Freeform surfaces from single curved panels

Abstract

Motivated by applications in architecture and manufacturing, we discuss the problem of covering a freeform surface by single curved panels. This leads to the new concept of semi-discrete surface representation, which constitutes a link between smooth and discrete surfaces. The basic entity we are working with is the developable strip model. It is the semi-discrete equivalent of a quad mesh with planar faces, or a conjugate parametrization of a smooth surface. We present a B-spline based optimization framework for efficient computing with D-strip models. In particular we study conical and circular models, which semi-discretize the network of principal curvature lines, and which enjoy elegant geometric properties. Together with geodesic models and cylindrical models they offer a rich source of solutions for surface panelization problems.

Anamorphic Projection: Analogical/Digital Algorithms

“…If the surface is particularly extended or complex, it is extremely expensive or even impossible to manufacture it as a single piece; instead, applying one of the methods previously described, approximating it to a polyhedral surface, can produce an unsatisfactory overall outcome, thanks to the different directions of the normals to the faces (Baglioni 2012). In fact, the surface would show discontinuous reflection and refraction phenomena, quite different from the outcome predicted by the designer (Pottmann et al 2008).…”

Width-bounded geodesic strips for surface tiling

“…This results from the straightness property of geodesics -a geodesic over a smooth surface curves only in the direction of the surface normal -and it motivates the use of geodesic marching for straight strip tiling of surfaces. With the recent work of Pottmann et al [20] which can convert sets of geodesic curves on a surface into adjoining strips to approximate the surface, we are left to address the problem of computing an optimal set of geodesics fulfilling the various constraints.…”

“…Figure 2 shows a large curved roof covered with metal panels trimmed, placed and joined to tile the roof surface. Given a mesh of this surface and the material width, our algorithm (in conjunction with a strip construction algorithm like Pottmann [20]) can be used to prescribe the trimming and placement of the material strips necessary to panel the roof.…”

“…Recent work of Pottmann et al [20] takes as input a set of curves on a mesh surface and produces a set of adjoining developable strips (constrained by the input curves) which together provide a good surface approximation. If one of these curve families consists of geodesics, then this algorithm can be adapted to produce strips with approximately straight development.…”

“…If one of these curve families consists of geodesics, then this algorithm can be adapted to produce strips with approximately straight development. Making use of this work affords us the luxury of addressing our strip tiling problem by only focusing on covering the surface with geodesics and then deferring to Pottmann et al [20] for constructing the actual material strips to approximate the surface. Specifically, we frame the problem we wish to solve as follows: on a given mesh patch, find a Strips produced via planar mesh parameterization.…”