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 ) .
Highlights
The theory of surfaces has an important role in mathematics, physics, biology, architecture, see e.g., the classical books [1,2] and papers [3,4,5,6,7,8,9].A minimal surface in the three-dimensional Euclidean space E3, in higher dimensions, is a regular surface for which the mean curvature vanishes identically
Considering the Weierstrass data as (ψ, f, g) = (2, 1 − z−m, zn ), we introduce a two-parameter family of Henneberg-type minimal surface that we call Hm,n for positive integers (m, n) by using the Weierstrass representation in the four-dimensional Euclidean space E4
We study a two-parameter family of Henneberg-type minimal surfaces using the Weierstrass representation in E4
Summary
The theory of surfaces has an important role in mathematics, physics, biology, architecture, see e.g., the classical books [1,2] and papers [3,4,5,6,7,8,9]. A Henneberg surface [4,5,6], obtained by the Weierstrass representation [8,9] is well-known classical minimal surface in E3. We study a two-parameter family of Henneberg-type minimal surfaces using the Weierstrass representation in E4.
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