Design of Developable Surfaces Using Optimal Control

Abstract

This paper formulates the developable surface design problem in an optimal control setting. Given a regular curve b(t) on the unit sphere corresponding to a one-parameter family of rulings, and two base curve endpoints a0,a1∊R3, we consider the problem of constructing a base curve a(t) such that a(t0)=a0,a(t1)=a1, and the resulting surface f(s,t)=a(t)+sb(t) is developable. We formulate the base curve design problem as an optimal control problem, and derive solutions for objective functions that reflect various practical aspects of developable surface design, e.g., minimizing the arc length of the base curve, keeping the line of regression distant from the base curve, and approximating a given arbitrary ruled surface by a developable surface. By drawing upon the large body of available results for the optimal control of linear systems with quadratic criteria, our approach provides a flexible method for designing developable surfaces that are optimized for various criteria.