Classification of polynomial minimal surfaces

Minimal surfaces are widely applied in Computer-Aided Design and architecture due to their elegant shapes and remarkable geometric properties. On the basis of the theoretical results of Pythagorean hodograph (PH) curves, we provide a complete classification of the polynomial surface of degrees three to five. We point out the existence of a unique cubic minimal surface, one family of quartic polynomial minimal surfaces, and three families of quintic polynomial minimal surfaces up to similarities and linear reparametrization. We give explicit expressions for each family of polynomial minimal surfaces with parameters that can be used to adjust their shapes. We also prove that an isoparametric curve that is a planar PH curve always exists for any polynomial minimal surfaces. Finally we apply the classification to some existing minimal surfaces.