Describe the parametric surface $S$ parametrized by $$\begin{equation*} \mathbf{r}(u,v) =( 3+\cos v) \cos u\,\mathbf{i} +( 3+\cos v) \sin u\,\mathbf{j}+\sin v\mathbf{k} \end{equation*}$$

where $0\leq u\leq 2\pi$ and $0\leq v\leq 2\pi .$

Solution We begin by finding some $u$-coordinate curves. Since the parametrization involves trigonometric functions, we choose $0,\dfrac{\pi }{2},\pi ,$ and $\dfrac{3\pi }{2}.$ Then the $u$-coordinate curves are $$\begin{equation*} \begin{array}{rcl@{\qquad}rrcl} \mathbf{r}(u,0) &=&4\cos u\,\mathbf{i}+4\sin u\,\mathbf{j} & \mathbf{r}\left( u,\dfrac{\pi }{2}\right) &=&3\cos u\,\mathbf{i}+3\sin u\,\mathbf{j+k}\\[5pt] \mathbf{r}( u,\pi ) &=&2\cos u\,\mathbf{i}+2\sin u\,\mathbf{j} & \mathbf{r}\left( u,\dfrac{3\pi }{2}\right) &=&3\cos u\,\mathbf{i}+3\sin u\,\mathbf{j-k} \end{array} \end{equation*}$$

We identify the graphs of each of these curves by using parametric equations. For example, for $\mathbf{r}(u,0)$, we have $x=4\cos u,$ $y=4\sin u,$ and $$x^{2}+y^{2}=( 4\cos u) ^{2}+( 4\sin u)^{2}=4^{2}( \cos ^{2}u+\sin ^{2}u) =4^{2}$$

Figure 42 A torus centered at the origin.

a circle of radius $4$ centered at the origin. Similarly, $\mathbf{r=r} ( u,\pi )$ is a circle of radius $2$ centered at the origin. The graphs of $\mathbf{r}\left( u,\dfrac{\pi }{2}\right)$ and $\mathbf{r}\left( u,\dfrac{3\pi }{2}\right)$ are both circles of radius $3,$ but the center of $\mathbf{r=r}\left( u,\dfrac{\pi }{2}\right)$ is at $( 0,0,1)$ and the center of $\mathbf{r=r}\left( u,\dfrac{3\pi }{2}\right)$ is at $( 0,0,-1) .$ Do you see why? See the blue circles in Figure 42.

Using $0,\dfrac{\pi }{2},\pi ,$ and $\dfrac{3\pi }{2}$ for $u,$ we obtain the $v$-coordinate curves: $$\begin{array}{rcl@{\qquad}llcl} \mathbf{r}( 0,v) &=&( 3+\cos v) \mathbf{i}+\sin v\,\mathbf{k} & \mathbf{r}\left( \dfrac{\pi }{2},v\right) =( 3+\cos v) \,\mathbf{j}+\sin v\,\mathbf{k} \\[5pt] \mathbf{r}( \pi ,v) &=&-( 3+\cos v) \,\mathbf{i}+\sin v\, \mathbf{k} & \mathbf{r}\left( \dfrac{3\pi }{2},v\right) =-( 3+\cos v) \,\mathbf{j}+\sin v\,\mathbf{k} \end{array}$$

Again, we identify the graphs of these curves as circles. Each graph is a circle of radius $1,$ but the centers of the circles $\mathbf{r=r}(0,v) ,$ $\mathbf{r=r}( \pi ,v) ,$ $\mathbf{r=r}\left( \dfrac{\pi }{2},v\right)$, and $\mathbf{r=r}\left( \dfrac{3\pi }{2},v\right)$ are $( 3,0,0)$, $( -3,0,0) ,$ $(0,3,0) ,$ and $( 0,-3,0) ,$ respectively. See the red circles in Figure 42.

The surface $S$ is a torus. This particular torus results from revolving a circle of radius $1$ about a circle of radius $3$ centered at the origin.