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 bt 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 at such that at0=a0,at1=a1, and the resulting surface fs,t=at+sbt 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.

Analysis and Simulation of Long-Range Correlations in Curved Space

“…Park et al 36 described an interpolative optimal control problem, generating a developable surface from two points and a curve of tangent directions connecting them. Several other approaches are briefly reviewed by Subag and Elber.…”

Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches

“…In general, a surface is developable if and only if the Gaussian curvature of every point on it is zero -this is the constraint that we want to preserve during the surface optimization. Research related to Computer Aided Geometric Design, in particular those concerning the design and approximation of developable surfaces, can be found in [18][19][20][21][22][23][24][25][26][27]. Most of them are in terms of NURBS or its special case -B-spline or Bézier surfaces [18][19][20][21][22][23][24].…”

“…In the work of [22][23][24], the approximation methods are used to design developable B-Spline surfaces based on projective geometry. Other approaches are based on alternative perspective: Redont [25] constructs developable surfaces by specifying tangent planes along a geodesic of a surface, Randrup [26] approximates a given surface by cylinders in its Gaussian image, and Park et al [27] design developable surfaces by the methods from optimal control theory.…”

Recognition and reconstruction of developable surfaces from point clouds

“…There is quite a lot of literature on modelling with developable surfaces, see for instance [1,2,3,7,12,15]. Bspline representations and the dual representation are well known and have been used for interpolation and approximation tasks.…”

“…By investigating the magnitude of the eigenvalues λ i of the eigenvectors h i and the coefficients of H i we can classify the type of surface D. Let H i be given by the equations (12). An example is displayed in Figure 6.…”