Minimal Surface Of Revolution
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a minimal surface of revolution or minimum surface of revolution is a
surface of revolution A surface of revolution is a Surface (mathematics), surface in Euclidean space created by rotating a curve (the ''generatrix'') one full revolution (unit), revolution around an ''axis of rotation'' (normally not Intersection (geometry), intersec ...
defined from two
points A point is a small dot or the sharp tip of something. Point or points may refer to: Mathematics * Point (geometry), an entity that has a location in space or on a plane, but has no extent; more generally, an element of some abstract topologica ...
in a
half-plane In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example ...
, whose boundary is the axis of revolution of the surface. It is generated by 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 ...
that lies in the half-plane and connects the two points; among all the surfaces that can be generated in this way, it is the one that minimizes 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 ...
. 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.


Relation to minimal surfaces

A minimal surface of revolution is a subtype of minimal surface. A minimal surface is defined not as a surface of minimal area, but as a surface with a mean curvature of 0. Since a mean curvature of 0 is a
necessary condition In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If then ", is necessary for , because the truth of ...
of a surface of minimal area, all minimal surfaces of revolution are minimal surfaces, but not all minimal surfaces are minimal surfaces of revolution. As a point forms 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 ...
when rotated about an axis, finding the minimal surface of revolution is equivalent to finding the minimal surface passing through two circular wireframes. A physical realization of a minimal surface of revolution is
soap film Soap films are thin layers of liquid (usually water-based) surrounded by air. For example, if two soap bubbles come into contact, they merge and a thin film is created in between. Thus, foams are composed of a network of films connected by Plat ...
stretched between two parallel circular
wire file:Sample cross-section of high tension power (pylon) line.jpg, Overhead power cabling. The conductor consists of seven strands of steel (centre, high tensile strength), surrounded by four outer layers of aluminium (high conductivity). Sample d ...
s: the soap film naturally takes on the shape with least surface area.


Catenoid solution

If the half-plane containing the two points and the axis of revolution is given
Cartesian coordinate In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...
s, making the axis of revolution into the ''x''-axis of the coordinate system, then the curve connecting the points may be interpreted as the
graph of a function In mathematics, the graph of a function f is the set of ordered pairs (x, y), where f(x) = y. In the common case where x and f(x) are real numbers, these pairs are Cartesian coordinates of points in a plane (geometry), plane and often form a P ...
. If the Cartesian coordinates of the two given points are (x_1,y_1), (x_2,y_2), then the area of the surface generated by a nonnegative
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
f may be expressed mathematically as :2\pi\int_^ f(x) \sqrt dx and the problem of finding the minimal surface of revolution becomes one of finding the function that minimizes this integral, subject to the
boundary conditions In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
that f(x_1)=y_1 and f(x_2)=y_2. In this case, the optimal curve will necessarily be a
catenary In physics and geometry, a catenary ( , ) is the curve that an idealized hanging chain or wire rope, cable assumes under its own weight when supported only at its ends in a uniform gravitational field. The catenary curve has a U-like shape, ...
. The axis of revolution is the directrix of the catenary, and the minimal surface of revolution will thus be a catenoid.


Goldschmidt solution

Solutions based on discontinuous functions may also be defined. In particular, for some placements of the two points the optimal solution is generated by a discontinuous function that is nonzero at the two points and zero everywhere else. This function leads to a surface of revolution consisting of two circular disks, one for each point, connected by a degenerate line segment along the axis of revolution. This is known as a Goldschmidt solution after German mathematician
Carl Wolfgang Benjamin Goldschmidt Carl Wolfgang Benjamin Goldschmidt (4 August 1807 – 15 February 1851) was a German astronomer, mathematician, and physicist of Jewish descent who was a professor of astronomy at the University of Göttingen. He is also known as Benjamin Goldsc ...
, who announced his discovery of it in his 1831 paper "Determinatio superficiei minimae rotatione curvae data duo puncta jungentis circa datum axem ortae" ("Determination of the surface-minimal rotation curve given two joined points about a given axis of origin"). To continue the physical analogy of soap film given above, these Goldschmidt solutions can be visualized as instances in which the soap film breaks as the circular wires are stretched apart. However, in a physical soap film, the connecting line segment would not be present. Additionally, if a soap film is stretched in this way, there is a range of distances within which the catenoid solution is still feasible but has greater area than the Goldschmidt solution, so the soap film may stretch into a configuration in which the area is a
local minimum In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...
but not a global minimum. For distances greater than this range, the catenary that defines the catenoid crosses the ''x''-axis and leads to a self-intersecting surface, so only the Goldschmidt solution is feasible..


References

{{reflist Minimal surfaces