Focal Curves of Closed Toroidal Curves

TL;DR

This paper introduces a method to derive the focal curve of a non-planar space curve on a right generalized cylinder from a given plane curve, with applications to engineering curves like epicycloids and hypocycloids.

Contribution

It presents a novel approach to construct and analyze the focal curves of cylindrical curves based on the curvature of the base plane curve, including parametric representations and projections.

Findings

Derived the Frenet-Serret system for cylindrical and focal curves.

Obtained parametric forms for the orthogonal projection of focal curves.

Demonstrated the method on classical engineering curves.

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. The paper discusses relations between the base curve and the non-planar space curve on the right generalized cylinder. Based on these relations, a method for obtaining a new space curve from a given plane curve parameterized about an arbitrary parameter is presented. At first, we define a non-planar space curve on the right generalized cylinder whose base curve is the considered plane curve, parameterized about an arbitrary parameter. Later on, we examine the focal curve of the obtained cylindrical curve which is also a non-planar curve. The Frenet-Seret system of the cylindrical curve and its focal curve are expressed in terms of the signed curvature of the abovementioned plane curve and its derivatives. Finally, we obtained the parametric representation of orthogonal projection of the focal curve onto the Euclidean plane via before mentioned plane curve and its derivatives. That curve is called generalized focal curve of a plane curve. The proposed method is demonstrated for several closed plane curves used in engineering practice. These curves include: epicycloid, hypocycloid and a curve that is orthogonal projection of toroidal helix onto the Euclidean plane.