 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 formula_1, formula_2, then the area of the surface generated by a nonnegative differentiable function formula_3 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 formula_5 and formula_6.In this case, the optimal curve will necessarily be a catenary.