The Contribution of Discrete Differential Geometry to Contemporary Architecture Helmut Pottmann Vienna University of Technology, Austria 1
Project in Seoul, Hadid Architects 2
Lilium Tower Warsaw, Hadid Architects 3
Project in Baku, Hadid Architects 4
Discrete Surfaces in Architecture triangle meshes 5
Zlote Tarasy, Warsaw Waagner-Biro Stahlbau AG Chapter 19 - Discrete Freeform Structures 6 6
Visible mesh quality 7
Geometry in architecture The underlying geometry representation may greatly contribute to the aesthetics and has to meet manufacturing constraints Different from typical graphics applications 8
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nodes in the support structure triangle mesh: generically nodes of valence 6; `torsion : central planes of beams not co-axial torsion-free node 10
quad meshes in architecture Schlaich Bergermann hippo house, Berlin Zoo quad meshes with planar faces (PQ meshes) are preferable over triangle meshes (cost, weight, node complexity, ) 11
Discrete Surfaces in Architecture Helmut Pottmann 1, Johannes Wallner 2, Alexander Bobenko 3 Yang Liu 4, Wenping Wang 4 1 TU Wien 2 TU Graz 3 TU Berlin 4 University of Hong Kong 12
Previous work 13
Previous work Difference geometry (Sauer, 1970) Quad meshes with planar faces (PQ meshes) discretize conjugate curve networks. Example 1: translational net Example 2: principal curvature lines 14
PQ meshes A PQ strip in a PQ mesh is a discrete model of a tangent developable surface. Differential geometry tells us: PQ meshes are discrete versions of conjugate curve networks 15
Previous work Discrete Differential Geometry: Bobenko & Suris, 2005: integrable systems circular meshes: discretization of the network of principal curvature lines (R. Martin et al. 1986) 16
Computing PQ meshes 17
Computational Approach Computation of a PQ mesh is based on a nonlinear optimization algorithm: Optimization criteria planarity of faces aesthetics (fairness of mesh polygons) proximity to a given reference surface Requires initial mesh: ideal if taken from a conjugate curve network. 18
subdivision & optimization Refine a coarse PQ mesh by repeated application of subdivision and PQ optimization PQ meshes via Catmull-Clark subdivision and PQ optimization 19
Opus (Hadid Architects) 20
Opus (Hadid Architects) 21
OPUS (Hadid Architects) 22
Conical Meshes 23
Conical meshes Liu et al. 06 Another discrete counterpart of network of principal curvature lines PQ mesh is conical if all vertices of valence 4 are conical: incident oriented face planes are tangent to a right circular cone 24
Conical meshes Cone axis: discrete surface normal Offsetting all face planes by constant distance yields conical mesh with the same set of discrete normals 25
Offset meshes 26
Offset meshes 27
Offset meshes 28
Offset meshes 29
normals of a conical mesh neighboring discrete normals are coplanar conical mesh has a discretely orthogonal support structure 30
Multilayer constructions 31
Computing conical meshes angle criterion add angle criterion to PQ optimization alternation with subdivision as design tool 32
subdivision-based design combination of Catmull-Clark subdivision and conical optimization; design: Benjamin Schneider 33
design by Benjamin Schneider 34
Mesh Parallelism and Nodes 35
supporting beam layout beams: prismatic, symmetric with respect to a plane optimal node (i.e. without torsion): central planes of beams pass through node axis Existence of a parallel mesh whose vertices lie on the node axes geometric support structure 36
Parallel meshes M M* meshes M, M* with planar faces are parallel if they are combinatorially equivalent and corresponding edges are parallel 37
Geometric support structure Connects two parallel meshes M, M* 38
Computing a supporting beam layout given M, construct a beam layout find parallel mesh S which approximates a sphere solution of a linear system initialization not required 39
Triangle meshes Parallel triangle meshes are scaled versions of each other Triangle meshes possess a support structure with torsion free nodes only if they represent a nearly spherical shape 40
Triangle Meshes beam layout with optimized nodes We can minimize torsion in the supporting beam layout for triangle meshes 41
Mesh optimization Project YAS island (Asymptote, Gehry Technologies, Schlaich Bergermann, Waagner Biro, ) 42
Mesh smoothing Project YAS island (quad mesh with nonplanar faces) 43
Mesh smoothing Project YAS island (quad mesh with nonplanar faces) 44
Optimized nodes Torsion minimization also works for quadrilateral meshes with nonplanar faces 45
Optimized nodes Torsion minimization also works for quadrilateral meshes with nonplanar faces 46
YAS project, mockup 47
Offset meshes 48
node geometry Employing beams of constant height: misalignment on one side 49
Cleanest nodes Perfect alignment on both sides if a mesh with edge offsets is used 50
Cleanest nodes 51
Offset meshes at constant distance d from M is called an offset of M. different types, depending on the definition of vertex offsets: edge offsets: distance of corresponding (parallel) edges is constant =d face offsets: distance of corresponding faces (parallel planes) is constant =d 52
Exact offsets For offset pair define Gauss image Then: vertex offsets vertices of S lie in S 2 (if M quad mesh, then circular mesh) edge offsets edges of S are tangent to S 2 face offsets face planes of S are tangent to S 2 (M is a conical mesh) 53
Meshes with edge offsets 54
Edge offset meshes M has edge offsets iff it is parallel to a mesh S whose edges are tangent to S 2 55
Koebe polyhedra Meshes with planar faces and edges tangent to S 2 have a beautiful geometry; known as Koebe polyhedra. Closed Koebe polyhedra defined by their combinatorics up to a Möbius transform computable as minimum of a convex function (Bobenko and Springborn) 56
vertex cones Edges emanating from a vertex in an EO mesh are contained in a cone of revolution whose axis serves as node axis. Simplifies the construction of the support structure 57
Laguerre geometry Laguerre geometry is the geometry of oriented planes and oriented spheres in Euclidean 3- space. L-trafo preserves or. planes, or. spheres and contact; simple example: offsetting operation or. cones of revolution are objects of Laguerre geometry (envelope of planes tangent to two spheres) If we view an EO mesh as collection of vertex cones, an L-trafo maps an EO mesh M to an EO mesh. 58
Example: discrete CMC surface M (hexagonal EO mesh) and Laguerre transform M 59
Hexagonal EO mesh 60
Planar hexagonal panels Planar hexagons non-convex in negatively curved areas Phex mesh layout largely unsolved initial results by Y. Liu and W. Wang 61
Single curved panels, ruled panels and semi-discrete representations 62
developable surfaces in architecture (nearly) developable surfaces F. Gehry, Guggenheim Museum, Bilbao 63
surfaces in architecture single curved panels 64
D-strip models One-directional limit of a PQ mesh: developable strip model (D-strip model) semi-discrete surface representation 65
Principal strip models I Circular strip model as limit of a circular mesh 66
Principal strip models II Conical strip model as limit of a conical mesh 67
Principal strip models III Conversion: conical model to circular model 68
Conical strip model 69
Conical strip model 70
Multi-layer structure D-strip model on top of a PQ mesh 71
Project in Cagliari, Hadid Architects
Approximation by ruled surface strips 73
Semi-discrete model: smooth union of ruled strips 74
Circle packings on surfaces M. Höbinger 75
Project in Budapest, Hadid Architects 76
Selfridges, Birmingham Architects: Future Systems 77
Optimization based on triangulation 78
Circle packings on surfaces 79
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Conclusion architecture poses new challenges to geometric design and computing: paneling, supporting beam layout, offsets, Main problem: rationalization of freeform shapes, i.e., the segmentation into panels which can be manufactured at reasonable cost; depends on material and manufacturing technology many relations to discrete differential geometry semi-discrete representations interesting as well architectural geometry: a new research direction in geometric computing 84
Acknowledgements S. Brell-Cokcan, S. Flöry, Y. Liu, A. Schiftner, H. Schmiedhofer, B. Schneider, J. Wallner, W. Wang Funding Agencies: FWF, FFG Waagner Biro Stahlbau AG, Vienna Evolute GmbH, Vienna RFR, Paris Zaha Hadid Architects, London 85
Literature Architectural Geometry H. Pottmann, A. Asperl, M. Hofer, A. Kilian Publisher: BI Press, 2007 ISBN: 978-1-934493-04-5 725 pages, 800 color figures www.architecturalgeometry.at 86
D-strip models via subdivision 87