Darboux Frame of Pancake Curve in Euclidean 3-Space

Abstract

In this paper, we focus on such a rare curve: “Pancake curve” which is named by Robert Ferréol. We investigate the Darboux frame of the curve and we calculate the curvatures: geodesic curvature, normal curvature, and geodesic torsion. We discuss two cases, for instance pancake curve as an intersection of Plücker’s conoid and unit cylinder, and intersection of hyperbolic paraboloid and unit cylinder. Moreover, we give the relations between the Frenet and Darboux curvatures. Finally, we give some of examples related with Frenet and Darboux frame and curvatures.