The Contribution of Discrete Differential Geometry to Contemporary Architecture

Transcription

1 The Contribution of Discrete Differential Geometry to Contemporary Architecture Helmut Pottmann Vienna University of Technology, Austria 1

2 Project in Seoul, Hadid Architects 2

3 Lilium Tower Warsaw, Hadid Architects 3

4 Project in Baku, Hadid Architects 4

5 Discrete Surfaces in Architecture triangle meshes 5

6 Zlote Tarasy, Warsaw Waagner-Biro Stahlbau AG Chapter 19 - Discrete Freeform Structures 6 6

7 Visible mesh quality 7

8 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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10 nodes in the support structure triangle mesh: generically nodes of valence 6; `torsion : central planes of beams not co-axial torsion-free node 10

11 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

12 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

13 Previous work 13

14 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

15 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

16 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

17 Computing PQ meshes 17

18 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

19 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

20 Opus (Hadid Architects) 20

21 Opus (Hadid Architects) 21

22 OPUS (Hadid Architects) 22

23 Conical Meshes 23

24 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

25 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

26 Offset meshes 26

27 Offset meshes 27

28 Offset meshes 28

29 Offset meshes 29

30 normals of a conical mesh neighboring discrete normals are coplanar conical mesh has a discretely orthogonal support structure 30

31 Multilayer constructions 31

32 Computing conical meshes angle criterion add angle criterion to PQ optimization alternation with subdivision as design tool 32

33 subdivision-based design combination of Catmull-Clark subdivision and conical optimization; design: Benjamin Schneider 33

34 design by Benjamin Schneider 34

35 Mesh Parallelism and Nodes 35

36 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

37 Parallel meshes M M* meshes M, M* with planar faces are parallel if they are combinatorially equivalent and corresponding edges are parallel 37

38 Geometric support structure Connects two parallel meshes M, M* 38

39 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

40 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

41 Triangle Meshes beam layout with optimized nodes We can minimize torsion in the supporting beam layout for triangle meshes 41

42 Mesh optimization Project YAS island (Asymptote, Gehry Technologies, Schlaich Bergermann, Waagner Biro, ) 42

43 Mesh smoothing Project YAS island (quad mesh with nonplanar faces) 43

44 Mesh smoothing Project YAS island (quad mesh with nonplanar faces) 44

45 Optimized nodes Torsion minimization also works for quadrilateral meshes with nonplanar faces 45

46 Optimized nodes Torsion minimization also works for quadrilateral meshes with nonplanar faces 46

47 YAS project, mockup 47

48 Offset meshes 48

49 node geometry Employing beams of constant height: misalignment on one side 49

50 Cleanest nodes Perfect alignment on both sides if a mesh with edge offsets is used 50

51 Cleanest nodes 51

52 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

53 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

54 Meshes with edge offsets 54

55 Edge offset meshes M has edge offsets iff it is parallel to a mesh S whose edges are tangent to S 2 55

56 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

57 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

58 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

59 Example: discrete CMC surface M (hexagonal EO mesh) and Laguerre transform M 59

60 Hexagonal EO mesh 60

61 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

62 Single curved panels, ruled panels and semi-discrete representations 62

63 developable surfaces in architecture (nearly) developable surfaces F. Gehry, Guggenheim Museum, Bilbao 63

64 surfaces in architecture single curved panels 64

65 D-strip models One-directional limit of a PQ mesh: developable strip model (D-strip model) semi-discrete surface representation 65

66 Principal strip models I Circular strip model as limit of a circular mesh 66

67 Principal strip models II Conical strip model as limit of a conical mesh 67

68 Principal strip models III Conversion: conical model to circular model 68

69 Conical strip model 69

70 Conical strip model 70

71 Multi-layer structure D-strip model on top of a PQ mesh 71

72 Project in Cagliari, Hadid Architects

73 Approximation by ruled surface strips 73

74 Semi-discrete model: smooth union of ruled strips 74

75 Circle packings on surfaces M. Höbinger 75

76 Project in Budapest, Hadid Architects 76

77 Selfridges, Birmingham Architects: Future Systems 77

78 Optimization based on triangulation 78

79 Circle packings on surfaces 79

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84 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

85 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

86 Literature Architectural Geometry H. Pottmann, A. Asperl, M. Hofer, A. Kilian Publisher: BI Press, 2007 ISBN: pages, 800 color figures 86

87 D-strip models via subdivision 87