Solid of revolution

A portion of the curve x=2+cos z rotated around the z axis

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 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 its center point generates a sphere, and if the circle is rotated about a coplanar axis, not crossing the circle, then 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 $A_{y}$ is given by the integral

$A_{y}=2\pi \int _{a}^{b}x(t)\ {\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,$

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), a ≤ x ≤ b, 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 a ≤ y ≤ b). 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 $[0,\pi ]$. Its area is therefore

${\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {\left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}}$

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

${\begin{aligned}A&{}=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx\\&{}=2\pi r\int _{-r}^{r}\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx\\&{}=2\pi r\int _{-r}^{r}\,dx\\&{}=4\pi r^{2}\,\end{aligned}}$

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.

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 $f(u)\sin v$ and $f(u)\cos v$. For example, to rotate a function $y=f(x)$ around the x-axis starting from the top of the $xz$-plane, parameterize it as ${\vec {r}}(u,v)=\langle u,f(u)\sin v,f(u)\cos v\rangle$ for $u=x$ and $v\in [0,2\pi ]$ .

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 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.

See also

• Channel surface, a generalisation of a surface of revolution
• Gabriel's Horn
• Liouville surface, another generalization of a surface of revolution
• Solid of revolution
• Surface integral

External links

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• "Surface de révolution" at Encyclopédie des Formes Mathématiques Remarquables