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 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 ${\displaystyle (x_{1},y_{1})}$, ${\displaystyle (x_{2},y_{2})}$, then the area of the surface generated by a nonnegative differentiable function ${\displaystyle f}$ may be expressed mathematically as

${\displaystyle 2\pi \int _{x_{1}}^{x_{2}}f(x){\sqrt {1+f'(x)^{2}}}dx}$

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 that ${\displaystyle f(x_{1})=y_{1}}$ and ${\displaystyle f(x_{2})=y_{2}}$.^[5] In this case, the optimal curve will necessarily be a catenary.^[1][5] The axis of revolution is the directrix of the catenary, and the minimal surface of revolution will thus be a catenoid.^[1][6][7]

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^[5][8] 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").^[9]

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.^[10]