Geometry to Contemporary Architecture Helmut Pottmann Vienna - - PowerPoint PPT Presentation

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The Contribution of Discrete Differential Geometry to Contemporary Architecture

Helmut Pottmann

Vienna University of Technology, Austria

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Project in Seoul, Hadid Architects

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Lilium Tower Warsaw, Hadid Architects

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Project in Baku, Hadid Architects

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Discrete Surfaces in Architecture

triangle meshes

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Chapter 19 - Discrete Freeform Structures 6

Zlote Tarasy, Warsaw

Waagner-Biro Stahlbau AG

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Visible mesh quality

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Geometry in architecture

 The underlying geometry representation

may greatly contribute to the aesthetics and has to meet manufacturing constraints

 Different from typical graphics applications

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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

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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,…)

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Discrete Surfaces in Architecture

Helmut Pottmann1, Johannes Wallner2, Alexander Bobenko3 Yang Liu4, Wenping Wang4

1 TU Wien 2 TU Graz 3 TU Berlin 4 University of Hong Kong

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Previous work

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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

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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

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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)

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Computing PQ meshes

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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.

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subdivision & optimization

 Refine a coarse PQ mesh by repeated

application of subdivision and PQ optimization

PQ meshes via Catmull-Clark subdivision and PQ optimization

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Opus (Hadid Architects)

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Opus (Hadid Architects)

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OPUS (Hadid Architects)

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Conical Meshes

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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

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Conical meshes

 Cone axis: discrete

surface normal

 Offsetting all face planes

by constant distance yields conical mesh with the same set of discrete normals

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Offset meshes

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Offset meshes

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Offset meshes

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Offset meshes

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normals of a conical mesh

 neighboring discrete

normals are coplanar

 conical mesh has a

discretely orthogonal support structure

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Multilayer constructions

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Computing conical meshes

 angle criterion  add angle criterion to PQ

• ptimization

 alternation with

subdivision as design tool

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subdivision-based design

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

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design by Benjamin Schneider

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Mesh Parallelism and Nodes

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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

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Parallel meshes

 meshes M, M* with planar faces are parallel if

they are combinatorially equivalent and corresponding edges are parallel M M*

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Geometric support structure

 Connects two parallel meshes M, M*

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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

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Triangle meshes

 Parallel triangle

meshes are scaled versions of each

• ther

 Triangle meshes

possess a support structure with torsion free nodes only if they represent a nearly spherical shape

Triangle Meshes – beam layout with optimized nodes

• We can minimize torsion in the supporting beam layout for

triangle meshes

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Mesh optimization

• Project YAS island (Asymptote, Gehry Technologies, Schlaich

Bergermann, Waagner Biro, …)

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Mesh smoothing

• Project YAS island (quad mesh with nonplanar faces)

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Mesh smoothing

• Project YAS island (quad mesh with nonplanar faces)

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Optimized nodes

• Torsion minimization also works for quadrilateral meshes

with nonplanar faces

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Optimized nodes

• Torsion minimization also works for quadrilateral meshes

with nonplanar faces

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YAS project, mockup

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Offset meshes

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node geometry

Employing beams

• f constant height:

misalignment on

• ne side

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Cleanest nodes

 Perfect alignment on both sides if a mesh with

edge offsets is used

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Cleanest nodes

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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

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Exact offsets

 For offset pair define

Gauss image

 Then:

 vertex offsets

vertices of S lie in S2 (if M quad mesh, then circular mesh)

 edge offsets

edges of S are tangent to S2

 face offsets

face planes of S are tangent to S2 (M is a conical mesh)

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Meshes with edge offsets

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Edge offset meshes

 M has edge offsets iff it is parallel to a

mesh S whose edges are tangent to S2

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Koebe polyhedra

 Meshes with planar faces and

edges tangent to S2 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)

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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

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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 .

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

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Hexagonal EO mesh

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Planar hexagonal panels Planar hexagons

 non-convex in

negatively curved areas

 Phex mesh layout

largely unsolved

 initial results by

• Y. Liu and W. Wang

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Single curved panels, ruled panels and semi-discrete representations

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developable surfaces in architecture

 (nearly) developable surfaces

• F. Gehry, Guggenheim Museum, Bilbao

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surfaces in architecture

single curved panels

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D-strip models

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

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Principal strip models I

Circular strip model as limit of a circular mesh

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Principal strip models II

Conical strip model as limit of a conical mesh

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Principal strip models III

Conversion: conical model to circular model

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Conical strip model

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Conical strip model

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Multi-layer structure

D-strip model on top of a PQ mesh

Project in Cagliari, Hadid Architects

Approximation by ruled surface strips

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Semi-discrete model: smooth union of ruled strips

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Circle packings on surfaces

• M. Höbinger

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Project in Budapest, Hadid Architects

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Selfridges, Birmingham

Architects: Future Systems

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Optimization based on triangulation

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Circle packings on surfaces

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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

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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

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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

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D-strip models via subdivision