DefinePK

Abstract

Geometric constructions are widely used in computer graphics and engineering drawing. A right generalized cylinder is a ruled surface whose base curve is a plane curve perpendicular to the rulings. In this paper, relations between the base curve and the non-planar geodesics on the right generalized cylinder are discussed. Based on these relations, a method for obtaining a new space curve from a given unit speed plane curve is presented. Firstly, we consider a new plane curve with a constant speed one divided by the square root of two. The original unit speed plane curve and the new plane curve coincide as point sets. Their parameterizations and signed curvatures are closely related. Secondly, we define a unique unit speed non-planar geodesic on the right generalized cylinder whose base curve is the considered plane curve with a constant speed one divided by the square root of two. Finally, we examine the focal curve of the obtained geodesic which is also a non-planar curve. The curvature and torsion of the geodesic and its focal curve are expressed in terms of the signed curvature of the above-mentioned plane curve with a constant speed one divided by the square root of two. We discuss also other two invariants of the same space curves with respect to the direct similarities of the Euclidean 3-space. They are called a shape curvature and a shape torsion. In particular, it is shown that the shape torsion (the ratio of torsion and curvature) of the unit speed geodesic and its focal curve is equal to either +1 or -1. The proposed method is demonstrated for several plane curves used in engineering practice. These curves include: the circle, the logarithmic spiral, the in volute of a circle, and the catenary.