Implicit Equations of the Henneberg-Type Minimal Surface in the Four-Dimensional Euclidean Space

Abstract

Considering the Weierstrass data as (psi,f,g) = (2, 1 - z(-m),z(n)), we introduce a two-parameter family of Henneberg-type minimal surface that we call h(m,n) for positive integers (m, n) by using the Weierstrass representation in the four-dimensional Euclidean space E-4. We define in h(m,n) in (r,theta) coordinates for positive integers (m, n) with m not equal 1, n not equal -1, -m + n not equal -1, and also in (u,upsilon) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values (4, 2).