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Surface of revolution

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From Wikipedia, the free encyclopedia

Surface created by rotating a curve about an axis

A portion of the curvex = 2 + cos(z) rotated around thez-axis

Atorus as a square revolved around an axis parallel to one of its diagonals.

Asurface of revolution is asurface inEuclidean space created byrotating acurve (thegeneratrix) one fullrevolution around anaxis of rotation (normally notintersecting the generatrix, except at its endpoints).^[1]The volume bounded by the surface created by this revolution is thesolid of revolution.

Examples of surfaces of revolution generated by a straight line arecylindrical andconical 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 agreat circle, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates atorus which does not intersect itself (aring torus).

Properties

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The sections of the surface of revolution made by planes through the axis are calledmeridional 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 ofhyperboloids (of either one or two sheets) andelliptic paraboloids are surfaces of revolution. These may be identified as those quadratic surfaces all of whosecross sections perpendicular to the axis are circular.

Area formula

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If the curve is described by theparametric functionsx(t),y(t), witht ranging over some interval[a,b], and the axis of revolution is they-axis, then thesurface areaA_y is given by theintegral $A_{y}=2\pi \int _{a}^{b}x(t)\,{\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt,$ provided thatx(t) is never negative between the endpointsa andb. This formula is the calculus equivalent ofPappus's centroid theorem.^[3] The quantity ${\sqrt {\left({dx \over dt}\right)^{2}+\left({dy \over dt}\right)^{2}}}\,dt$ comes from thePythagorean theorem and represents a small segment of the arc of the curve, as in thearc length formula. The quantity2πx(t) is the path of (the centroid of) this small segment, as required by Pappus' theorem.

Likewise, when the axis of rotation is thex-axis and provided thaty(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 continuous curve is described by the functiony =f(x),a ≤x ≤b, then the integral becomes $A_{x}=2\pi \int _{a}^{b}y{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx=2\pi \int _{a}^{b}f(x){\sqrt {1+{\big (}f'(x){\big )}^{2}}}\,dx$ for revolution around thex-axis, and $A_{y}=2\pi \int _{a}^{b}x{\sqrt {1+\left({\frac {dy}{dx}}\right)^{2}}}\,dx$ for revolution around they-axis (provideda ≥ 0). These come from the above formula.^[5]

This can also be derived from multivariable integration. If a plane curve is given by $\langle x(t),y(t)\rangle$ then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by $\mathbf {r} (t,\theta )=\langle y(t)\cos(\theta ),y(t)\sin(\theta ),x(t)\rangle$ with $0\leq \theta \leq 2\pi$ . Then the surface area is given by thesurface integral $A_{x}=\iint _{S}dS=\iint _{[a,b]\times [0,2\pi ]}\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt.$

Computing the partial derivatives yields ${\frac {\partial \mathbf {r} }{\partial t}}=\left\langle {\frac {dy}{dt}}\cos(\theta ),{\frac {dy}{dt}}\sin(\theta ),{\frac {dx}{dt}}\right\rangle ,$ ${\frac {\partial \mathbf {r} }{\partial \theta }}=\langle -y\sin(\theta ),y\cos(\theta ),0\rangle$ and computing thecross product yields ${\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}=\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle =y\left\langle \cos(\theta ){\frac {dx}{dt}},\sin(\theta ){\frac {dx}{dt}},{\frac {dy}{dt}}\right\rangle$ where the trigonometric identity $\sin ^{2}(\theta )+\cos ^{2}(\theta )=1$ was used. With this cross product, we get ${\begin{aligned}A_{x}&=\int _{a}^{b}\int _{0}^{2\pi }\left\|{\frac {\partial \mathbf {r} }{\partial t}}\times {\frac {\partial \mathbf {r} }{\partial \theta }}\right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }\left\|y\left\langle y\cos(\theta ){\frac {dx}{dt}},y\sin(\theta ){\frac {dx}{dt}},y{\frac {dy}{dt}}\right\rangle \right\|\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\cos ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\sin ^{2}(\theta )\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}\int _{0}^{2\pi }y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ d\theta \ dt\\[1ex]&=\int _{a}^{b}2\pi y{\sqrt {\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}}}\ dt\end{aligned}}$ where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar.

For example, thespherical surface with unit radius is generated by the curvey(t) = sin(t),x(t) = cos(t), whent ranges over[0,π]. Its area is therefore ${\begin{aligned}A&{}=2\pi \int _{0}^{\pi }\sin(t){\sqrt {{\big (}\cos(t){\big )}^{2}+{\big (}\sin(t){\big )}^{2}}}\,dt\\&{}=2\pi \int _{0}^{\pi }\sin(t)\,dt\\&{}=4\pi .\end{aligned}}$

For the case of the spherical curve with radiusr,y(x) =√r² −x² rotated about thex-axis ${\begin{aligned}A&=2\pi \int _{-r}^{r}{\sqrt {r^{2}-x^{2}}}\,{\sqrt {1+{\frac {x^{2}}{r^{2}-x^{2}}}}}\,dx\\&=2\pi r\int _{-r}^{r}\,{\sqrt {r^{2}-x^{2}}}\,{\sqrt {\frac {1}{r^{2}-x^{2}}}}\,dx\\&=2\pi r\int _{-r}^{r}\,dx\\&=4\pi r^{2}\,\end{aligned}}$

Aminimal surface of revolution is the surface of revolution of the curve between two given points whichminimizes surface area.^[6] A basic problem in thecalculus of variations is finding the curve between two points that produces this minimal surface of revolution.^[6]

There are only twominimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): theplane and thecatenoid.^[7]

Coordinate expressions

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A surface of revolution given by rotating a curve described by $y=f(x)$ around the x-axis may be most simply described by $y^{2}+z^{2}=f(x)^{2}$ . This yields the parametrization in terms of $x {\displaystyle x}$ and $\theta$ as $(x,f(x)\cos(\theta ),f(x)\sin(\theta ))$ . If instead we revolve the curve around the y-axis, then the curve is described by $y=f({\sqrt {x^{2}+z^{2}}})$ , yielding the expression $(x\cos(\theta ),f(x),x\sin(\theta ))$ in terms of the parameters $x {\displaystyle x}$ and $\theta$ .

If x and y are defined in terms of a parameter $t {\displaystyle t}$ , then we obtain a parametrization in terms of $t {\displaystyle t}$ and $\theta$ . If $x {\displaystyle x}$ and $y {\displaystyle y}$ are functions of $t {\displaystyle t}$ , then the surface of revolution obtained by revolving the curve around the x-axis is described by $(x(t),y(t)\cos(\theta ),y(t)\sin(\theta ))$ , and the surface of revolution obtained by revolving the curve around the y-axis is described by $(x(t)\cos(\theta ),y(t),x(t)\sin(\theta ))$ .

Geodesics

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Meridians are always geodesics on a surface of revolution. Other geodesics are governed byClairaut's relation.^[8]

Toroids

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Main article:Toroid

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 acircle, then the object is called atorus.

References

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^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.
^^a ^bWeisstein, 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.