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

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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

SLIDE 2

Part I Nodes in steel/glass construction

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Steel–Glass constructions

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Steel–Glass constructions

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Nodes

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Nodes

• symmetry planes

intersect in node axis

= ⇒ no torsion

• here:

node with torsion

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Parallel meshes and node axes

• Ai

mi mi mi mi mi mi mi mi mi mi mi mi mi mi mi mi mi 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 M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′ M ′

meshes M , M ′ have parallel edges =

⇒

vertices mi, m′

i

span node axes Ai.

• ∃ axes ⇐

⇒

no torsion in nodes

• ∃ axes and M simply

connected =

⇒ ∃ M ′.

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Triangle meshes

• Triangle meshes M , M ′ parallel =

⇒ M ′ = o+λ·M

• For triangle meshes, node axes exist only in the trivial

ways

Ai = z ∨ mi

• r all Ai are parallel.
• Node complexity is higher for triangle meshes because

average # of edges in node equals 6

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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?

SLIDE 10

Part II Offsets – Beams in steel/glass constructions

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Offsets

• Offsets: meshes M , M ′ at “constant distance”
• E.g. such that edges of physically realized beams of

constant height are exactly aligned.

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Definition of offset meshes

• Consider parallel meshes M , M ′. Then M ′ is a
• vertex offset ⇐

⇒ mi − m′

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.

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Characterization of meshes with offsets

• S

M ′ = M + dS M

Lemma: A mesh M has a vertex/edge/face offset M ′

at distance d = 0 ⇐

⇒ S := (M ′ − M )/d (Note

S M ) has vertices/edges/faces tangent to S2.

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Characterization of meshes with offsets

• Lemma: A quad mesh M has a vertex offset M ′ =

M + dS ⇐

⇒ M is a circular mesh.

• Proof of “=

⇒”: ∃ S with vertices in S2 and parallel to

M =

⇒ faces of S have

circumcircle =

⇒

faces of M have circumcircle.

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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 S2 =

⇒ S conical = ⇒ M conical.

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Characterization of meshes with offsets

• M , M ′

S, λS

Lemma: M has

an edge offset M ′

⇐ ⇒ the space of

meshes parallel to

M contains a

Koebe polyhedron.

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Approximate offset meshes

• S = σ(M )

S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M + dS M

mj m′

j

si sj mi m′

i =

mi + dsi dist(M , M ′) ≈ const. ⇐ ⇒ M ′ = M + dS, where

S ≈ S2.

SLIDE 18

Part III Sample application

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Approximation problems

Assume that Φ is a surface. We ask:

• Is there a quad-dominant mesh (triangle mesh, hexag-
• nal mesh) approximating Φ?
• Is there a circular/conical mesh approximating Φ?
• are exactly constant beam heights possible?
• are approximately constant beam heights possible?

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Approximation problems

Assume that Φ is a surface. We ask:

• Is there a quad-dominant mesh (triangle mesh, hexag-
• nal mesh) approximating Φ? YES
• Is there a circular/conical mesh approximating Φ?
• are exactly constant beam heights possible?
• are approximately constant beam heights possible?

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Approximation problems

Assume that Φ is a surface. We ask:

• Is there a quad-dominant mesh (triangle mesh, hexag-
• nal 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?

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Approximation problems

Assume that Φ is a surface. We ask:

• Is there a quad-dominant mesh (triangle mesh, hexag-
• nal 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?

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Approximation problems

Assume that Φ is a surface. We ask:

• Is there a quad-dominant mesh (triangle mesh, hexag-
• nal 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)

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Beam layout

• For applications: Beams follow parallel meshesM , M ′
• Node axes

= ⇒ ‘no torsion’

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Processing pipeline demonstration

• A quadrilateral mesh with planar faces is a discrete

network of conjugate curves

• =

⇒ discuss various ‘principal curvature’ lines.

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Processing pipeline demonstration

• Construct mesh which follows conjugate curve network
• Planarize, e.g. with
• faces(

αj − (n − 2)π)2 → min

[Liu et al. 2006]

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S M and M ′ = M + dS −

− − − − − →

Processing pipeline demonstration

• Find S ≈ S2 in the space P(M ) by minimizing a

quadratic functional.

SLIDE 28

Part IV Edge offset meshes

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Koebe polyhedra and EO meshes

• S

S S S S S S S S S S S S S S S S

e e e e e e e e e e e e e e e e e si si si si si si si si si si si si si si si si si sj sj sj sj sj sj sj sj sj sj sj sj sj sj sj sj sj Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fk Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl Fl te te te te te te te te te te te te te te te te te csi csi csi csi csi csi csi csi csi csi csi csi csi csi csi csi csi csj csj csj csj csj csj csj csj csj csj csj csj csj csj csj csj csj cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk cFk

• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi
• Γi

si si si si si si si si si si si si si si si si si S2

Each vertex si has an associated cone Γi with axis through o which contains edges.

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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¨

• bius transformation dα on the unit

sphere, we have α(P(S)) = P(dα(S)).

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L-Invariance of the edge offset property

S

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L-Invariance of the edge offset property

S M

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L-Invariance of the edge offset property

S M

dα(S)

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L-Invariance of the edge offset property

S M

dα(S) α(M )

SLIDE 35

Part V Curvature theory

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Gauss image meshes

• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• mi
• 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

= mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi = mi + dsi

S M

The meshS (“Gauss image”) defined by parallel meshes

M , M ′ via M ′ = M + dS, defines normal vectors

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Mixed areas

• P

P P P P P P P P P P P P P P P P p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p0 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p1 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p2 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 p3 Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q1 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q2 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3 q3

• A (P + dQ) = A (P) + 2d A (P, Q) + d2 A (Q)
• A (P, Q) = 1

4

k−1

• i=0
• det(pi, qi+1) + det(qi, pi+1)

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Areas of offset surfaces

• Offsets Φd of a smooth surface Φ have the area

A (Φd) = ∫Φ

• 1 − 2dH(x) + d2K(x)
• dx

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Areas of offset surfaces

• Offsets Φd of a smooth surface Φ have the area

A (Φd) = ∫Φ

• 1 − 2dH(x) + d2K(x)
• dx
• Consider offsets M +dS of a mesh M , where M S:

HF := −A (F, σ(F))

A (F)

, KF := A (σ(F))

A (F)

= ⇒

A (M + dS) =

• F(1 − 2dHF + d2KF) A (F).

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Areas of offset surfaces

• Offsets Φd of a smooth surface Φ have the area

A (Φd) = ∫Φ

• 1 − 2dH(x) + d2K(x)
• dx
• Consider offsets M +dS of a mesh M , where M S:

HF := −A (F, σ(F))

A (F)

, KF := A (σ(F))

A (F)

= ⇒

A (M + dS) =

• F(1 − 2dHF + d2KF) A (F).
• this analogy shows that HF and KF can be considered

as mean and Gaussian curvatures of the face F .

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Defining principal ‘face curvatures’

• M

S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M ) S = σ(M )

we must have κ1 + κ2 = 2H, κ1κ2 = K, i.e.,

κ1, κ2 are definable ⇐ ⇒ H2 − K ≥ 0

• Lemma: F or σ(F) strictly convex

= ⇒ H2

F − KF ≥ 0.

• Proof: f (t) := A (F + tσ(F)) has a zero

⇐ ⇒ H2 − K ≥ 0 ⇐ ⇒

A (F, σ(F))2 − A (F) A (σ(F)) ≥ 0

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Face curvatures and edge curvatures

M

e e e e e e e e e e e e e e e e e Ce mi mi mi mi mi mi mi mi mi mi mi mi mi mi mi mi mi mj mj mj mj mj mj mj mj mj mj mj mj mj mj mj mj mj Ai Aj ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e ∆e

S

Σ si si si si si si si si si si si si si si si si si sj sj sj sj sj sj sj sj sj sj sj sj sj sj sj sj sj

• κe∆e

κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e κe∆e σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) σ(e) F F F F F F F F F F F F F F F F F − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e3 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e1 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e2 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4 − → e4

m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m1 = (0, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m2 = (1, 0) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m4 = (0, 1) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) m3 = s(1 − γ, γ) (1 − γ, γ)

σ(F) −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe3− → e3 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe1− → e1 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe2− → e2 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4 −κe4− → e4

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Face curvatures and edge curvatures

• Consider a quadrilateral F with edges e1, . . . , e4 and

edge curvatures κe1, . . . , κe4.

= ⇒ HF = (1 − γ)κe3 + κe4 2 + γκe1 + κe2 2 , KF = (1 − γ)κe3κe4 + γκe1κe2.

• Recall that we have an affine coordinate system with

m1 = (0, 0), m2 = (1, 0), m4 = (0, 1), m3 = s(1 − γ, γ)

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Minimal surfaces

• Definition. A polyhedral surface M is minimal w.r.t.

a Gauss image S parallel to M

⇐ ⇒ all mean face

curvatures vanish.

• This is equivalent to saying that for each face F of M

and its corresponding Gauss image σ(F) in S we have

A (F, σ(F)) = 0.

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Parallel quads with zero mixed area

• Qij

Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Qij Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

Q∗

ij

S S S S S S S S S S S S S S S S S M M M M M M M M M M M M M M M M M

1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 1∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 2∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 3∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 4∗ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

S M

• Prop. For parallel quads P = 1234, Q = 1∗2∗3∗4∗,

A (P, Q) = 0 ⇐

⇒ 13 2∗4∗ and 24 1∗3∗

• (Koenigs nets)

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Parallel polygons with zero mixed area

• p0

p1 p2 p3 p4 p5 p6 p7 q0 q1 q2 q3 q4 q5 q6 q7

• Prop. (C. M¨

uller) Parallel polygons P, Q (2k vertices) have A (P, Q) = 0, if one of {Li | i even }, {Li | i odd} is concurrent, where Li = qi + [pi−1 − pi+1].

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Parallel hexagons with zero mixed area

• p0

p1 p2 p3 p4 p5 q0 q1 q2 q3 q4 q5 d0 d1

• Prop. (C. M¨

uller) For parallel hexagons P, Q, A (P, Q) =

0 ⇐ ⇒ {Li | i even } or {Li | i odd} are concurrent.

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Minimal surfaces

• S

M S is a Koebe polyhedron and M is a discrete minimal

surface w.r.t. the Gauss image S

= ⇒ M has the

edge offset property.

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Minimal surfaces

• Mesh has edge
• ffset property
• can be realized

with beams of constant height (3D printout)

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Christoffel transform

• Both constructions of discrete minimal surface of [Bobenko

Pinkall 1996] and [Bobenko Hoffmann Suris 2006] lead to parallel quadrilaterals with vanishing mixed area.

• Definition: Meshes M M ′ are Christoffel trans-

forms of each other ⇐

⇒ corr. mixed areas vanish.

• Definition: The mesh M is a discrete minimal surface

⇐ ⇒ it is a Christoffel dual of its Gauss image S.

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Reciprocal-parallel meshes

• Prop. A (quad or hex) mesh possesses a Christoffel

dual ⇐

⇒ one or both of the two meshes consisting

• f the diagonals pi−1, pi+1 of each face p0, p1, . . . are

in static equilibrium.

• Proof: The geometric conditions equivalent to vanish-

ing mixed area yield reciprocal-parallel meshes, i.e., systems of forces.

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Reciprocal-parallel meshes

F(i, j) pi,j pi,j

1

pi,j

2

pi,j

3

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

4

pi,j

5

qi,j

1

qi,j

2

qi,j qi,j

3

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

qi,j

5

di,j

1

di+1,j

1

qi+1,j

3

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

di+1,j+1

1

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

qi,j

4

SLIDE 53

36/36

Conclusion

• Use parallel meshes to define node axes (beam layout)
• encode offset properties in Gauss image (also parallel)
• Koebe polyhedra =

⇒ EO meshes (Lag. geom.)

• curvatures via mixed areas,

discrete minimal surfaces

• reciprocal-parallel meshes