Surface of revolution

Surface of revolution
A surface of revolution is a surface in Euclidean space created by rotating a curve (the generatrix) one full revolution around an axis of rotation (normally not intersecting the generatrix, except at its endpoints).
The volume bounded by the surface created by this revolution is the solid of revolution.
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).
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 is given by the integral
provided that is never negative between the endpoints and . This formula is the calculus equivalent of Pappus's centroid theorem. The quantity
comes from the Pythagorean theorem and represents a small segment of the arc of the curve, as in the arc length 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
If the continuous curve is described by the function then the integral becomes
for revolution around the -axis, and
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 then its corresponding surface of revolution when revolved around the x-axis has Cartesian coordinates given by with . Then the surface area is given by the surface integral
.
Computing the partial derivatives yields
,
and computing the cross product yields
where the trigonometric identity was used. With this cross product, we get
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
For the case of the spherical curve with radius , rotated about the -axis
A minimal surface of revolution is the surface of revolution of the curve between two given points which minimizes surface area. A basic problem in the calculus of variations 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 around the x-axis may be most simply described by . This yields the parametrization in terms of and as . If instead we revolve the curve around the y-axis, then the curve is described by , yielding the expression in terms of the parameters and .
If x and y are defined in terms of a parameter , then we obtain a parametrization in terms of and . If and are functions of , then the surface of revolution obtained by revolving the curve around the x-axis is described by , and the surface of revolution obtained by revolving the curve around the y-axis is described by .
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, then the object is called a torus.
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, 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
Surface integral
Translation surface (differential geometry)
References
External links
Integral calculus
Surfaces of revolution