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

Examples of surfaces generated by a straight line are cylindrical and conical surfaces when the line is coplanar with the axis, as well as hyperboloids of one sheet when the line is skew to the axis. A circle that is rotated about a diameter generates a sphere and if the circle is rotated about an coplanar axis other than the diameter it generates a torus.

Area formula

If the curve is described by the parametric functions $x(t)$, $y(t)$, with $t$ ranging over some interval $[a,b]$, and the axis of revolution is the $y$ axis, then the area $A$ is given by the integral

provided that $x(t)$ is never negative. This formula is the calculus equivalent of Pappus's centroid theorem. The quantity

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity $2\backslash pi\; x(t)$ is the path of (the centroid of) this small segment, as required by Pappus's theorem.

If the curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes

for revolution around the x-axis, and

for revolution around the y-axis. These come from the above formula.

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

For the case of the spherical curve with radius r, $y(x)\; =\; \backslash sqrt\{r^2\; -\; x^2\}$ rotated about the x-axis

Rotating a function

To generate a surface of revolution out of any 2-dimensional scalar function $y=f(x)$, simply make $u$ the function's parameter, set the axis of rotation's function to simply $u$, then use $v$ to rotate the function around the axis by setting the other two functions equal to $f(u)\backslash sin\; v$ and $f(u)\backslash cos\; v$ conversely. For example, to rotate a function $y=f(x)$ around the x-axis starting from the top of the $xz$-plane, parameterize it as $\backslash vec\; r(u,v)=\backslash langle\; u,f(u)\backslash sin\; v,f(u)\backslash cos\; v\backslash rangle$ for $u\backslash in\; x$ and $v\backslash in[0,2\backslash pi]$ .

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 surface of revolutions 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.

See also

• Solid of revolution

• Gabriel's Horn

• Channel surface, a generalisation of a surface of revolution

• Liouville surface, another generalization of a surface of revolution