Sweeping surface due to rotation minimizing Darboux frame in Euclidean 3-space E³

Abstract

In this paper, we address a new version of Darboux frame using a common tangent vector field to a surface along a curve and call this frame the rotation minimizing Darboux frame (RMDF). Then, we give the parametric equation due to the RMDF frame for a sweeping surface and show that the parametric curves on this surface are curvature lines. Consequently, necessary and sufficient conditions for sweeping surfaces to be developable ruled surfaces are derived. Also, we analyze the conditions when the resulting developable surface is a cylinder, cone or tangential surface. We also provide some examples to illustrate the main results.

Keywords

• darboux frame
• singularity and convexity
• developable surface