# Definition:Surface of Revolution

## Definition

### Intuitive Definition

Let $F$ be a plane figure.

Let $L$ be a straight line in the same plane as $F$.

Let $F$ be rotated one full turn about $L$ in $3$-dimensional space.

The surface of revolution generated by $F$ around $L$ is the surface of the solid of revolution generated by $F$ around $L$.

### Formal Definition

Let $H = \set {\tuple {x, y} : y \in \R_{> 0}} \subset \R^2$ be the open upper half-plane.

Let $F \subset H$ be a $1$-dimensional embedded submanifold.

The surface of revolution is the subset $S_F \subseteq \R^3$ such that:

$S_F = \set {\tuple {x, y, z} : \tuple {\sqrt{x^2 + y^2}, z} \in F}$

where $F$ is the generating curve of the surface of revolution.

## Also see

• Results about surfaces of revolution can be found here.