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, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
created by rotating a
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
(the
generatrix In geometry, a generatrix () or describent is a point, curve or surface that, when moved along a given path, generates a new shape. The path directing the motion of the generatrix motion is called a directrix or dirigent. Examples A cone can be ...
) around an
axis of rotation Rotation around a fixed axis is a special case of rotational motion. The fixed-axis hypothesis excludes the possibility of an axis changing its orientation and cannot describe such phenomena as wobbling or precession. According to Euler's rota ...
. 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. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
, 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 (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
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 paraboloid In geometry, a paraboloid is a quadric surface that has exactly one axis of symmetry and no center of symmetry. The term "paraboloid" is derived from parabola, which refers to a conic section that has a similar property of symmetry. Every plan ...
s 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 area is given by the
integral In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
: 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 In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The ...
. The quantity :\sqrt 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 may refer to: Business * Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s * Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services * ...
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. For example, the
spherical surface A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
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 In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in t ...
is the surface of revolution of the curve between two given points which minimizes
surface area The surface area 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 definition of arc ...
. 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 functions and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
is finding the curve between two points that produces this minimal surface of revolution. There are only two
minimal surfaces of revolution In mathematics, a minimal surface of revolution or minimum surface of revolution is a surface of revolution defined from two points in a half-plane, whose boundary is the axis of revolution of the surface. It is generated by a curve that lies in ...
(
surfaces of revolution A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) around an axis of rotation. Examples of surfaces of revolution generated by a straight line are cylindrical and conical surfaces depending on ...
which are also minimal surfaces): the
plane Plane(s) most often refers to: * Aero- or airplane, a powered, fixed-wing aircraft * Plane (geometry), a flat, 2-dimensional surface Plane or planes may also refer to: Biology * Plane (tree) or ''Platanus'', wetland native plant * ''Planes' ...
and the
catenoid In geometry, a catenoid is a type of surface, arising by rotating a catenary curve about an axis (a surface of revolution). It is a minimal surface, meaning that it occupies the least area when bounded by a closed space. It was formally describe ...
.


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

Meridian Meridian or a meridian line (from Latin ''meridies'' via Old French ''meridiane'', meaning “midday”) may refer to Science * Meridian (astronomy), imaginary circle in a plane perpendicular to the planes of the celestial equator and horizon * ...
s are always geodesics on a surface of revolution. Other geodesics are governed by
Clairaut's relation In classical differential geometry, Clairaut's relation, named after Alexis Claude de Clairaut, is a formula that characterizes the great circle paths on the unit sphere. The formula states that if γ is a parametrization of a great circle then ...
.


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 points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is const ...
, then the object is called a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
.


Applications

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 In geometry and topology, a channel or canal surface is a surface formed as the envelope of a family of spheres whose centers lie on a space curve, its '' directrix''. If the radii of the generating spheres are constant, the canal surface is ca ...
, a generalisation of a surface of revolution *
Gabriel's Horn Gabriel's horn (also called Torricelli's trumpet) is a particular geometry, geometric figure that has infinite surface area but finite volume. The name refers to the Christian tradition where the archangel Gabriel blows the horn to announce Last ...
*
Generalized helicoid In geometry, a generalized helicoid is a surface in Euclidean space generated by rotating and simultaneously displacing a curve, the ''profile curve'', along a line, its ''axis''. Any point of the given curve is the starting point of a circular hel ...
*
Lemon (geometry) In geometry, a lemon is a geometric shape, constructed as the surface of revolution of a circular arc of angle less than half of a full circle, rotated about an axis passing through the endpoints of the lens (or arc). The surface of revolution o ...
, surface of revolution of a circular arc * Liouville surface, another generalization of a surface of revolution * Solid of revolution *
Spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has cir ...
*
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, one may ...
*
Translation surface (differential geometry) In differential geometry a translation surface is a surface that is generated by translations: * For two space curves c_1, c_2 with a common point P, the curve c_1 is shifted such that point P is moving on c_2. By this procedure curve c_1 gene ...


References


External links

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