Helmut Pottmann
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.
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Index Terms
- Freeform surfaces from single curved panels
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Reviews
Joseph J. O'Rourke
Freeform surfaces are increasingly used in spectacular steel and glass architectural designs. Constructing such a surface from flat pieces renders it polyhedral, diminishing the aesthetics of curved surfaces. Constructing it from doubly curved panels is expensive. An attractive middle ground is singly curved panels, the topic of this paper. The goal is to partition the surface into developable strips (D-strips), each of which may be locally developed on the plane and singly bent back to its three-dimensional (3D) form. The partition is semidiscrete in the sense that there is a finite number of strips, but each is a smooth, developable surface. Pottmann et al. argue that it is best to start with a fully discrete planar quad model that naturally provides a conjugate network of curves. From this, they optimize the strip boundary curves-modeled as B-splines-to achieve various architecturally desirable properties. They show that conical meshes are particularly attractive. Offsetting each face of a conical mesh by a constant distance produces a parallel conical mesh that makes them useful for multilayer constructions. A conical D-strip can be converted to a canal (swept-sphere) surface that then leads to a natural partition into panels bounded by arc splines, convenient for realizing as beams. Their optimization framework also supports finding geodesic D-strips that develop approximately straight, another manufacturing plus. The paper is so dense with detail that, if expanded to be self-contained, it would triple the length beyond the ten-page SIGGRAPH limit. The stunning images leave no doubt that the authors have opened a door to a fruitful melding of geometry and architectural aesthetics. Online Computing Reviews Service
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