Solved: Investigation Use a computer algebra system to graph the torus for each set of

Chapter 15, Problem 53

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Use a computer algebra system to graph the torus

\(\mathbf{r}(u, v)=(a+b \cos v) \cos u \mathbf{i}+\)

                             \((a+b \cos v) \sin u \mathbf{j}+b \sin v \mathbf{k}\)

for each set of values of a and b, where \(0 \leq u \leq 2 \pi\) and \(0 \leq v \leq 2 \pi\). Use the results to describe the effects of a and b on the shape of the torus.

(a) a = 4,    b = 1

(b) a = 4,     b = 2

(c) a = 8,     b = 1

(d) a = 8,     b = 3

Text Transcription:

r(u, v) = (a + b cos v) cos ui +

(a + b cos v) sin uj + b sin vk

0 leq u leq 2 pi

0 leq v leq 2 pi