Minimal surface of revolution

Catenoid solution

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

${\displaystyle 2\pi {\int }_{x_1}^{x_2}f(x)\sqrt{1+f^{\prime }(x{)}^2}dx}$

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 ${\displaystyle f(x_2)=y_2}$. In this case, the optimal curve will necessarily be a catenary. 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, 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 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.