Smooth quasi-developable surfaces bounded by smooth curves

TL;DR

This paper introduces a novel variational method for generating smooth, quasi-developable surfaces bounded by smooth curves, ensuring maximum developability and no self-intersections, with applications in industrial design.

Contribution

It presents the first continuous solution approach for quasi-developable surfaces that explores the full input curve space, improving smoothness and developability over discrete methods.

Findings

Produces smooth quasi-developable surfaces with maximum developability.

Guarantees surfaces are exactly bounded by input curves and free of self-intersections.

Provides an algorithm to represent surfaces as B-spline surfaces.

Abstract

Computing a quasi-developable strip surface bounded by design curves finds wide industrial applications. Existing methods compute discrete surfaces composed of developable lines connecting sampling points on input curves which are not adequate for generating smooth quasi-developable surfaces. We propose the first method which is capable of exploring the full solution space of continuous input curves to compute a smooth quasi-developable ruled surface with as large developability as possible. The resulting surface is exactly bounded by the input smooth curves and is guaranteed to have no self-intersections. The main contribution is a variational approach to compute a continuous mapping of parameters of input curves by minimizing a function evaluating surface developability. Moreover, we also present an algorithm to represent a resulting surface as a B-spline surface when input curves are B-spline curves.