On A Class Of Linear Weingarten Surfaces

We consider a class of linear Weingarten surfaces of revolution whose principal curvatures, meridional k(mu) and parallel k(pi), satisfy the relation k(mu) = (n + 1) k(pi), n = 0, 1, 2, . . . . The first two members of this class of surfaces are the sphere (n = 0) and the Mylar balloon (n = 1). Elsewhere the Mylar balloon has been parameterized via the Jacobian and Weierstrassian elliptic functions and elliptic integrals. Here we derive six alternative parameterizations describing the third type of surfaces when n = 2. The so obtained explicit formulas are applied for the derivation of the basic geometrical characteristics of this surface.