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D. Pellis, M. Kilian, H. Pottmann, M. Pauly:
"Computational design of Weingarten surfaces";
ACM Transactions on Graphics, 40 (2021), 4; 1 - 11.

In this paper we study Weingarten surfaces and explore their potential for fabrication-aware design in freeform architecture. Weingarten surfaces are characterized by a functional relation between their principal curvatures that implicitly defines approximate local congruences on the surface. These symmetries can be exploited to simplify surface paneling of double-curved architectural skins through mold re-use.

We present an optimization approach to find a Weingarten surface that is close to a given input design. Leveraging insights from differential geometry, our method aligns curvature isolines of the surface in order to contract the curvature diagram from a 2D region into a 1D curve. The unknown functional curvature relation then emerges as the result of the optimization. We show how a robust and efficient numerical shape approximation method can be implemented using a guided projection approach on a high-order B-spline representation. This algorithm is applied in several design studies to illustrate how Weingarten surfaces define a versatile shape space for fabrication-aware exploration in freeform architecture. Our optimization algorithm provides the first practical tool to compute general Weingarten surfaces with arbitrary curvature relation, thus enabling new investigations into a rich, but as of yet largely unexplored class of surfaces.

Computing methodologies; Shape modeling; fabrication-aware design, computational differential geometry, geometric optimization, Weingarten surface, architectural geometry, paneling, mold reduction

http://dx.doi.org/10.1145/3450626.3459939

https://dmg.tuwien.ac.at/geom/ig/publications/weingarten/weingarten.pdf