Geometry of Multi-layer Freeform Structures for Architecture (joint - PowerPoint PPT Presentation

Geometry of Multi-layer Freeform Structures for Architecture (joint work with A. Bobenko, Liu Yang, C. M¨ uller, H. Pottmann, W. Wang) Johannes Wallner Institut f¨ ur Geometrie, TU Graz July 18, 2007 Discrete Differential Geometry Berlin 2007

Part I Nodes in steel/glass construction

1/36 Steel–Glass constructions

1/36 Steel–Glass constructions

2/36 Nodes

3/36 Nodes • symmetry planes intersect in node axis = ⇒ no torsion • here: node with torsion

4/36 Parallel meshes and node axes meshes M , M ′ have parallel edges = ⇒ • vertices m i , m ′ A i i span node axes A i . m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m i m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ m ′ i i i i i i i i i i i i i i i i i • ∃ axes ⇐ ⇒ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ no torsion in nodes M M M M M M M M M M M M M M M M M • ∃ axes and M simply ⇒ ∃ M ′ . connected =

5/36 Triangle meshes ⇒ M ′ = o + λ · M • Triangle meshes M , M ′ parallel = • For triangle meshes, node axes exist only in the trivial ways A i = z ∨ m i or all A i are parallel. • Node complexity is higher for triangle meshes because average # of edges in node equals 6

6/36 Conclusion: • To avoid torsion in nodes, beams of a steel/glass con- struction should follow two parallel meshes M , M ′ . • Triangle meshes have only a few parallel meshes, also average number of edges per vertex is high • = ⇒ Quad or hex meshes are better suited for building construction?

Part II Offsets – Beams in steel/glass constructions

7/36 Offsets • Offsets: meshes M , M ′ at “constant distance” • E.g. such that edges of physically realized beams of constant height are exactly aligned.

8/36 Definition of offset meshes • Consider parallel meshes M , M ′ . Then M ′ is a ⇒ � m i − m ′ • vertex offset ⇐ i � = d = const . • edge offsets ⇐ ⇒ distance of lines carrying corre- sponding edges equals d . • face offsets ⇐ ⇒ distance of planes carrying corre- sponding faces equals d .

9/36 Characterization of meshes with offsets Lemma: A mesh M has a vertex/edge/face offset M ′ • ⇒ S := ( M ′ − M ) /d (Note at distance d � = 0 ⇐ S � M ) has vertices/edges/faces tangent to S 2 . S M M ′ = M + d S

10/36 Characterization of meshes with offsets • Lemma: A quad mesh M has a vertex offset M ′ = M + d S ⇐ ⇒ M is a circular mesh. ⇒ ”: ∃ S with vertices in S 2 and parallel to • Proof of “ = ⇒ faces of S have M = circumcircle = ⇒ faces of M have circumcircle.

11/36 Characterization of meshes with offsets • Lemma: A mesh M has a face offset M ′ at distance d > 0 ⇐ ⇒ M is a conical mesh. • Proof: Let S = ( M ′ − M ) /d . Faces of S are tangent to S 2 = ⇒ S conical = ⇒ M conical.

12/36 Characterization of meshes with offsets Lemma: M has • M , M ′ an edge offset M ′ ⇐ ⇒ the space of meshes parallel to M contains a Koebe polyhedron. S , λ S

13/36 Approximate offset meshes ⇒ M ′ = M + d S , where dist( M , M ′ ) ≈ const. ⇐ • S ≈ S 2 . S = σ ( M ) S = σ ( M ) S = σ ( M ) S = σ ( M ) S = σ ( M ) M + d S M + d S M + d S S = σ ( M ) S = σ ( M ) M + d S M + d S S = σ ( M ) S = σ ( M ) S = σ ( M ) M + d S M + d S S = σ ( M ) S = σ ( M ) M + d S M + d S M + d S S = σ ( M ) S = σ ( M ) M + d S M + d S S = σ ( M ) S = σ ( M ) S = σ ( M ) M + d S M + d S M + d S M + d S M + d S M m ′ s j j s i m j m ′ i = m i + d s i m i

Part III Sample application

14/36 Approximation problems Assume that Φ is a surface. We ask: • Is there a quad-dominant mesh (triangle mesh, hexag- onal mesh) approximating Φ ? • Is there a circular/conical mesh approximating Φ ? • are exactly constant beam heights possible? • are approximately constant beam heights possible?

14/36 Approximation problems Assume that Φ is a surface. We ask: • Is there a quad-dominant mesh (triangle mesh, hexag- onal mesh) approximating Φ ? YES • Is there a circular/conical mesh approximating Φ ? • are exactly constant beam heights possible? • are approximately constant beam heights possible?

14/36 Approximation problems Assume that Φ is a surface. We ask: • Is there a quad-dominant mesh (triangle mesh, hexag- onal mesh) approximating Φ ? YES • Is there a circular/conical mesh approximating Φ ? YES • are exactly constant beam heights possible? • are approximately constant beam heights possible?

14/36 Approximation problems Assume that Φ is a surface. We ask: • Is there a quad-dominant mesh (triangle mesh, hexag- onal mesh) approximating Φ ? YES • Is there a circular/conical mesh approximating Φ ? YES • are exactly constant beam heights possible? NO • are approximately constant beam heights possible?

14/36 Approximation problems Assume that Φ is a surface. We ask: • Is there a quad-dominant mesh (triangle mesh, hexag- onal mesh) approximating Φ ? YES • Is there a circular/conical mesh approximating Φ ? YES • are exactly constant beam heights possible? NO • are approximately constant beam heights possible? YES (generic answers)

15/36 Beam layout • For applications: Beams follow parallel meshes M , M ′ • Node axes = ⇒ ‘no torsion’

16/36 Processing pipeline demonstration • A quadrilateral mesh with planar faces is a discrete network of conjugate curves • = ⇒ discuss various ‘principal curvature’ lines.

17/36 Processing pipeline demonstration • Construct mesh which follows conjugate curve network � α j − ( n − 2) π ) 2 → min � • Planarize, e.g. with faces ( [Liu et al. 2006]

18/36 Processing pipeline demonstration • Find S ≈ S 2 in the space P ( M ) by minimizing a quadratic functional. S M and M ′ = M + d S − − − − − − →

Part IV Edge offset meshes

19/36 Koebe polyhedra and EO meshes Each vertex s i has an associated cone Γ i with axis • through o which contains edges. S 2 c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j c s j F l F l F l F l F l F l F l F l F l F l F l F l F l F l F l F l F l s j s j s j s j s j s j s j s j s j s j s j s j s j s j s j s j s j t e t e t e t e t e t e t e t e t e t e t e t e t e t e t e t e t e e e e e e e e e e e e e e e e e e � � � � � � � � � � � � � � Γ i Γ i Γ i � � � Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i Γ i F k F k F k c F k c F k c F k c s i c s i c s i F k F k F k F k c F k c F k c F k c F k c s i c s i c s i c s i F k F k F k c F k c F k c F k c s i c s i c s i F k F k F k F k c F k c F k c F k c F k c s i c s i c s i c s i F k F k F k c F k c F k c F k c s i c s i c s i S S S S S S S S S S S S S S S S S s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i s i

20/36 L-Invariance of the edge offset property • Cones as envelopes of planes are objects of Laguerre geometry. Therefore so are edge offset meshes. • Prop. (H. Pottmann) With the notation P ( S ) for the meshes parallel to a Koebe polyhedron S , for any Laguerre transformation α in the space of planes and induced M¨ obius transformation dα on the unit sphere, we have α ( P ( S )) = P ( dα ( S )) .

21/36 L-Invariance of the edge offset property S

21/36 L-Invariance of the edge offset property S M

21/36 L-Invariance of the edge offset property S M dα ( S )

21/36 L-Invariance of the edge offset property S M dα ( S ) α ( M )

Part V Curvature theory

22/36 Gauss image meshes The mesh S (“Gauss image”) defined by parallel meshes • M , M ′ via M ′ = M + d S , defines normal vectors ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m ′ ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i ◦ m i i i i i i i i i i i i i i i i i i S = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i = m i + d s i M

23/36 Mixed areas A ( P + dQ ) = A ( P ) + 2 d A ( P, Q ) + d 2 A ( Q ) • k − 1 � � � • A ( P, Q ) = 1 det( p i , q i +1 ) + det( q i , p i +1 ) 4 i =0 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 p 3 q 3 q 3 q 3 q 2 q 2 q 2 q 3 q 3 q 2 q 2 q 3 q 3 q 2 q 2 q 3 q 3 q 3 q 2 q 2 q 2 q 3 q 3 q 2 q 2 q 3 q 3 q 2 q 2 q 3 q 3 q 3 q 2 q 2 q 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 p 2 P Q P P P P Q Q Q Q P P Q Q P P P Q Q Q P P Q Q P P Q Q P P P Q Q Q o o o o o o o o o o o o o o o o o p 0 p 0 p 0 p 1 p 1 p 1 q 0 q 0 q 0 q 1 q 1 q 1 p 0 p 0 p 1 p 1 q 0 q 0 q 1 q 1 p 0 p 0 p 1 p 1 q 0 q 0 q 1 q 1 p 0 p 0 p 0 p 1 p 1 p 1 q 0 q 0 q 0 q 1 q 1 q 1 p 0 p 0 p 1 p 1 q 0 q 0 q 1 q 1 p 0 p 0 p 1 p 1 q 0 q 0 q 1 q 1 p 0 p 0 p 0 p 1 p 1 p 1 q 0 q 0 q 0 q 1 q 1 q 1