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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 Wien2 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 netExample 2: principalcurvature 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 optimization

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 other

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 of constant height: misalignment on one 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 andsemi-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 ontop 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