Homework 7
Math 352, Fall 2014
Due Date: Friday, October 24
1. In Conoid.gif, the black point has coordinates (0, 0, sin t) at time t. The red line initially
lies along the x-axis, and rotates at a rate of 1 rad/sec in the horizontal direction while
also moving vertically.
(a) Find parametric equations for the surface traced out by the red line.
(b) Find a Cartesian equation for this surface. Your answer should be a polynomial
equation involving x, y, and z.
2. The unit circle in the xy-plane begins rotating around the y-axis at a rate of 1 rad/sec,
while simultaneously moving in the y direction at a rate of 1 unit/sec, as shown in
TwistingCircle.gif. Find parametric equations for the surface traced out by the circle.
3. Let T be the trefoil knot parameterized by
~x(t) =
2 sin 2t − sin t, 2 cos 2t + cos t, sin 3t .
Find parametric equations for any surface of finite area whose boundary is T . (One
such surface is shown in TrefoilSurface.png. Make sure to include bounds on u and v
in your parameterization.
4. Let P be the parabola y = x2 in the xy-plane, and let L be the line y = x − 1 in the
xy-plane. Find parametric equations for the surface of revolution obtained by rotating
P around L.