Surfaces in Cylindrical Coordinates

Many objects have cylindrical symmetry (for example, grain silos, water towers, footballs, soda cans, and the cooling towers of a nuclear power plant). Therefore it is natural to use cylindrical coordinates to analyze these objects mathematically.

In the following, you may want to use Maple to plot some surfaces in polar coordinates. The command plot3d with the option coords=cylindrical will be useful for plotting functions of the form r=f(theta,z).

Recall that if a surface is parametrized by

( r(s,t), theta(s,t), z(s,t) )

then the tangent plane is spanned by the two vectors

Question #1

In cylindrical coordinates, find r as a function of z and theta in order to parametrize and plot the following surfaces for theta=0..2*Pi and z=-1..1.

A cylinder
A vase (Hint: Try a cubic function of z. You do not need to worry about the bottom of the vase, ie, the vase doesn't need to hold water. We've provided an additional hint if you need it.)

For each surface,

Compute the tangent plane to the surface at theta=Pi and z=0.
Sketch the tangent plane on the surface.

Next: Saddle Surfaces
Previous: Introduction

Frederick J. Wicklin<fjw@geom.umn.edu>

Document Created: Sat Jan 7 1995
Last modified: Wed Jan 18 12:48:16 1995