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. A basic problem in the calculus of variations 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 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.

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

(

x

1

,

y

1

)

{\displaystyle (x_{1},y_{1})}

,

(

x

2

,

y

2

)

{\displaystyle (x_{2},y_{2})}

, then the area of the surface generated by a nonnegative differentiable function

f

{\displaystyle f}

may be expressed mathematically as

2

π

∫

x

1

x

2

f

(

x

)

1

+

f

′

(

x

)

2

d

x

{\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

f

(

x

1

)

=

y

1

{\displaystyle f(x_{1})=y_{1}}

and

f

(

x

2

)

=

y

2

{\displaystyle f(x_{2})=y_{2}}

.