Ruled Free Forms
The geometry of ruled surfaces is highly relevant for freeform architectural design, owing to the fact that any negatively curved freeform shape can be approximated by a smooth union of ruled surface strips. This information is important for the efficient
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Florin Isvoranu Evolute GmbH, Vienna
Helmut Pottmann King Abdullah University of Science and Technology
Johannes Wallner Graz University of Technology
Abstract. The geometry of ruled surfaces is highly relevant for jreeform architectural design, owing to the fact that any negatively curved freeform shape can be approximated by a smooth union of ruled surface strips. This information is important for the efficient manufacturing of both skins and underconstructions. The present paper discusses how to algorithmically solve this approximation problem. In particular we emphasize the special class of conoidal ruled surfaces which suggest new and simple ways of surface generation.
1 Introduction This paper reports on a study of piecewise-ruled surfaces and their use in freeform architecture. The notion "piecewise-ruled" means a watertight union of ruled strips, each of which is defined by two curves in space joined by straight line segments. The basic geometric problem to be solved is how to approximate a given freeform surface by a piecewise-ruled surface with the additional requirement that the latter is itself as smooth as possible. The importance of this geometric question comes from manufacturing processes [Flory and Pottmann 2010]: a ruled strip is comparatively easy to make, owing to its straight defining elements. Therefore freeform skins or underconstructions benefit enormously from the possibility of being manufactured as a piecewise-ruled surface. We also consider additional functional requirements: Strip boundaries might be geodesic, so they can be realized by elements made by bending [Pottmann et al. 2010], or planar (again, for simple manufacturing). Other properties having to do with aesthetics are constant strip width, or the angle between rulings and strip boundaries. We further show some interesting applications of the concept of conoidal ruled surfaces, see Figure 1.
L. Hesselgren et al. (eds.), Advances in Architectural Geometry 2012 © Springer-Verlag/Wien 2013
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S. Flbry, Y. Nagai, F. Isvoranu, H. Pottmann, and J. Wallner
Figure 1: Conoidal ruled sUifaces. The left hand image symbolically illustrates a conoidal surface, which by definition has rulings parallel to a certain fixed plane. Here spines of stacked books indicate the rulings, which are parallel to a horizontal reference plane. The right hand image illustrates the use of conoidal ruled strips in freeform architectural design: Each strip is made from wooden planks of simple geometry, which are stacked quite in the manner of the stack of books at left.
Previous work. The mathematical problem of approximating surfaces by ruled surfaces has been considered mostly in the context of approximation with B-spline surfaces and because of its importance for NC flank milling, see e.g. [Sprott and Ravani 2008], [Senatore et al. 2008], and [Flory 2010, Ch. 5] and the references therein. Most work however considers neither aesthetics nor functional properties along boundaries. Work in connection with architectural geometry has been done by [Flory
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