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

Abstract

Considering the Weierstrass data as ( ψ , 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 H m , n in ( r , θ ) coordinates for positive integers ( m , n ) with m ≠ 1 , n ≠ - 1 , - m + n ≠ - 1 , and also in ( u , v ) coordinates, and then we obtain implicit algebraic equations of the Henneberg-type minimal surface of values ( 4 , 2 ) .

A new characterization of Aminov surface with regards to its Gauss map in E^4

“…In [3,4], the authors handled Aminov surfaces according to their curvatures in − 4 dimensional Euclidean and Minkowski spaces. The other studies about some surfacesin can be found in [9,10,11,12]. In this study, we evaluate Aminov surfaces with regards to their Gauss maps in .…”

(ζ−m, ζm)-Type Algebraic Minimal Surfaces in Three-Dimensional Euclidean Space

“…Lie [10] studied algebraic minimal surfaces and gave a table for these kinds of surfaces. See also [6,[16][17][18][19][20][21][22][23][24] for details.…”