The Construction of a Minimal Helicoid Supported by Two Skew Lines

Abstract

In this article, we addressed a problem involving a subfield of mathematics known as Differential Geometry, specifically focusing on minimal helicoids whose projection onto a plane perpendicular to the helicoid axis does not form a circle. The methodology employed was based on consulting books and articles related to the field, as well as reflecting on the concepts associated with the subject matter, with the aim of obtaining a method capable of parameterizing these helicoids from any two skew lines. The study was divided into two phases, with the first phase being responsible for verifying the mean curvature of the helicoid and the second phase focused on obtaining a method capable of parameterizing them. To achieve this, we introduced in the first moment, the definition of a ruled surface and the necessary concepts and propositions to verify that the helicoid is a minimal surface. It is worth noting that these concepts can easily be found in introductory materials on Differential Geometry, with the innovation of this work lying in the method obtained in the second phase. Furthermore, we presented a chapter on the applications of helicoids in architecture, exemplifying their aesthetic and structural potential in both this field and civil construction.