surface of revolution

A surface swept out by spinning a profile curve around an axis, using the rotation angle as one parameter and position along the curve as the other.

A surface of revolution uses the angle of rotation as one parameter and the position along the original curve as the other. Revolving $y=f(x)\ge0$ about the $x$ -axis keeps $x$ as one parameter and spins the $(y,z)$ -cross section by $\theta$ ; revolving a curve about the $z$ -axis keeps $z$ as one parameter, $\mathbf{r}(z,\theta) = \langle f(z)\cos\theta,\, f(z)\sin\theta,\,z\rangle$ . Every point at a fixed value of the "height" parameter traces out a full circle of radius $f(\cdot)$ as $\theta$ ranges over $[0,2\pi]$ .

Examples

Revolving $y=x^2$ , $0\le x\le 1.5$ , about the $x$ -axis gives $\mathbf{r}(x,\theta) = \langle x,\; x^2\cos\theta,\; x^2\sin\theta \rangle$ .
A vase on a potter's wheel, a lampshade, and a wine glass are all surfaces of revolution.