Surface Of Revolution
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A surface of revolution is a
surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
created by rotating a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
(the '' generatrix'') one full
revolution In political science, a revolution (, 'a turn around') is a rapid, fundamental transformation of a society's class, state, ethnic or religious structures. According to sociologist Jack Goldstone, all revolutions contain "a common set of elements ...
around an ''
axis of rotation Rotation or rotational/rotary motion is the circular movement of an object around a central line, known as an ''axis of rotation''. A plane figure can rotate in either a clockwise or counterclockwise sense around a perpendicular axis intersect ...
'' (normally not intersecting the generatrix, except at its endpoints). The volume bounded by the surface created by this revolution is the ''
solid of revolution In geometry, a solid of revolution is a Solid geometry, solid figure obtained by rotating a plane figure around some straight line (the ''axis of revolution''), which may not Intersection (geometry), intersect the generatrix (except at its bound ...
''. 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 In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Discussion Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
, and if the circle is rotated around an axis that does not intersect the interior of a circle, then it generates a
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
which does not intersect itself (a ring torus).


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. 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 , , with ranging over some interval , and the axis of revolution is the -axis, then the
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
is given by the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
A_y = 2 \pi \int_a^b x(t) \, \sqrt \, dt, provided that is never negative between the endpoints and . This formula is the calculus equivalent of Pappus's centroid theorem. The quantity \sqrt \, dt comes from the
Pythagorean theorem In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
and represents a small segment of the arc of the curve, as in the
arc length Arc length is the distance between two points along a section of a curve. Development of a formulation of arc length suitable for applications to mathematics and the sciences is a problem in vector calculus and in differential geometry. In the ...
formula. The quantity is the path of (the centroid of) this small segment, as required by Pappus' theorem. Likewise, when the axis of rotation is the -axis and provided that is never negative, the area is given by A_x = 2 \pi \int_a^b y(t) \, \sqrt \, dt. If the continuous curve is described by the function , , then the integral becomes A_x = 2\pi\int_a^b y \sqrt \, dx = 2\pi\int_a^bf(x)\sqrt \, dx for revolution around the -axis, and A_y =2\pi\int_a^b x \sqrt \, dx for revolution around the ''y''-axis (provided ). These come from the above formula. 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(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 the
surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, o ...
A_x = \iint_S dS = \iint_ \left\, \frac \times \frac\right\, \ d\theta\ dt = \int_a^b \int_0^ \left\, \frac \times \frac\right\, \ d\theta\ dt. Computing the partial derivatives yields \frac = \left\langle \frac \cos(\theta), \frac \sin(\theta), \frac \right\rangle, \frac = \langle -y \sin(\theta), y \cos(\theta), 0 \rangle and computing the
cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...
yields \frac \times \frac = \left\langle y \cos(\theta)\frac, y \sin(\theta)\frac, y \frac \right\rangle = y \left\langle \cos(\theta)\frac, \sin(\theta)\frac, \frac \right\rangle where the trigonometric identity \sin^2(\theta) + \cos^2(\theta) = 1 was used. With this cross product, we get \begin A_x &= \int_a^b \int_0^ \left\, \frac \times \frac\right\, \ d\theta\ dt \\ ex&= \int_a^b \int_0^ \left\, y \left\langle y \cos(\theta)\frac, y \sin(\theta)\frac, y \frac \right\rangle \right\, \ d\theta\ dt \\ ex&= \int_a^b \int_0^ y \sqrt\ d\theta\ dt \\ ex&= \int_a^b \int_0^ y \sqrt\ d\theta\ dt \\ ex&= \int_a^b 2\pi y \sqrt\ dt \end where the same trigonometric identity was used again. The derivation for a surface obtained by revolving around the y-axis is similar. For example, the spherical surface with unit radius is generated by the curve , , when ranges over . Its area is therefore \begin A &= 2 \pi \int_0^\pi \sin(t) \sqrt \, dt \\ &= 2 \pi \int_0^\pi \sin(t) \, dt \\ &= 4\pi. \end For the case of the spherical curve with radius , rotated about the -axis \begin A &= 2 \pi \int_^ \sqrt\,\sqrt\,dx \\ &= 2 \pi r\int_^ \,\sqrt\,\sqrt\,dx \\ &= 2 \pi r\int_^ \,dx \\ &= 4 \pi r^2\, \end A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes
surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...
. A basic problem in the
calculus of variations The calculus of variations (or variational calculus) is a field of mathematical analysis that uses variations, which are small changes in Function (mathematics), functions and functional (mathematics), functionals, to find maxima and minima of f ...
is finding the curve between two points that produces this minimal surface of revolution. There are only two minimal surfaces of revolution ( surfaces of revolution which are also minimal surfaces): the plane and the catenoid.


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 by y^2+z^2 = f(x)^2. This yields the parametrization in terms of 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), yielding the expression (x \cos(\theta), f(x), x \sin(\theta)) in terms of the parameters x 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 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

Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation.


Toroids

A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid. 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 A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, then the object is called a
torus In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
.


See also

* Channel surface, a generalisation of a surface of revolution * Gabriel's Horn * Generalized helicoid * Lemon (geometry), surface of revolution of a circular arc * Liouville surface, another generalization of a surface of revolution *
Spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
*
Surface integral In mathematics, particularly multivariable calculus, a surface integral is a generalization of multiple integrals to integration over surfaces. It can be thought of as the double integral analogue of the line integral. Given a surface, o ...
* Translation surface (differential geometry)


References


External links

* * {{DEFAULTSORT:Surface Of Revolution Integral calculus Surfaces of revolution