Research and Investigate a Surface

You may work alone or in a group of up to 3 people.

Surfaces will be assigned on a first come-first-served basis in the choice selection feature on ASULearn.

2.1.14 helicoid
2.1.16 Enneper's Surface
p. 80:
2.2.4 Mobius strip
p. 114-117:
3.2.11 hyperboloid of 2 sheets
3.2.12 hyperboloid of 1 sheet
3.2.13 elliptic paraboloid
3.2.14 hyperbolic paraboloid
3.2.16 saddle
3.2.17 Kuen's Surface
3.2.19 cone
p. 120-121:
3.3.2 torus
3.3.9 pseudosphere
p. 218 in the print book:
color picture ellipsoid
p. 168-170:
4.2.8 Henneberg's Surface
4.2.8 Catalan's surface
4.2.8 Scherk's Fifth Surface
p. 197:
4.9.1 catenoid
strake

If the book has parameterizations you should use that one. If not, you can find one on Wolfram's MathWorld.

Other surfaces can be chosen if they have a reasonable parametrization(s) that can be used for the Maple worksheets

Explore the following questions via researching our textbook and elsewhere (keep track of ALL your references for # 15) 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 components with your classmates (see #16).
1. Print a picture of the surface. Be sure to provide any picture references (and any other references) in #15.
2. In bullet point format, summarize the physically interesting features of your surface.
3. Research a few historical mathematicians who are related to your surface, including at least one mathematician from a country outside of the U.S. when possible (it could be someone who laid groundwork on the surface, or peripheral but connected work). In bullet point format, include relevent dates, names and their contributions to your surface.
4. In bullet point format, summarize the significance of your surface in historical or current research, and (if possible) real-life applications [try to find as many as you can here].
5. Search MathSciNet for current journal articles related to your surface. Choose one you find interesting and write down the full bibliographic reference from the MathSciNet database.
  
  What is MathSciNet? Historically, mathematicians communicated by letters, during visits, or by reading each other's published articles or books once such means became available. For example, Marin Mersenne had approximately 200 correspondents. Some mathematical concepts were developed in parallel by mathematicians working in different areas of the world who were not aware of each others progress. In an effort to increase the accessibility of mathematics research articles, reviews began appearing in print journals like Zentralblatt fur Mathematik, which originated in 1931, and Mathematical Reviews, which originated in 1940. Since the 1980s, electronic versions of these reviews have allowed researchers to search for publications. In October 2015 MathSciNet, the electronic version of Mathematical Reviews, listed over 3.2 million items.
6. 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
7. Provide parametrization(s) for your surface that can be used for the Maple worksheets
8. 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).
9. 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.
10. 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 #6 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)
11. 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.
12. Write the metric form (ds/dt)² for your surface and compare it to the flat metric form. Does the Pythagorean theorem hold for your surface?
13. 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]
14. What kind(s) of Gauss curvature is possible on your surface (positive, negative, zero)?
15. Discuss Gauss curvature intuition for one interesting point on your surface (from a curvature perspective, like the intuition we did for the torus).
16. Use the Maple file Gauss Curvature and Mean Curvature to explore the shape operator a bit. Is the rate of change of the surface normal u a multiple of either x_u or x_v individually?
17. If you used any references other than our text, the Maple files or me, then give proper credit.
18. You will turn in all of the above. In addition, prepare a short presentation for your classmates based on the following components:
  1. 1
  2. 2
  3. one mathematician from #3 and their contributions to your surface
  4. one real-life application from #4
  5. 5
  6. 7
  7. 12
  8. 15
  9. 17