Ruled Surfaces and Their Geometric Invariants via the Orthogonal Modified Frame in Minkowski 3-Space

Ruled surfaces in Minkowski 3-space play a crucial role in differential geometry and have significant applications in physics and engineering. This study explores the fundamental properties of ruled surfaces via orthogonal modified frame in Minkowski space ${\mathrm{E}}_1^3$, focusing on their minimality, developability, and curvature characteristics. We examine the necessary and sufficient conditions for a ruled surface to be minimal, considering the mean curvature and its implications. Furthermore, we analyze the developability of such surfaces, determining the conditions under which they can be locally unfolded onto a plane without distortion. The Gaussian and mean curvatures of ruled surfaces in Minkowski space are computed and discussed, providing insights into their geometric behavior. Special attention is given to spacelike, timelike, and lightlike rulings, highlighting their unique characteristics. This research contributes to the broader understanding of the geometric properties of ruled surfaces within the framework of Minkowski geometry.