Pin Merging in Planar Body Frameworks Rudi Penne rudi.penne@kdg.be - - PowerPoint PPT Presentation

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Pin Merging in Planar Body Frameworks Rudi Penne rudi.penne@kdg.be - - PowerPoint PPT Presentation

Oct 30, 2022 314 likes •1.54k views

Rigidity Workshop, Toronto 2011 Pin Merging in Planar Body Frameworks Rudi Penne rudi.penne@kdg.be Karel de Grote-Hogeschool University of Antwerp pin merging in planar body frameworks p. 1/25 Meet a 3 2 T -graph pin merging in planar

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SLIDE 1

Rigidity Workshop, Toronto 2011

Pin Merging in Planar Body Frameworks

Rudi Penne

rudi.penne@kdg.be

Karel de Grote-Hogeschool University of Antwerp

pin merging in planar body frameworks – p. 1/25

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SLIDE 2

Meet a 3

2T-graph

pin merging in planar body frameworks – p. 2/25

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SLIDE 3

Meet a 3

2T-graph

7 6 5 3 4 1 2

pin merging in planar body frameworks – p. 2/25

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SLIDE 4

Meet a 3

2T-graph

7 6 5 3 4 1 2

Definition:

G = (V, E) is a 3

2T-graph if

E = FR ∪ FY ∪ FG

with covering trees FR ∪ FY , FR ∪ FG and FY ∪ FG.

pin merging in planar body frameworks – p. 2/25

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SLIDE 5

Meet a 3

2T-graph

7 6 5 3 4 1 2

Definition:

G = (V, E) is a 3

2T-graph if

E = FR ∪ FY ∪ FG

with covering trees FR ∪ FY , FR ∪ FG and FY ∪ FG.

pin merging in planar body frameworks – p. 2/25

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SLIDE 6

Meet a 3

2T-graph

7 6 5 3 4 1 2

Definition:

G = (V, E) is a 3

2T-graph if

E = FR ∪ FY ∪ FG

with covering trees FR ∪ FY , FR ∪ FG and FY ∪ FG.

pin merging in planar body frameworks – p. 2/25

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SLIDE 7

Meet a 3

2T-graph

7 6 5 3 4 1 2

Definition:

G = (V, E) is a 3

2T-graph if

E = FR ∪ FY ∪ FG

with covering trees FR ∪ FY , FR ∪ FG and FY ∪ FG.

pin merging in planar body frameworks – p. 2/25

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SLIDE 8

The double of a 3

2T-graph

Definition: 2G: the double of a graph:

pin merging in planar body frameworks – p. 3/25

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SLIDE 9

The double of a 3

2T-graph

Definition: 2G: the double of a graph:

= multigraph obtained from G = (V, E) by doubling each edge in E

pin merging in planar body frameworks – p. 3/25

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SLIDE 10

The double of a 3

2T-graph

Definition: 2G: the double of a graph:

= multigraph obtained from G = (V, E) by doubling each edge in E

Property: G is a 3

2T-graph

⇐ ⇒ 2G is the union of 3 spanning trees

pin merging in planar body frameworks – p. 3/25

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SLIDE 11

The double of a 3

2T-graph

Definition: 2G: the double of a graph:

= multigraph obtained from G = (V, E) by doubling each edge in E

Property: G is a 3

2T-graph

⇐ ⇒ 2G is the union of 3 spanning trees (Indeed: 2G = TRY ∪ TRG ∪ TY G with TRY = FR ∪ FY , TRG = FR ∪ FG and TY G = FY ∪ FG. )

pin merging in planar body frameworks – p. 3/25

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SLIDE 12

The double of a 3

2T-graph

Definition: 2G: the double of a graph:

= multigraph obtained from G = (V, E) by doubling each edge in E

Property: G is a 3

2T-graph

⇐ ⇒ 2G is the union of 3 spanning trees (Indeed: 2G = TRY ∪ TRG ∪ TY G with TRY = FR ∪ FY , TRG = FR ∪ FG and TY G = FY ∪ FG. )

Conclusion: (Nash-Williams, Tutte)

G = (V, E) is a 3

2T-graph ⇐

⇒

  • 1. 2|E| = 3|V | − 3
  • 2. ∀∅ = E′ ⊂ E : 2|E′| ≤ 3|V ′| − 3

pin merging in planar body frameworks – p. 3/25

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SLIDE 13

Graphs as framework design

G = (V, E): given graph (e.g. 3

2 T-graph)

pin merging in planar body frameworks – p. 4/25

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SLIDE 14

Graphs as framework design

G = (V, E): given graph (e.g. 3

2 T-graph)

V ↔ rigid bodies in the plane E ↔ revolute pins connecting body pairs

pin merging in planar body frameworks – p. 4/25

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SLIDE 15

Graphs as framework design

G = (V, E): given graph (e.g. 3

2 T-graph)

V ↔ rigid bodies in the plane E ↔ revolute pins connecting body pairs

1 2 3 4 5 6 7 6 3 4 1 2 7 5

pin merging in planar body frameworks – p. 4/25

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SLIDE 16

Graphs as framework design

G = (V, E): given graph (e.g. 3

2 T-graph)

V ↔ rigid bodies in the plane E ↔ revolute pins connecting body pairs

1 2 3 4 5 6 7 6 3 4 1 2 7 5

Remark: pins have degree 2 in generic realizations

pin merging in planar body frameworks – p. 4/25

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SLIDE 17

Results in general dimensions

d: dimension workspace

pin merging in planar body frameworks – p. 5/25

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SLIDE 18

Results in general dimensions

d: dimension workspace

G = (V, E): design for body-and-hinge framework V ↔ rigid bodies in d-space E ↔ hinges attaching body pairs

pin merging in planar body frameworks – p. 5/25

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SLIDE 19

Results in general dimensions

d: dimension workspace

G = (V, E): design for body-and-hinge framework V ↔ rigid bodies in d-space E ↔ hinges attaching body pairs

D = d+1

d−1

  • : dimension space of hinges

Theorem: G can be realized as inf. rigid body-and-hinge

framework in I Rd iff. (D − 1)G contains D edge-disjoint spanning trees. (Tay-Whiteley)

pin merging in planar body frameworks – p. 5/25

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SLIDE 20

Results in general dimensions

d: dimension workspace

G = (V, E): design for body-and-hinge framework V ↔ rigid bodies in d-space E ↔ hinges attaching body pairs

D = d+1

d−1

  • : dimension space of hinges

Theorem: G can be realized as inf. rigid body-and-hinge

framework in I Rd iff. (D − 1)G contains D edge-disjoint spanning trees. (Tay-Whiteley)

Special case (d = 2): 3

2T-graph is a minimal design for inf. rigid

body-and-pin framework in the plane.

pin merging in planar body frameworks – p. 5/25

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SLIDE 21

Non-generic realizations

G = (V, E): design for body-and-hinge framework in I Rd

pin merging in planar body frameworks – p. 6/25

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SLIDE 22

Non-generic realizations

G = (V, E): design for body-and-hinge framework in I Rd Assume: (D − 1)G contains D edge-disjoint spanning trees

pin merging in planar body frameworks – p. 6/25

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SLIDE 23

Non-generic realizations

G = (V, E): design for body-and-hinge framework in I Rd Assume: (D − 1)G contains D edge-disjoint spanning trees

Katoh-Tanigawa: ∃ rigid realizations with plates

pin merging in planar body frameworks – p. 6/25

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SLIDE 24

Non-generic realizations

G = (V, E): design for body-and-hinge framework in I Rd Assume: (D − 1)G contains D edge-disjoint spanning trees

Katoh-Tanigawa: ∃ rigid realizations with plates

Special case: 3

2T-graph can be realized as inf. rigid frameworks

in the plane with collinear pins for each body. (Jackson-Jordán)

pin merging in planar body frameworks – p. 6/25

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SLIDE 25

Non-generic realizations

G = (V, E): design for body-and-hinge framework in I Rd Assume: (D − 1)G contains D edge-disjoint spanning trees

Katoh-Tanigawa: ∃ rigid realizations with plates General incidences: (allowing coinciding hinges = "multi-hinges")

count criterium on incidence graph Kb,h but no tree decomposition (Tay(?), 1987) (Tanigawa, 2011):

pin merging in planar body frameworks – p. 6/25

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SLIDE 26

Non-generic realizations

G = (V, E): design for body-and-hinge framework in I Rd Assume: (D − 1)G contains D edge-disjoint spanning trees

Katoh-Tanigawa: ∃ rigid realizations with plates General incidences: (allowing coinciding hinges = "multi-hinges")

count criterium on incidence graph Kb,h but no tree decomposition (Tay(?), 1987) (Tanigawa, 2011): ∃ rigid realization ⇐ ⇒ ∃ I ⊂ (D − 1)E(Kb,h) s.t. 1) |I| = D · b + (D − 1) · h − D 2) ∀F ⊂ I: F ≤ D · B(F) + (D − 1) · H(F) − D

pin merging in planar body frameworks – p. 6/25

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SLIDE 27

Pin merging

pin merging in planar body frameworks – p. 7/25

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SLIDE 28

Pin merging

= non-generic planar realization of G = (V, E) as body framework such that certain attachments (in E) are realized as coinciding pins:

pin merging in planar body frameworks – p. 7/25

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SLIDE 29

Pin merging

= non-generic planar realization of G = (V, E) as body framework such that certain attachments (in E) are realized as coinciding pins:

7 6 5 3 4 1 2 2 7 6 5 4 3 1

pin merging in planar body frameworks – p. 7/25

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SLIDE 30

Pin merging

= non-generic planar realization of G = (V, E) as body framework such that certain attachments (in E) are realized as coinciding pins:

7 6 5 3 4 1 2 2 7 6 5 4 3 1

Question:

What pin mergings in 3

2T-graphs preserve inf. rigidity?

pin merging in planar body frameworks – p. 7/25

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SLIDE 31

Hypergraphs and merged pins

Example:

pin merging in planar body frameworks – p. 8/25

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SLIDE 32

Hypergraphs and merged pins

Example:

7 6 5 3 4 1 2

constraint: bodies 1,2,4,7 attached by one pin

pin merging in planar body frameworks – p. 8/25

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SLIDE 33

Hypergraphs and merged pins

Example:

7 6 5 3 4 1 2

constraint: bodies 1,2,4,7 attached by one pin

⇒ edges 12, 42, 72 clustered as hyperedge

pin merging in planar body frameworks – p. 8/25

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SLIDE 34

Hypergraphs and merged pins

Example:

7 6 5 3 4 1 2

constraint: bodies 1,2,4,7 attached by one pin

⇒ edges 12, 42, 72 clustered as hyperedge ⇒ framework design = hypergraph

pin merging in planar body frameworks – p. 8/25

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SLIDE 35

Hypergraphs and merged pins

Example:

7 6 5 3 4 1 2

constraint: bodies 1,2,4,7 attached by one pin

⇒ edges 12, 42, 72 clustered as hyperedge ⇒ framework design = hypergraph

7 3 6 5 4 1 2 1 3 4 2 7 5 6

pin merging in planar body frameworks – p. 8/25

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SLIDE 36

Hypergraphs and merged pins

Example:

7 6 5 3 4 1 2

constraint: bodies 1,2,4,7 attached by one pin

⇒ edges 12, 42, 72 clustered as hyperedge ⇒ framework design = hypergraph

7 3 6 5 4 1 2 1 3 4 2 7 5 6

Notice: this pin merge causes non-trivial motions.

pin merging in planar body frameworks – p. 8/25

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SLIDE 37

Intermezzo: weights of hyperedges

pin merging in planar body frameworks – p. 9/25

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SLIDE 38

Intermezzo: weights of hyperedges

4 2 1 3 5 6 7

G = ({1, 2, 3, 4, 5, 6, 7}, {e1, e2, e3, e4})

pin merging in planar body frameworks – p. 9/25

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SLIDE 39

Intermezzo: weights of hyperedges

4 2 1 3 5 6 7

G = ({1, 2, 3, 4, 5, 6, 7}, {e1, e2, e3, e4}) hyperedges:

e1 = {2, 7} e2 = {2, 4} e3 = {3, 4, 5} e4 = {1, 3, 6}

pin merging in planar body frameworks – p. 9/25

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SLIDE 40

Intermezzo: weights of hyperedges

4 2 1 3 5 6 7

G = ({1, 2, 3, 4, 5, 6, 7}, {e1, e2, e3, e4}) hyperedges:

e1 = {2, 7} e2 = {2, 4} e3 = {3, 4, 5} e4 = {1, 3, 6}

Definition: weight hyperedge: w(e) = |e| − 1

pin merging in planar body frameworks – p. 9/25

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SLIDE 41

Intermezzo: weights of hyperedges

4 2 1 3 5 6 7

G = ({1, 2, 3, 4, 5, 6, 7}, {e1, e2, e3, e4}) hyperedges:

e1 = {2, 7} e2 = {2, 4} e3 = {3, 4, 5} e4 = {1, 3, 6}

Definition: weight hyperedge: w(e) = |e| − 1

w(e1) = w(e2) = 1, w(e3) = w(e4) = 2

pin merging in planar body frameworks – p. 9/25

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SLIDE 42

Intermezzo: weights of hyperedges

4 2 1 3 5 6 7

G = ({1, 2, 3, 4, 5, 6, 7}, {e1, e2, e3, e4}) hyperedges:

e1 = {2, 7} e2 = {2, 4} e3 = {3, 4, 5} e4 = {1, 3, 6}

Definition: weight hyperedge: w(e) = |e| − 1

w(e1) = w(e2) = 1, w(e3) = w(e4) = 2

Application: Hypertree:

pin merging in planar body frameworks – p. 9/25

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SLIDE 43

Intermezzo: weights of hyperedges

4 2 1 3 5 6 7

G = ({1, 2, 3, 4, 5, 6, 7}, {e1, e2, e3, e4}) hyperedges:

e1 = {2, 7} e2 = {2, 4} e3 = {3, 4, 5} e4 = {1, 3, 6}

Definition: weight hyperedge: w(e) = |e| − 1

w(e1) = w(e2) = 1, w(e3) = w(e4) = 2

Application: Hypertree:

G connected and no hypercycles ⇐ ⇒ G connected and w(E) = |V | − 1 ⇐ ⇒ w(E) = |V | − 1 and for each ∅ = E′ ⊂ E : w(E′) ≤ | ∪ E′| − 1

pin merging in planar body frameworks – p. 9/25

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SLIDE 44

3 2HT-Hypergraphs

4 2 1 3 5 6 7

pin merging in planar body frameworks – p. 10/25

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SLIDE 45

3 2HT-Hypergraphs

4 2 1 3 5 6 7

3 2HT-decomposition: 3 colours for hyperedges

pin merging in planar body frameworks – p. 10/25

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SLIDE 46

3 2HT-Hypergraphs

4 2 1 3 5 6 7

3 2HT-decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree

pin merging in planar body frameworks – p. 10/25

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SLIDE 47

3 2HT-Hypergraphs

4 2 1 3 5 6 7

3 2HT-decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree

spanning hypertree:

pin merging in planar body frameworks – p. 10/25

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SLIDE 48

3 2HT-Hypergraphs

4 2 1 3 5 6 7

3 2HT-decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree

spanning hypertree:

4 2 1 3 5 6 7

pin merging in planar body frameworks – p. 10/25

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SLIDE 49

3 2HT-Hypergraphs

4 2 1 3 5 6 7

3 2HT-decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree

spanning hypertree:

4 2 1 3 5 6 7

Consequence: 3

2HT-hypergraph G = (V, E) ⇒

2 · w(E) = 3|V | − 3

for each ∅ = E′ ⊂ E : 2 · w(E′) ≤ 3| ∪ E′| − 3

pin merging in planar body frameworks – p. 10/25

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SLIDE 50

3 2HT-Hypergraphs

4 2 1 3 5 6 7

3 2HT-decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree

spanning hypertree:

4 2 1 3 5 6 7

Consequence: 3

2HT-hypergraph G = (V, E) ⇒

2 · w(E) = 3|V | − 3

for each ∅ = E′ ⊂ E : 2 · w(E′) ≤ 3| ∪ E′| − 3

(3/2,3/2)-hypertight (?)

pin merging in planar body frameworks – p. 10/25

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SLIDE 51

3 2HT versus 3 2T

hypergraph G = (V, E)

4 2 1 3 5 6 7

→

hosting

←

clustering

graph G2 = (V, E2)

4 2 1 3 5 6 7

pin merging in planar body frameworks – p. 11/25

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SLIDE 52

3 2HT versus 3 2T

hypergraph G = (V, E)

4 2 1 3 5 6 7

total weight w(E) →

hosting

←

clustering

graph G2 = (V, E2)

4 2 1 3 5 6 7

number of edges |E2|

pin merging in planar body frameworks – p. 11/25

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SLIDE 53

3 2HT versus 3 2T

hypergraph G = (V, E)

4 2 1 3 5 6 7

total weight w(E) →

coloured hosting

←

monochromatic clustering

graph G2 = (V, E2)

7 6 5 3 4 1 2

number of edges |E2|

3 2HT-decomposition

→

always

←

sometimes

3 2T-decomposition

pin merging in planar body frameworks – p. 11/25

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SLIDE 54

3 2HT-obstructions

1 3 4 2 7 5 6

(3/2, 3/2)-hypertight

pin merging in planar body frameworks – p. 12/25

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SLIDE 55

3 2HT-obstructions

1 3 4 2 7 5 6

(3/2, 3/2)-hypertight yet not 3

2HT-decomposable

pin merging in planar body frameworks – p. 12/25

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SLIDE 56

3 2HT-obstructions

1 3 4 7 5 6 2

(3/2, 3/2)-hypertight yet not 3

2HT-decomposable

Property: 3

2HT ⇒ no (hyper)leaves

pin merging in planar body frameworks – p. 12/25

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SLIDE 57

3 2HT-obstructions

1 3 4 7 5 6 2

(3/2, 3/2)-hypertight yet not 3

2HT-decomposable

Property: 3

2HT ⇒ no (hyper)leaves

Conjecture: leaf-free + (3/2,3/2)-hypertight ⇐

⇒

3 2HT

pin merging in planar body frameworks – p. 12/25

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SLIDE 58

3 2HT-obstructions

1 3 4 7 5 6 2

(3/2, 3/2)-hypertight yet not 3

2HT-decomposable

Property: 3

2HT ⇒ no (hyper)leaves

Conjecture: leaf-free + (3/2,3/2)-hypertight ⇐

⇒

3 2HT Lucky guess: leaf-free + (D/(D-1),D/(D-1))-hypertight ⇐ ⇒

D D−1 HT

pin merging in planar body frameworks – p. 12/25

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SLIDE 59

3 2HT-Hypergraphs as rigid frameworks

  • Definition. Planar framework realization of hypergraph G =

body-and-pin realization where hyperedges represent merged pins.

pin merging in planar body frameworks – p. 13/25

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SLIDE 60

3 2HT-Hypergraphs as rigid frameworks

  • Definition. Planar framework realization of hypergraph G =

body-and-pin realization where hyperedges represent merged pins.

Theorem:

pin merging in planar body frameworks – p. 13/25

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SLIDE 61

3 2HT-Hypergraphs as rigid frameworks

  • Definition. Planar framework realization of hypergraph G =

body-and-pin realization where hyperedges represent merged pins.

Theorem:

G contains 3

2HT ⇒ realizable as inf. rigid planar body-and-pin

framework.

pin merging in planar body frameworks – p. 13/25

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SLIDE 62

3 2HT-Hypergraphs as rigid frameworks

  • Definition. Planar framework realization of hypergraph G =

body-and-pin realization where hyperedges represent merged pins.

Theorem:

G contains 3

2HT ⇒ realizable as inf. rigid planar body-and-pin

framework.

  • Proof. Specialisation of rigidity matrix. Valid for general

dimensions.

pin merging in planar body frameworks – p. 13/25

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SLIDE 63

3 2HT-Hypergraphs as rigid frameworks

  • Definition. Planar framework realization of hypergraph G =

body-and-pin realization where hyperedges represent merged pins.

Theorem:

G contains 3

2HT ⇒ realizable as inf. rigid planar body-and-pin

framework.

  • Proof. Specialisation of rigidity matrix. Valid for general

dimensions.

Conjecture: The converse holds.

pin merging in planar body frameworks – p. 13/25

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SLIDE 64

The rigidity matrix I

Realization of hypergraph G = (V, E) as body and pin framework in the plane: F = (G, P) P : E → I R2 : e → P(e) = (xe, ye)

pin merging in planar body frameworks – p. 14/25

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SLIDE 65

The rigidity matrix I

Realization of hypergraph G = (V, E) as body and pin framework in the plane: F = (G, P) P : E → I R2 : e → P(e) = (xe, ye) Choose host graph G2 = (V, E2).

pin merging in planar body frameworks – p. 14/25

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SLIDE 66

The rigidity matrix I

Realization of hypergraph G = (V, E) as body and pin framework in the plane: F = (G, P) P : E → I R2 : e → P(e) = (xe, ye) Choose host graph G2 = (V, E2). For each edge ij ∈ E2 with {i, j} ⊂ e ∈ E:

JXij = (0, . . . , 0, 1, 0 . . . , 0, −1, 0, . . . , 0|0, . . . , 0|0, . . . , 0, −xe, 0, . . . , 0, xe, 0, . . . , 0) JYij = (0, . . . , 0|0, . . . , 0, 1, 0 . . . , 0, −1, 0, . . . , 0|0, . . . , 0, −ye, 0, . . . , 0, ye, 0, . . . , 0) (non-zero entries in positions i and j in subsequences of length |V |)

pin merging in planar body frameworks – p. 14/25

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SLIDE 67

The rigidity matrix I

Realization of hypergraph G = (V, E) as body and pin framework in the plane: F = (G, P) P : E → I R2 : e → P(e) = (xe, ye) Choose host graph G2 = (V, E2). For each edge ij ∈ E2 with {i, j} ⊂ e ∈ E:

JXij = (0, . . . , 0, 1, 0 . . . , 0, −1, 0, . . . , 0|0, . . . , 0|0, . . . , 0, −xe, 0, . . . , 0, xe, 0, . . . , 0) JYij = (0, . . . , 0|0, . . . , 0, 1, 0 . . . , 0, −1, 0, . . . , 0|0, . . . , 0, −ye, 0, . . . , 0, ye, 0, . . . , 0) (non-zero entries in positions i and j in subsequences of length |V |)

⇒ 2w(E) × |V | matrix M(G2, P).

pin merging in planar body frameworks – p. 14/25

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SLIDE 68

The rigidity matrix II

Given hypergraph G = (V, E) with realization F = (G, P)

pin merging in planar body frameworks – p. 15/25

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SLIDE 69

The rigidity matrix II

Given hypergraph G = (V, E) with realization F = (G, P)

Let Ci = (ai, bi, ci) with i = 1, . . . , v = |V | and put γ = (a1, . . . , av, b1, . . . , bv, c1, . . . , cv) ∈ I R3v

pin merging in planar body frameworks – p. 15/25

slide-70
SLIDE 70

The rigidity matrix II

Given hypergraph G = (V, E) with realization F = (G, P)

Let Ci = (ai, bi, ci) with i = 1, . . . , v = |V | and put γ = (a1, . . . , av, b1, . . . , bv, c1, . . . , cv) ∈ I R3v

Property: The Ci are centers of motion for bodies of F ⇐

⇒ M(G2, P) · γT = 0 for any host G2 ⇐ ⇒ M(G2, P) · γT = 0 for every host G2

pin merging in planar body frameworks – p. 15/25

slide-71
SLIDE 71

The rigidity matrix II

Given hypergraph G = (V, E) with realization F = (G, P)

Let Ci = (ai, bi, ci) with i = 1, . . . , v = |V | and put γ = (a1, . . . , av, b1, . . . , bv, c1, . . . , cv) ∈ I R3v

Property: The Ci are centers of motion for bodies of F ⇐

⇒ M(G2, P) · γT = 0 for any host G2 ⇐ ⇒ M(G2, P) · γT = 0 for every host G2

Remarks:

rank M(G2, P) independent from host F inf. rigid ⇐ ⇒ rank M(G2, P) = 3|V | − 3 F isostatic ⇐ ⇒ M(G2, P) has independent rows and 2 · w(E) = 3|V | − 3

pin merging in planar body frameworks – p. 15/25

slide-72
SLIDE 72

Independent hypergraphs

pin merging in planar body frameworks – p. 16/25

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SLIDE 73

Independent hypergraphs

Definition: Hypergraph G = (V, E) is 2-independent iff. there is

a realization P such that for some (hence for every) host G2 the rows of M(G2, P) are linearly independent.

pin merging in planar body frameworks – p. 16/25

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SLIDE 74

Independent hypergraphs

Definition: Hypergraph G = (V, E) is 2-independent iff. there is

a realization P such that for some (hence for every) host G2 the rows of M(G2, P) are linearly independent.

Count criterion: Hypergraph G = (V, E) without leaves is

2-independent iff.

∀∅ = E′ ⊂ E : 2 · w(E′) ≤ 3| ∪ E′| − 3

pin merging in planar body frameworks – p. 16/25

slide-75
SLIDE 75

Independent hypergraphs

Definition: Hypergraph G = (V, E) is 2-independent iff. there is

a realization P such that for some (hence for every) host G2 the rows of M(G2, P) are linearly independent.

Count criterion: Hypergraph G = (V, E) without leaves is

2-independent iff.

∀∅ = E′ ⊂ E : 2 · w(E′) ≤ 3| ∪ E′| − 3

Proof:

necessary: corank of each row subset ≥ 3 sufficient: Laman’s Theorem

pin merging in planar body frameworks – p. 16/25

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SLIDE 76

Independent hypergraphs

Definition: Hypergraph G = (V, E) is 2-independent iff. there is

a realization P such that for some (hence for every) host G2 the rows of M(G2, P) are linearly independent.

Count criterion: Hypergraph G = (V, E) without leaves is

2-independent iff.

∀∅ = E′ ⊂ E : 2 · w(E′) ≤ 3| ∪ E′| − 3

Proof:

necessary: corank of each row subset ≥ 3 sufficient: Laman’s Theorem

  • Remark. Our count is equivalent to the Tay-Tanigawa criterion

for d = 2 (extra condition: no leaves).

pin merging in planar body frameworks – p. 16/25

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SLIDE 77

Results and conjectures: overview

G = (V, E): hypergraph with no isolated vertices

pin merging in planar body frameworks – p. 17/25

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SLIDE 78

Results and conjectures: overview

G = (V, E): hypergraph with no isolated vertices G realizable as inf. rigid planar body-pin framework

pin merging in planar body frameworks – p. 17/25

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SLIDE 79

Results and conjectures: overview

G = (V, E): hypergraph with no isolated vertices G realizable as inf. rigid planar body-pin framework

  • no hyperleaves

and G contains ( 3

2, 3 2)-hypertight

subgraph

pin merging in planar body frameworks – p. 17/25

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SLIDE 80

Results and conjectures: overview

G = (V, E): hypergraph with no isolated vertices G realizable as inf. rigid planar body-pin framework

  • no hyperleaves

and G contains ( 3

2, 3 2)-hypertight

subgraph ⇐ ⇑ G contains

3 2HT-decomposition

pin merging in planar body frameworks – p. 17/25

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SLIDE 81

Results and conjectures: overview

G = (V, E): hypergraph with no isolated vertices G realizable as inf. rigid planar body-pin framework

  • no hyperleaves

and G contains ( 3

2, 3 2)-hypertight

subgraph ⇐ ⇒ ⇑ ⇓ G contains

3 2HT-decomposition

(conjectured)

pin merging in planar body frameworks – p. 17/25

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SLIDE 82

Matrix proof

Proposition: G is 3

2HT ⇒ G is independent.

pin merging in planar body frameworks – p. 18/25

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SLIDE 83

Matrix proof

Proposition: G is 3

2HT ⇒ G is independent.

Proof:

hypergraph G → host graph G2 → M(G2, X, Y)

variables (X, Y) = (Xe, Ye, . . .) for each hyperedge e

pin merging in planar body frameworks – p. 18/25

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SLIDE 84

Matrix proof

Proposition: G is 3

2HT ⇒ G is independent.

Proof:

hypergraph G → host graph G2 → M(G2, X, Y)

3 2HT ⇒ 2G2 = T1 ∪ T2 ∪ T3 (doubled) edges hosting the same hyperedge belong to the same trees

pin merging in planar body frameworks – p. 18/25

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SLIDE 85

Matrix proof

Proposition: G is 3

2HT ⇒ G is independent.

Proof:

hypergraph G → host graph G2 → M(G2, X, Y)

3 2HT ⇒ 2G2 = T1 ∪ T2 ∪ T3

Rearrange rows of M(G2, X, Y): (T2 = T2x ∪ T2y)        I(T1) 0(T1) X(T1) I(T2x) 0(T2x) X(T2x) 0(T2y) I(T2y) Y (T2y) 0(T3) I(T3) Y (T3)       

pin merging in planar body frameworks – p. 18/25

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SLIDE 86

Matrix proof (continued)

Specialization (X, Y): e covered by T1 ⇒ Xe = 0 e not covered by T1 ⇒ Xe = 1 e covered by T3 ⇒ Ye = 0 e not covered by T1 ⇒ Ye = 1

pin merging in planar body frameworks – p. 19/25

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SLIDE 87

Matrix proof (continued)

Specialization (X, Y): e covered by T1 ⇒ Xe = 0 e not covered by T1 ⇒ Xe = 1 e covered by T3 ⇒ Ye = 0 e not covered by T1 ⇒ Ye = 1 ⇒ M(G2, X, Y) becomes: M =      I(T1) 0(F1) 0(F1) I(T2x) 0(T2x) −I(T2x) 0(T2y) I(T2y) −I(T2y) 0(T3) I(T3) 0(T3)     

pin merging in planar body frameworks – p. 19/25

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SLIDE 88

Matrix proof (continued)

Specialization (X, Y): e covered by T1 ⇒ Xe = 0 e not covered by T1 ⇒ Xe = 1 e covered by T3 ⇒ Ye = 0 e not covered by T1 ⇒ Ye = 1 ⇒ M(G2, X, Y) becomes: M =      I(T1) 0(F1) 0(F1) I(T2x) 0(T2x) −I(T2x) 0(T2y) I(T2y) −I(T2y) 0(T3) I(T3) 0(T3)     

Observe: rows M lin. independent.

pin merging in planar body frameworks – p. 19/25

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SLIDE 89

Matrix proof (continued)

Specialization (X, Y): e covered by T1 ⇒ Xe = 0 e not covered by T1 ⇒ Xe = 1 e covered by T3 ⇒ Ye = 0 e not covered by T1 ⇒ Ye = 1 ⇒ M(G2, X, Y) becomes: M =      I(T1) 0(F1) 0(F1) I(T2x) 0(T2x) −I(T2x) 0(T2y) I(T2y) −I(T2y) 0(T3) I(T3) 0(T3)     

Observe: rows M lin. independent. Q.E.D.

pin merging in planar body frameworks – p. 19/25

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SLIDE 90

Generalization to higher dimensions

d: dimension workspace

pin merging in planar body frameworks – p. 20/25

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SLIDE 91

Generalization to higher dimensions

d: dimension workspace

hypergraph G = (V, E): design for body-and-hinge framework

V ↔ rigid bodies in d-space hyperedges: collecting bodies attached by 1 common hinge

pin merging in planar body frameworks – p. 20/25

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SLIDE 92

Generalization to higher dimensions

d: dimension workspace

hypergraph G = (V, E): design for body-and-hinge framework

V ↔ rigid bodies in d-space hyperedges: collecting bodies attached by 1 common hinge

D = d+1

d−1

  • : dimension space of hinges

Theorem: G contains a

D D−1HT-decomposition

⇒ realizable as inf. rigid body-hinge framework in d-space.

pin merging in planar body frameworks – p. 20/25

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SLIDE 93

Generalization to higher dimensions

d: dimension workspace

hypergraph G = (V, E): design for body-and-hinge framework

V ↔ rigid bodies in d-space hyperedges: collecting bodies attached by 1 common hinge

D = d+1

d−1

  • : dimension space of hinges

Theorem: G contains a

D D−1HT-decomposition

⇒ realizable as inf. rigid body-hinge framework in d-space.

Proof: cf. d = 2.

pin merging in planar body frameworks – p. 20/25

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SLIDE 94

Generalization to higher dimensions

d: dimension workspace

hypergraph G = (V, E): design for body-and-hinge framework

V ↔ rigid bodies in d-space hyperedges: collecting bodies attached by 1 common hinge

D = d+1

d−1

  • : dimension space of hinges

Theorem: G contains a

D D−1HT-decomposition

⇒ realizable as inf. rigid body-hinge framework in d-space.

Proof: cf. d = 2.

Theorem: Assume no leaves. G is d-independent iff.

∀∅ = E′ ⊂ E : (D − 1) · w(E′) ≤ D| ∪ E′| − D.

  • Proof. Tay-Tanigawa count for rigidity.

pin merging in planar body frameworks – p. 20/25

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SLIDE 95

Generating 3

2T-decompositions

Given a 3

2T-graph G = (V, E):

pin merging in planar body frameworks – p. 21/25

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SLIDE 96

Generating 3

2T-decompositions

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG with covering trees TRY = FR ∪ FY , TRG = FR ∪ FG and TY G = FY ∪ FG.

7 6 5 3 4 1 2

pin merging in planar body frameworks – p. 21/25

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SLIDE 97

Generating 3

2T-decompositions

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG with covering trees TRY = FR ∪ FY , TRG = FR ∪ FG and TY G = FY ∪ FG.

7 6 5 3 4 1 2

Question: How do we find other decompositions?

pin merging in planar body frameworks – p. 21/25

slide-98
SLIDE 98

Generating 3

2T-decompositions

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG with covering trees TRY = FR ∪ FY , TRG = FR ∪ FG and TY G = FY ∪ FG.

7 6 5 3 4 1 2 7 6 5 3 4 1 2

Question: How do we find other decompositions?

pin merging in planar body frameworks – p. 21/25

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SLIDE 99

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

pin merging in planar body frameworks – p. 22/25

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SLIDE 100

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

pin merging in planar body frameworks – p. 22/25

slide-101
SLIDE 101

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

pin merging in planar body frameworks – p. 22/25

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SLIDE 102

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

Select a red edge e

7 6 5 3 4 1 2

e

pin merging in planar body frameworks – p. 22/25

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SLIDE 103

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

Select a red edge e Consider unique circuit γ in TY G + e

7 6 5 3 4 1 2

e

pin merging in planar body frameworks – p. 22/25

slide-104
SLIDE 104

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

Select a red edge e Consider unique circuit γ in TY G + e Select green edge d in γ such that unique circuit in TRY + d contains e

7 6 5 3 4 1 2

d e

pin merging in planar body frameworks – p. 22/25

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SLIDE 105

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

Select a red edge e Consider unique circuit γ in TY G + e Select green edge d in γ such that unique circuit in TRY + d contains e

7 6 5 3 4 1 2

d e

pin merging in planar body frameworks – p. 22/25

slide-106
SLIDE 106

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

Select a red edge e Consider unique circuit γ in TY G + e Select green edge d in γ such that unique circuit in TRY + d contains e Swap colours of e and d.

7 6 5 3 4 1 2

d e

pin merging in planar body frameworks – p. 22/25

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SLIDE 107

Generating 3

2T-decompositions II

colour swap: an edge for an edge!

(e.g. red edge turns green ⇐ ⇒ green edge turns red)

Algorithm:

Select a red edge e Consider unique circuit γ in TY G + e Select green edge d in γ such that unique circuit in TRY + d contains e Swap colours of e and d.

7 6 5 3 4 1 2

d e

Observe: TY G + e1 − e2 and TRY + e2 − e1 still trees!

pin merging in planar body frameworks – p. 22/25

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SLIDE 108

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG

pin merging in planar body frameworks – p. 23/25

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SLIDE 109

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG e ∈ E → γ(e): unique path in G without colour(e) connecting endpoints e

pin merging in planar body frameworks – p. 23/25

slide-110
SLIDE 110

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG e ∈ E → γ(e): unique path in G without colour(e) connecting endpoints e

  • Theorem. Let e be edge of 3

2T-graph G.

Choose any colour K different from colour(e).

pin merging in planar body frameworks – p. 23/25

slide-111
SLIDE 111

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG e ∈ E → γ(e): unique path in G without colour(e) connecting endpoints e

  • Theorem. Let e be edge of 3

2T-graph G.

Choose any colour K different from colour(e). There always exists an edge d in γ(e) s.t. 1) colour(d) = K 2) e belongs to γ(d)

pin merging in planar body frameworks – p. 23/25

slide-112
SLIDE 112

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG e ∈ E → γ(e): unique path in G without colour(e) connecting endpoints e

  • Theorem. Let e be edge of 3

2T-graph G.

Choose any colour K different from colour(e). There always exists an edge d in γ(e) s.t. 1) colour(d) = K 2) e belongs to γ(d)

  • Proof. (assume colour(e) = R, K = G)

Suppose for every green edge d of γ(e): e ∈ γ(d)

pin merging in planar body frameworks – p. 23/25

slide-113
SLIDE 113

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG e ∈ E → γ(e): unique path in G without colour(e) connecting endpoints e

  • Theorem. Let e be edge of 3

2T-graph G.

Choose any colour K different from colour(e). There always exists an edge d in γ(e) s.t. 1) colour(d) = K 2) e belongs to γ(d)

  • Proof. (assume colour(e) = R, K = G)

Suppose for every green edge d of γ(e): e ∈ γ(d) ⇒ e ∪ γ(e) ∪ γ(d1) ∪ γ(d2) ∪ . . . contains a red-yellow cycle.

pin merging in planar body frameworks – p. 23/25

slide-114
SLIDE 114

Colour swap condition

Given a 3

2T-graph G = (V, E):

E = FR ∪ FY ∪ FG e ∈ E → γ(e): unique path in G without colour(e) connecting endpoints e

  • Theorem. Let e be edge of 3

2T-graph G.

Choose any colour K different from colour(e). There always exists an edge d in γ(e) s.t. 1) colour(d) = K 2) e belongs to γ(d)

  • Proof. (assume colour(e) = R, K = G)

Suppose for every green edge d of γ(e): e ∈ γ(d) ⇒ e ∪ γ(e) ∪ γ(d1) ∪ γ(d2) ∪ . . . contains a red-yellow cycle.

QED

pin merging in planar body frameworks – p. 23/25

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SLIDE 115

Brain tapas

Is every 3

2T-decomposition reachable by colour swaps?

pin merging in planar body frameworks – p. 24/25

slide-116
SLIDE 116

Brain tapas

Is every 3

2T-decomposition reachable by colour swaps?

Is every minimally rigid design for planar body-pin frameworks (allowing multi-pins) obtained from a

3 2HT-decomposition?

pin merging in planar body frameworks – p. 24/25

slide-117
SLIDE 117

Brain tapas

Is every 3

2T-decomposition reachable by colour swaps?

Is every minimally rigid design for planar body-pin frameworks (allowing multi-pins) obtained from a

3 2HT-decomposition?

Equivalently: is ( 3

2, 3 2)-hypertightness sufficient for a 3 2HT?

pin merging in planar body frameworks – p. 24/25

slide-118
SLIDE 118

Brain tapas

Is every 3

2T-decomposition reachable by colour swaps?

Is every minimally rigid design for planar body-pin frameworks (allowing multi-pins) obtained from a

3 2HT-decomposition?

Equivalently: is ( 3

2, 3 2)-hypertightness sufficient for a 3 2HT?

Is every minimally rigid design for spatial body-hinge frameworks (allowing multi-hinges) obtained from a

6 5HT-decomposition?

pin merging in planar body frameworks – p. 24/25

slide-119
SLIDE 119

Brain tapas

Is every 3

2T-decomposition reachable by colour swaps?

Is every minimally rigid design for planar body-pin frameworks (allowing multi-pins) obtained from a

3 2HT-decomposition?

Equivalently: is ( 3

2, 3 2)-hypertightness sufficient for a 3 2HT?

Is every minimally rigid design for spatial body-hinge frameworks (allowing multi-hinges) obtained from a

6 5HT-decomposition?

Is ( 6

5, 6 5)-hypertightness sufficient for a 6 5HT?

pin merging in planar body frameworks – p. 24/25

slide-120
SLIDE 120

Brain tapas

Is every 3

2T-decomposition reachable by colour swaps?

Is every minimally rigid design for planar body-pin frameworks (allowing multi-pins) obtained from a

3 2HT-decomposition?

Equivalently: is ( 3

2, 3 2)-hypertightness sufficient for a 3 2HT?

Is every minimally rigid design for spatial body-hinge frameworks (allowing multi-hinges) obtained from a

6 5HT-decomposition?

Is ( 6

5, 6 5)-hypertightness sufficient for a 6 5HT?

Generalization to spatial body-pin frameworks? (allowing

multi-pins, and body pairs sharing 2 pins)

pin merging in planar body frameworks – p. 24/25

slide-121
SLIDE 121

Any answers?

pin merging in planar body frameworks – p. 25/25