MATH1020 Calculus II	Maple Lab 8
Section 1-4	Due: Tuesday, November 16, 1999.
http://www.math.rpi.edu/~mitchj/math1020

This lab consists of two independent questions. In the first question, we consider the graphs of some quadric surfaces and in the second we examine the another family of curves. Be sure to follow the lab guidelines as described on the class handouts.

1.

Surfaces described by equations that involve terms quadratic in x, y, and z are called quadric surfaces. For example, the equation 5x²+3y²+z²=1 describes an ellipsoid. This surface can be plotted using Maple by plotting its parametric representation:

> x:=cos(t)/sqrt(5); y:=sin(t)*cos(s)/sqrt(3); z:=sin(t)*sin(s);
> plot3d([x,y,z],t=0..Pi,s=0..2*Pi);

Note that every choice of s and t gives a point on the surface, and every point on the surface corresponds to some choice of s and t. We use the fact that

$$\cos^2(t) + \sin^2(t) = 1 \mbox{ for all } t.$$

You will need to also use the equality

$$\cosh^2(s) - \sinh^2(s) = 1 \mbox{ for all } s$$

in what follows. Note that an appropriate range for s will include both positive and negative numbers.

Classify and use Maple to plot the quadric surfaces described by the equations

(a)

$2x\sp2-5y\sp2-3z\sp2=1$.

(b)

$x\sp2+4y\sp2=2z$.

(c)

$x\sp2-3y-4z\sp2+4z=1$.

In each case, you will need to figure out an appropriate parameterization of the surface. The text will be helpful to classify the surfaces.

2.

(Stewart, page 832.) Members of the family of surfaces given in spherical coordinates by the equation

$$\rho = 1 + 0.2 \sin(m\theta) \sin(n\phi)$$

have been suggested as models for tumors, and have been called bumpy spheres and wrinkled spheres. Investigate this family of surfaces using Maple, assuming that m and n are integers. What roles do m and n play in the shape of the surfaces?

To plot this surface with m=1 and n=2 in Maple, use the commands:

> rho := 1 + 0.2 * sin(m*theta) * sin(n*phi);
> m:=1; n:= 2; plot3d(rho,theta=0..2*Pi,phi=0..Pi,coords=spherical);