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 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.[1] A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.[1]
A minimal surface of revolution is a subtype of minimal surface.[1] A minimal surface is defined not as a surface of minimal area, but as a surface with a mean curvature of 0.[2] Since a mean curvature of 0 is a necessary condition 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 when rotated about an axis, finding the minimal surface of revolution is equivalent to finding the minimal surface passing through two circular wireframes.[1] A physical realization of a minimal surface of revolution is soap film stretched between two parallel circular wires: the soap film naturally takes on the shape with least surface area.[3] [4]
If the half-plane containing the two points and the axis of revolution is given Cartesian coordinates, 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. If the Cartesian coordinates of the two given points are
(x1,y1)
(x2,y2)
f
| x2 | |
| 2\pi\int | |
| x1 |
f(x)\sqrt{1+f'(x)2}dx
f(x1)=y1
f(x2)=y2
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[7] after German mathematician Carl Wolfgang Benjamin Goldschmidt,[4] 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").[8]
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.[4] 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 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.[9]