A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation.[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 around any diameter generates a sphere of which it is then a great circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a torus which does not intersect itself (a ring torus).
A portion of the curve x = 2 + cos z rotated around the z-axis
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.[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
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 Ay is given by the integral
\( {\displaystyle A_{y}=2\pi \int _{a}^{b}x(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,}\)
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 2πx(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]
\( {\displaystyle A_{x}=2\pi \int _{a}^{b}y(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt.}\)
If the continuous curve is described by the function y = f(x), a ≤ x ≤ b, then the integral becomes
\( {\displaystyle A_{x}=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=2\pi \int _{a}^{b}f(x){\sqrt {1+{\big (}f'(x){\big )}^{2}}}\,dx}\)
for revolution around the x-axis, and
\( {\displaystyle A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx}\)
for revolution around the y-axis (provided a ≥ 0). These come from the above formula.[5]
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,π]. Its area is therefore
\( {\displaystyle {\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {{\big (}\cos(t){\big )}^{2}+{\big (}\sin(t){\big )}^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}}}\)
For the case of the spherical curve with radius r, y(x) = √r2 − x2 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.[6] A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.[6]
There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid.[7]
Coordinate expressions
A surface of revolution given by rotating a curve described by y=f(x) around the x-axis may be most simply described in cylindrical coordinates by \( {\displaystyle r=f(z)} \). In Cartesian coordinates, this yields the parametrization in terms of z and \( \theta \) as \( {\displaystyle (f(z)\cos(\theta ),f(z)\sin(\theta ),z)} \). If instead we revolve the curve around the y-axis, then the curve is described in cylindrical coordinates by \( {\displaystyle z=f(r)} \), yielding the expression \( {\displaystyle (r\cos(\theta ),r\sin(\theta ),f(r))} \) in terms of the parameters r and \( \theta \).
If x and y are defined in terms of a parameter t, then we obtain a parametrization in terms of t and \( \theta \). If x and y are functions of t, then the surface of revolution obtained by revolving the curve around the x-axis is described in cylindrical coordinates by the parametric equation \( {\displaystyle (r,\theta ,z)=(y(t),\theta ,x(t))} \), and the surface of revolution obtained by revolving the curve around the y-axis is described by \( {\displaystyle (r,\theta ,z)=(x(t),\theta ,y(t))} \). In Cartesian coordinates, these (respectively) become \( {\displaystyle (y(t)\cos(\theta ),y(t)\sin(\theta ),x(t))} \) and \( {\displaystyle (x(t)\cos(\theta ),x(t)\sin(\theta ),y(t))} \). The above formulae for surface area then follow by taking the surface integral of the constant function 1 over the surface using these parametrizations.
Geodesics on a surface of revolution
Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation.[8]
Toroids
Main article: Toroid
A toroid generated from a square
A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid.[9] For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus.
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
Lemon (geometry), surface of revolution of a circular arc
Liouville surface, another generalization of a surface of revolution
Solid of revolution
Surface integral
Generalized helicoid
Translation surface (differential geometry)
References
Middlemiss; Marks; Smart. "15-4. Surfaces of Revolution". Analytic Geometry (3rd ed.). p. 378. LCCN 68015472.
Wilson, W.A.; Tracey, J.I. (1925), Analytic Geometry (Revised ed.), D.C. Heath and Co., p. 227
Thomas, George B. "6.7: Area of a Surface of Revolution; 6.11: The Theorems of Pappus". Calculus (3rd ed.). pp. 206–209, 217–219. LCCN 69016407.
Singh, R.R. (1993). Engineering Mathematics (6 ed.). Tata McGraw-Hill. p. 6.90. ISBN 0-07-014615-2.
Swokowski, Earl W. (1983), Calculus with analytic geometry (Alternate ed.), Prindle, Weber & Schmidt, p. 617, ISBN 0-87150-341-7
Weisstein, Eric W. "Minimal Surface of Revolution". MathWorld.
Weisstein, Eric W. "Catenoid". MathWorld.
Pressley, Andrew. “Chapter 9 - Geodesics.” Elementary Differential Geometry, 2nd ed., Springer, London, 2012, pp. 227–230.
Weisstein, Eric W. "Toroid". MathWorld.
External links
Weisstein, Eric W. "Surface of Revolution". MathWorld.
"Surface de révolution". Encyclopédie des Formes Mathématiques Remarquables (in French).
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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