Question

Investigation 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) \(a=4, \quad b=1\) (b) \(a=4, \quad b=2\) (c) \(a=8, \quad b=1\) (d) \(a=8, \quad b=3\)

Short answer

The shape and size of the torus change with different values of 'a' and 'b'. Parameter 'a' affects the size of the torus and its center's distance from the origin, while 'b' influences the thickness of the torus

Step-by-step solution

Input Function into System

Input the given function \( r(u, v) = (a + b \cos v) \cos u \mathbf{i} + (a + b \cos v) \sin u \mathbf{j} + b \sin v \mathbf{k}\) into the computer algebra system and set the limits of \(u\) and \(v\) to be within the interval \([0, 2\pi]\).

Plug in Values for a and b - Case a=4, b=1

Substitute \(a=4\) and \(b=1\) into the equation on the computer algebra system and then plot the graph. Observe the shape of the torus.

Plug in Values for a and b - Case a=4, b=2

Similarly, substitute \(a=4\) and \(b=2\) into the equation on the computer algebra system and plot the graph. Again, observe the shape of the torus and notice any changes in its shape compared to the graph from Step 2.

Plug in Values for a and b - Case a=8, b=1

Substitute \(a=8\) and \(b=1\) into the equation and plot the graph. Observe the shape of the torus and notice any changes in its shape compared to the graphs from previous steps.

Plug in Values for a and b - Case a=8, b=3

Finally, substitute \(a=8\) and \(b=3\) into the equation and plot the graph. Observe the shape of the torus and notice any changes in its shape compared to the graphs from previous steps.

Analyze the Results

After plotting all four graphs, analyze the effects of 'a' and 'b' on the shape of the torus. The parameter 'a' determines the overall size and the distance from the center of the torus, while 'b' influences the thickness of the torus.