answers may build upon one another, ie using part a in another part, and/or your answers in #4 above).
   1. l
   2. m
   3. n
   4. Gauss Curvature of a surface X(u,v)
   5. Mean Curvature of a surface X(u,v)
9. Use the Maple file Gauss Curvature and Mean Curvature to calculate
   1. Normal to the surface
   2. Unit Normal to the surface
   3. E, F and G for your surface
   4. l, m, n for your surface
   5. Gauss curvature for your surface
   6. Mean curvature for your surface
   and print what Maple provides.
10. Write the metric form (ds/dt)² for your surface
11. Set up, but do not solve, a surface area integral using E, F and G. Explain what your limits of integration would be to find the surface area for the entire surface. In addition, if your surface has infinite surface area, cap it off somewhere, and explain what the limits would be and what the capped picture would look like. [For instance for a parametrization of the x-y plane [u,v,0], u and v range from negative infinity to positive infinity, but a capped version of the plane could be a square with u and v from -1..1]
12. Discuss instrinsic Gauss curvature arguments (positive, negative, or zero) for a few interesting points on your surface (from a curvature perspective).
13. If you used any references other than our text, the Maple files or me, then give proper credit.