basic problem in the calculus of variations is finding the curve between two points that produces this minimal surface of revolution. There are only two minimal surfaces of revolution (surfaces of revolution which are also minimal surfaces): the plane and the catenoid. Coordinate expressions A surface of revolution given by rotating a curve described by around the x-axis may be most simply described by . This yields the parametrization in terms of and as . If instead we revolve the curve around the y-axis, then the curve is described by , yielding the expression in terms of the parameters and . If x and y are defined in terms of a parameter , then we obtain a parametrization in terms of and . If and are functions of , then the surface of revolution obtained by revolving the curve around the x-axis is described by , and the surface of revolution obtained by revolving the curve around the y-axis is described by . Geodesics Meridians are always geodesics on a surface of revolution. Other geodesics are governed by Clairaut's relation. Toroids A surface of revolution with a hole in, where the axis of revolution does not intersect the surface, is called a toroid. For example, when a rectangle is rotated around an axis parallel to one of its edges, then a hollow square-section ring is produced. If the revolved figure is a circle, then the object is called a torus. Applications The use of surfaces of revolution is essential in many fields in physics and engineering. When certain objects are designed digitally, revolutions like these can be used to determine surface area without the use of measuring the length and radius of the object being designed. See also Channel surface, a generalisation of a surface of revolution Gabriel's Horn Generalized helicoid Lemon (geometry), surface of revolution of a circular arc Liouville surface, another generalization of a surface of revolution Solid of revolution Spheroid Surface integral Translation surface (differential geometry) References External links Integral calculus Surfaces of revolution