# Parametric surfaces - Parameterization of torus

A rotational surface area is created when a curve in the $xz$-plane, with parameterization $\def\i{\pmb{i}}\def\k{\pmb k}$ $r=x(t)\i + z(t)\k$ , $t \in [a,b]$, rotates around the $z$-axis. This surface is parameterized by;

$$t \mapsto \bigl( x(t)\cos\theta, x(t)\sin\theta, z(t)\bigr)^t$$

$t \in [a,b]$, $\theta \in [0,2\pi)$.
Use the above information to parameterize the torus that is created when a circle $(x-3)^2+z^2=4$ rotates around the $z$-axis.

The above is the first of a number of exercises on parametric surfaces which I will attempt to complete. I have not recieved any solved examples or specific guidance on the subject so I am hoping that someone here could show me how this is done.

I apologize for any spelling errors and the general lack of LaTeX.

Help is much appreciated!

## 1 Answer

So we have to parametrize the given circle $(x-3)^2 + z^2 = 4$, in the form $\def\i{\pmb i}\def\k{\pmb k}$$r(t) = x(t)\i + z(t) \k$$ on some$t$-intervall$[a,b]$, then we can apply the given information. Keeping in mind, the construction of sine and cosine, we have that$(\cos t, \sin t)$describes a circle of radius 1 around$0$, we have a circle of radius$2 = \sqrt 4$around$(3,0)$, now$(2\cos t, 2\sin t)$is a circle of radius 2 around$0$, adding$(3,0)\$ we are done and obtain $$r(t) = (3 + 2\cos t)\i + 2\sin t \;\k, \qquad t \in [0,2\pi]$$ for the given circle. Now apply the given information.

• So I get: x(t,θ) = (8 + 2 cos t) cos θ, y(t,θ) = (8 + 2 cos t) sin θ and z(t,θ) = 2 sin t? Is there a way to include a picture? I have drawn the figuere in MATLAB and would like to check if it is accurate. – Soakr Oct 15 '13 at 14:45
• @Soakr: Looks fine. – martini Oct 15 '13 at 14:49