Representation and Geometric Construction of Catenoids and Helicoids

In this paper, the representation and geometric construction of the catenoids and helicoids are discussed. The tension representation of the catenoids and helicoids is derived by using triangle and hyperbolic triangle geometric spline. This kind of representation enables generating the catenoids and helicoids with subdivision rules. A constructive rule of the control net is given to define the catenoids and helicoids.