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Concept
Minimal surface of revolution
Summary
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. A basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution.
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 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. 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.
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 , , then the area of the surface generated by a nonnegative differentiable function may be expressed mathematically as
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 and . 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.
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