Surface of revolution

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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).[1]

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, and if the circle is rotated about an axis that does not intersect the circle, then it generates a torus.

Properties[edit]

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.[2]

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[edit]

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.[3] 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[4]

 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^bf(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 x(t) = sin(t), y(t) = cos(t), when t ranges over [0,\pi]. Its area is therefore

\begin{align}
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{align}

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

\begin{align}
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{align}

A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area.[5] A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.[5]

Rotating a function[edit]

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[edit]

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

Applications of surfaces of revolution[edit]

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[edit]

References[edit]

  1. ^ Analytic Geometry Middlemiss, Marks, and Smart. 3rd Edition Ch. 15 Surfaces and Curves, § 15-4 Surfaces of Revolution LCCN 68-15472 pp 378 ff.
  2. ^ Wilson, W.A.; Tracey, J.I. (1925), Analytic Geometry (Revised ed.), D.C. Heath and Co., p. 227 
  3. ^ Calculus, George B. Thomas, 3rd Edition, Ch. 6 Applications of the definite integral, §§ 6.7,6.11, Area of a Surface of Revolution pp 206-209, The Theorems of Pappus, pp 217-219 LCCN 69-16407
  4. ^ Singh (1993). Engineering Mathematics (6 ed.). Tata McGraw-Hill. p. 6.90. ISBN 0-07-014615-2. , Chapter 6, page 6.90
  5. ^ a b Weisstein, Eric W.. "Minimal Surface of Revolution". Mathworld. Wolfram Research. Retrieved 2012-08-29. 

External links[edit]