Surface_of_revolution

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

A surface of revolution is a surface created by rotating a curve lying on some plane (the generatrix) around a straight line (the axis of rotation) that lies on the same plane.

Examples of surfaces generated by a straight line are the cylindrical and conical surfaces. A circle that is rotated about a (coplanar) axis through the center generates a sphere. If the axis is coplanar and outside the circle it generates a toroidal surface.

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

$A\; =\; 2\; \backslash pi\; \backslash int\_a^b\; x(t)\; \backslash \; \backslash sqrt\{\backslash left(\{dx\; \backslash over\; dt\}\backslash right)^2\; +\; \backslash left(\{dy\; \backslash over\; dt\}\backslash right)^2\}\; \backslash ,\; dt,$

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

$\backslash left(\{dx\; \backslash over\; dt\}\backslash right)^2\; +\; \backslash left(\{dy\; \backslash over\; dt\}\backslash right)^2$

comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length formula. The quantity $2\backslash pi\; x(t)$ 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

$A=2\backslash pi\backslash int\_a^b\; y\; \backslash sqrt\{1+\backslash left(\backslash frac\{dy\}\{dx\}\backslash right)^2\}\; \backslash ,\; dx$

for revolution around the x-axis, and

$A=2\backslash pi\backslash int\_a^b\; x\; \backslash sqrt\{1+\backslash left(\backslash frac\{dx\}\{dy\}\backslash right)^2\}\; \backslash ,\; dy$

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

$A\; =\; 2\; \backslash pi\; \backslash int\_0^\backslash pi\; \backslash sin(t)\; \backslash sqrt\{\backslash left(\backslash cos(t)\backslash right)^2\; +\; \backslash left(\backslash sin(t)\backslash right)^2\}\; \backslash ,\; dt\; =\; 2\; \backslash pi\; \backslash int\_0^\backslash pi\; \backslash sin(t)\; \backslash ,\; dt\; =\; 4\backslash pi.$

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 $v\backslash in[0,2\backslash pi)$ .

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
• Canal surface - a generalisation of a surface of revolution
• Liouville surface - another generalization of a surface of revolution

This article is licensed under the GNU Free Documentation License. It uses material from Wikipedia