The algebraic surfaces of the minimal-maximal surfaces

Abstract

Considering soft computing, the Weierstrass data & zeta;-1/2, & zeta;1/2) gives two different minimal surface equations and figures. By using hard computing, we give the family of minimal and spacelike maximal surfaces S(m,n) for natural numbers m and n in Euclidean and Minkowski 3-spaces E3, E2,1, respectively. We obtain the classes and degrees of surfaces S(m,n). Considering the integral free form of Weierstrass, we define some algebraic functions for S(m,n). Indicating several maximal surfaces of value (m, n) are algebraic, we recall Weierstrass-type representations for maximal surfaces in E2,1, and give explicit parametrizations for spacelike maximal surfaces of value (m, n). Finally, we compute the implicit equations, degree, and class of the spacelike maximal surfaces S(0,1), S(1,1) and S(2,1) in terms of their cartesian or inhomogeneous tangential coordinates in E2,1.