Surfaces Assignment

Back

This is a group (2 people) assignment. You will investigate the surfaces below. For each surface you should

1. Determine the tangent vectors, T_u and T_v and use these to provide a normal to the surface. Do these exist for all points on the surface?
2. Compute the surface area for some interesting range of parameters (u,v). At least set up the integral in those cases in which you may not be able to analytically compute an answer. Verify your results in Maple.
3. Graph your surfaces and make any observations relevant to your computations.

Simplify your answer where possible and use enough points an a large enough domain to see smooth, interesting surfaces. You may want to look at surf2.mws for help in using Maple. [Right-click file and save to your TIMMY folder. Then open the file in Maple.]

Surfaces:

• Helicoid x = (u cos v, u sin v,v)
• Catenoid x = (-cosh u cos v, -cosh u sin v, u)
• Catalan's Surface x = (u-cosh v sin u, 1-cosh v cos u, 4 sin(u/2)sinh(v/2))
• Henneberg's Surface x = (2sinh u cos v-(2 sinh 3u cos 3v)/3, 2sinh u sin v
  +2 (sinh 3u sin 3v)/3, 2cosh 2u cos 2v)
• Enneper's Surface x = (u-(u³/3)+uv², v-(v³/3)+vu²,u²-v²)
• Scherk's Surface sin z = sinh x sinh y
• Explore some seashell plots. For examples of parametrizations leasding to seashell shapes, look  here, here, here, or here.
• Pick some other interesting surfaces from pages like Mathworld. Or you can check some galleries like the Virtual Math Museum.

All work should be typed with double-spacing and 12 pt font. You will be expected to use correct English grammar and punctuation. You will be graded on the evidence of work, mathematical detail and understanding, proper exposition and neatness. Your work should also be supported with properly labeled and embedded plots; i.e., insert plots into the report and do not attach a multitude of Maple worksheets without discussion. Any references used should be cited as well. These reports will count towards the project component of your grade.

Top