Geometric analysis of ruled surfaces constructed from integral curves in three-dimensional Euclidean space

Abstract

Ruled surfaces, defined by the motion of a straight line along a space curve, represent a fundamental class of surfaces in differential geometry with significant applications in engineering design, architectural modeling, and computer graphics. Despite their classical nature, the construction of ruled surfaces from integral curves, solutions to differential systems derived from Frenet frames, remains relatively unexplored in the literature. This paper presents a detailed geometric study of a new class of ruled surfaces constructed from integral curves associated with the Frenet frame of regular space curves with positive curvature. We focus on surfaces whose base curves are given by the integral binormal and integral normal curves of a given spatial curve. Explicit expressions for the fundamental forms, curvature properties, and striction curves are derived for six distinct types of surfaces. Necessary and sufficient conditions under which these surfaces are minimal or developable are established. A numerical example illustrates the theoretical results, highlighting potential applications in geometric modeling. This work extends the theory of ruled surfaces in differential geometry by introducing families based on integral curves and providing a complete geometric characterization via fundamental forms and curvature analysis.

Keywords

Ruled surfaces / Integral curves / Frenet frame / Euclidean space / Gaussian curvature / Mean curvature

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