Question

1. consider the circle $(y - 5)^2+z^2 = 9,x = 0$. let $s$ be the surface (torus) obtained by revolving this circle about the $z$-axis. find a parametric representation of $s$ with the parameters $\theta$ and $phi$ where $\theta$ and $phi$ are described as follows. if $(x,y,z)$ is any point on the surface then $\theta$ is the angle between the $x$-axis and the line joining $(0,0,0)$ and $(x,y,0)$ and $phi$ is the angle between the line joining $(x,y,z)$ and the center of the moving circle (which contains $(x,y,z)$) with the $xy$-plane.

Explanation:

Step1: Analyze the original circle

The circle in the \(yz -\)plane (\(x = 0\)) is given by \((y - 5)^{2}+z^{2}=9\). We can parametrize this circle as \(y=5 + 3\cos\phi\), \(z = 3\sin\phi\), where \(\phi\in[0,2\pi]\).

Step2: Apply the rotation about the \(z -\)axis

When we rotate a point \((0,y,z)\) about the \(z -\)axis, we use the rotation transformation. If we rotate the point \((0,y,z)\) (where \(y = 5+3\cos\phi\) and \(z = 3\sin\phi\)) about the \(z -\)axis by an angle \(\theta\), the \(x\) and \(y\) coordinates transform as follows:
The \(x\) coordinate is \(x=(5 + 3\cos\phi)\cos\theta\) and the \(y\) coordinate is \(y=(5 + 3\cos\phi)\sin\theta\), and \(z = 3\sin\phi\), where \(\theta\in[0,2\pi]\) and \(\phi\in[0,2\pi]\)

Answer:

\(x=(5 + 3\cos\phi)\cos\theta\), \(y=(5 + 3\cos\phi)\sin\theta\), \(z = 3\sin\phi\), \(\theta\in[0,2\pi]\), \(\phi\in[0,2\pi]\)