Parameterisation for a set

Question

I am struggling to find a parameterization for the following set :

$$F=\left\{(x,y,z)\in\mathbb R^3\middle| \left(\sqrt{x^2+y^2}-R\right)^2 + z^2 = r^2\right\} \quad\text{with }R>r$$

I also have to calculate the area.

I know its a circle so we express it in terms of the angle but my problem is with the $x$ and $y$ . They are not defined uniquely by the angle.
Please explain with details because it is more important for me to understand than the answer itself

Answer 1

One possibility is to begin by setting $\sqrt {x^2 + y^2} - R = r \cos u$ and $z = r \sin u$. The first equality implies that $x^2 + y^2 = (R + r \sin u)^2$, so now put $x = (R + r \sin u) \cos v$ and $y = (R + r \sin u) \sin v$. Take $u \in (- \frac \pi 2, \frac \pi 2)$ and $v \in (0, 2\pi)$.

Answer 2

For a given $z$ this gives a circle around the $z$ axis with radius $a:=\sqrt{x^2+y^2}$, subject to $(a-R)^2+z^2=r^2$ or $a^2-2Ra+R^2+z^2-r^2=0$ or

$$a=\frac{2R\pm\sqrt{4R^2-4R^2-4z^2+4r^2}}{2}=R\pm\sqrt{r^2-z^2}$$

So in fact you get two circles. You probably want $\lvert z\rvert\le\lvert r\rvert$. Let's use $b:=\pm\sqrt{r^2-z^2}$. Then $b^2+z^2=r^2$ so in e.g. the $x=0$ plane you have a circle of radius $r$ around the $x$ axis.

Going from $b$ to $a$, you take that circle and move its points perpendicular to the $z$ axis, by an offset of $R$. And then you rotate the resulting shape around the $z$ axis to go from $a$ to $x$ and $y$. You should get a torus.

To parametrize that, you need one parameter on the circle in the $x=0$ plane. Let's use an angle and call it $\theta$. So you have $z=r\sin\theta$ and $b=r\cos\theta$ with $-\pi<\theta<\pi$. Then you get $a=R+b=R+r\cos\theta$ as the radius of the circle in the plane for fixed $z$. And there you use $x=a\cos\varphi$ and $y=a\sin\varphi$ with $-\pi<\varphi<\pi$. Taken together:

\begin{align*} x&=(R+r\cos\theta)\cos\varphi \\ y&=(R+r\cos\theta)\sin\varphi \\ z&=r\sin\theta \end{align*}

If you'd rather avoid negative angles, you might as well choose either or both of the individual circle parametrizations from the $[0,2\pi]$ range instead of the $[-\pi,\pi]$ I suggested. And you may switch $\sin$ and $\cos$ for the same angle if you want.