# Surface of revolution

A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around a straight line in its plane (the axis).

Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on whether or not the line is parallel to the axis. A circle that is rotated about any diameter generates a sphere of which it is then a great circle, and if the circle is rotated about an axis that does not intersect the circle, then it generates a torus which does not intersect itself (a ring torus).

## Properties

The sections of the surface of revolution made by planes through the axis are called meridional sections. Any meridional section can be considered to be the generatrix in the plane determined by it and the axis.

The sections of the surface of revolution made by planes that are perpendicular to the axis are circles.

Some special cases of hyperboloids (of either one or two sheets) and elliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whose cross sections perpendicular to the axis are circular.

## Area formula

$A_{y}=2\pi \int _{a}^{b}x(t)\ {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,$ provided that $x(t)$ is never negative between the endpoints a and b. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

${\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}$ comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity $2\pi x(t)$ is the path of (the centroid of) this small segment, as required by Pappus' theorem.

Likewise, when the axis of rotation is the $x$ -axis and provided that $y(t)$ is never negative, the area is given by

$A_{x}=2\pi \int _{a}^{b}y(t)\ {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt.$ If the curve is described by the function y = f(x), axb, then the integral becomes

$A_{x}=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=2\pi \int _{a}^{b}f(x){\sqrt {1+\left(f'(x)\right)^{2}}}\,dx$ for revolution around the x-axis, and

$A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dx}{dy}}\right)^{2}}}\,dy$ for revolution around the y-axis (Using ayb). These come from the above formula.

For example, the spherical surface with unit radius is generated by the curve y(t) = sin(t), x(t) = cos(t), when t ranges over $[0,\pi ]$ . Its area is therefore

{\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {\left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}} {\begin{aligned}A&{}=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx\\&{}=2\pi r\int _{-r}^{r}\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx\\&{}=2\pi r\int _{-r}^{r}\,dx\\&{}=4\pi r^{2}\,\end{aligned}} A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area. A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.

There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.

## Geodesics on a surface of revolution

Geodesics on a surface of revolution are governed by Clairaut's relation.

## Applications of surfaces of revolution

The use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed.