Smarandache Ruled Surfaces According to the Modified Orthogonal Frame

In this work, we investigate the intriguing geometry of ruled surfaces in three-dimensional Euclidean space that are produced by Smarandache curves with a modified orthogonal frame. The TN-Smarandache, TB-Smarandache, and NB-Smarandache surfaces are three different kinds of ruled surfaces by using the modified orthogonal frame of an arbitrary regular curve. We formally analyze these surfaces to acquire theorems providing necessary and sufficient requirements for these surfaces to have minimality and developability, as well as nonexistence theorems for specific situations. This approach reveals unique geometric insights into the properties and behavior of these specific ruled surfaces. In order to visualize the structural details of each surface type, we provide an example and related images to demonstrate some results.