pott mann
DESCRIPTION
architectureTRANSCRIPT
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1GEOMETRIE
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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8GEOMETRIE
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
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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 Mbius 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
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Project in Cagliari, Hadid Architects
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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. Hbinger
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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. Flry, 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