Developable Ruled Surfaces Generated by the Curvature Axis of a Curve

TL;DR

This paper introduces a simple method to generate developable ruled surfaces from the curvature axes of curves, providing insights into their classification, singularities, and visualization techniques for practical design applications.

Contribution

It presents a straightforward approach to create developable ruled surfaces using curvature axes, enhancing understanding of their singularities and visualization in geometric modeling.

Findings

Generated ruled surfaces are developable.

At least one curve's curvature axis forms a line on the surface.

Visualizations use environmental maps with circular patterns.

Abstract

Ruled surfaces play an important role in various types of design, architecture, manufacturing, art, and sculpture. They can be created in a variety of ways, which is a topic that has been the subject of much discussion in mathematics and engineering journals. In geometric modelling, ideas are successful if they are not too complex for engineers and practitioners to understand and not too difficult to implement, because these specialists put mathematical theories into practice by implementing them in CAD/CAM systems. Some of these popular systems such as AutoCAD, Solidworks, CATIA, Rhinoceros 3D, and others are based on simple polynomial or rational splines and many other beautiful mathematical theories that have not yet been implemented due to their complexity. Based on this philosophy, in the present work, we investigate a simple way to generate ruled surfaces whose generators are the curvature axes of curves. We show that this type of ruled surface is a developable surface and that there is at least one curve whose curvature axis is a line on the given developable surface. In addition, we discuss the classifications of developable surfaces corresponding to space curves with singularities, while these curves and surfaces are most often avoided in practical design. Our research also contributes to the understanding of the singularities of developable surfaces and, in their visualisation, proposes the use of environmental maps with a circular pattern that creates flower-like structures around the singularities.