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

Intermezzo: weights of hyperedges G = ( { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , { e 1 , e 2 , e 3 , e 4 } ) hyperedges: 1 2 = { 2 , 7 } e 1 4 3 = { 2 , 4 } e 2 5 = { 3 , 4 , 5 } e 3 6 7 = { 1 , 3 , 6 } e 4 pin merging in planar body frameworks – p. 9/25

Intermezzo: weights of hyperedges G = ( { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , { e 1 , e 2 , e 3 , e 4 } ) hyperedges: 1 2 = { 2 , 7 } e 1 4 3 = { 2 , 4 } e 2 5 = { 3 , 4 , 5 } e 3 6 7 = { 1 , 3 , 6 } e 4 Definition: weight hyperedge: w ( e ) = | e | − 1 pin merging in planar body frameworks – p. 9/25

Intermezzo: weights of hyperedges G = ( { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , { e 1 , e 2 , e 3 , e 4 } ) hyperedges: 1 2 = { 2 , 7 } e 1 4 3 = { 2 , 4 } e 2 5 = { 3 , 4 , 5 } e 3 6 7 = { 1 , 3 , 6 } e 4 Definition: weight hyperedge: w ( e ) = | e | − 1 w ( e 1 ) = w ( e 2 ) = 1 , w ( e 3 ) = w ( e 4 ) = 2 pin merging in planar body frameworks – p. 9/25

Intermezzo: weights of hyperedges G = ( { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , { e 1 , e 2 , e 3 , e 4 } ) hyperedges: 1 2 = { 2 , 7 } e 1 4 3 = { 2 , 4 } e 2 5 = { 3 , 4 , 5 } e 3 6 7 = { 1 , 3 , 6 } e 4 Definition: weight hyperedge: w ( e ) = | e | − 1 w ( e 1 ) = w ( e 2 ) = 1 , w ( e 3 ) = w ( e 4 ) = 2 Application: Hypertree: pin merging in planar body frameworks – p. 9/25

Intermezzo: weights of hyperedges G = ( { 1 , 2 , 3 , 4 , 5 , 6 , 7 } , { e 1 , e 2 , e 3 , e 4 } ) hyperedges: 1 2 = { 2 , 7 } e 1 4 3 = { 2 , 4 } e 2 5 = { 3 , 4 , 5 } e 3 6 7 = { 1 , 3 , 6 } e 4 Definition: weight hyperedge: w ( e ) = | e | − 1 w ( e 1 ) = w ( e 2 ) = 1 , w ( e 3 ) = w ( e 4 ) = 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

2 HT -Hypergraphs 3 1 2 4 3 5 6 7 pin merging in planar body frameworks – p. 10/25

2 HT -Hypergraphs 3 1 2 4 3 5 6 7 3 2 HT -decomposition: 3 colours for hyperedges pin merging in planar body frameworks – p. 10/25

2 HT -Hypergraphs 3 1 2 4 3 5 6 7 3 2 HT -decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree pin merging in planar body frameworks – p. 10/25

2 HT -Hypergraphs 3 spanning hypertree: 1 2 4 3 5 6 7 3 2 HT -decomposition: 3 colours for hyperedges union 2 colours → spanning hypertree pin merging in planar body frameworks – p. 10/25

2 HT -Hypergraphs 3 spanning hypertree: 1 2 4 3 1 2 5 6 7 4 3 3 2 HT -decomposition: 5 6 3 colours for hyperedges 7 union 2 colours → spanning hypertree pin merging in planar body frameworks – p. 10/25

2 HT -Hypergraphs 3 spanning hypertree: 1 2 4 3 1 2 5 6 7 4 3 3 2 HT -decomposition: 5 6 3 colours for hyperedges 7 union 2 colours → spanning hypertree Consequence: 3 2 HT -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

2 HT -Hypergraphs 3 spanning hypertree: 1 2 4 3 1 2 5 6 7 4 3 3 2 HT -decomposition: 5 6 3 colours for hyperedges 7 union 2 colours → spanning hypertree Consequence: 3 2 HT -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

2 HT versus 3 3 2 T hypergraph G = ( V, E ) graph G 2 = ( V, E 2 ) → hosting 1 1 2 2 4 3 4 3 ← 5 5 clustering 6 6 7 7 pin merging in planar body frameworks – p. 11/25

2 HT versus 3 3 2 T hypergraph G = ( V, E ) graph G 2 = ( V, E 2 ) → hosting 1 1 2 2 4 3 4 3 ← 5 5 clustering 6 6 7 7 number of edges | E 2 | total weight w ( E ) pin merging in planar body frameworks – p. 11/25

2 HT versus 3 3 2 T hypergraph G = ( V, E ) graph G 2 = ( V, E 2 ) → coloured 1 2 hosting 1 2 ← 4 3 4 3 5 monochromatic 5 6 7 clustering 6 7 number of edges | E 2 | total weight w ( E ) → always 3 3 2 HT -decomposition 2 T -decomposition ← sometimes pin merging in planar body frameworks – p. 11/25

2 HT -obstructions 3 1 2 (3/2, 3/2)-hypertight 3 4 5 6 7 pin merging in planar body frameworks – p. 12/25

2 HT -obstructions 3 1 2 (3/2, 3/2)-hypertight 3 4 yet not 3 2 HT -decomposable 5 6 7 pin merging in planar body frameworks – p. 12/25

2 HT -obstructions 3 1 2 (3/2, 3/2)-hypertight 3 4 yet not 3 2 HT -decomposable 5 6 7 Property: 3 2 HT ⇒ no (hyper)leaves pin merging in planar body frameworks – p. 12/25

2 HT -obstructions 3 1 2 (3/2, 3/2)-hypertight 3 4 yet not 3 2 HT -decomposable 5 6 7 Property: 3 2 HT ⇒ no (hyper)leaves 3 Conjecture: leaf-free + (3/2,3/2)-hypertight ⇐ ⇒ 2 HT pin merging in planar body frameworks – p. 12/25

2 HT -obstructions 3 1 2 (3/2, 3/2)-hypertight 3 4 yet not 3 2 HT -decomposable 5 6 7 Property: 3 2 HT ⇒ no (hyper)leaves 3 Conjecture: leaf-free + (3/2,3/2)-hypertight ⇐ ⇒ 2 HT D Lucky guess: leaf-free + (D/(D-1),D/(D-1))-hypertight ⇐ ⇒ D − 1 HT pin merging in planar body frameworks – p. 12/25

2 HT -Hypergraphs as rigid frameworks 3 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

2 HT -Hypergraphs as rigid frameworks 3 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

2 HT -Hypergraphs as rigid frameworks 3 Definition. Planar framework realization of hypergraph G = body-and-pin realization where hyperedges represent merged pins. Theorem: G contains 3 2 HT ⇒ realizable as inf. rigid planar body-and-pin framework. pin merging in planar body frameworks – p. 13/25

2 HT -Hypergraphs as rigid frameworks 3 Definition. Planar framework realization of hypergraph G = body-and-pin realization where hyperedges represent merged pins. Theorem: G contains 3 2 HT ⇒ 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

2 HT -Hypergraphs as rigid frameworks 3 Definition. Planar framework realization of hypergraph G = body-and-pin realization where hyperedges represent merged pins. Theorem: G contains 3 2 HT ⇒ 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

The rigidity matrix I Realization of hypergraph G = ( V, E ) as body and pin framework in the plane: F = ( G, P ) R 2 : e �→ P ( e ) = ( x e , y e ) P : E → I pin merging in planar body frameworks – p. 14/25

The rigidity matrix I Realization of hypergraph G = ( V, E ) as body and pin framework in the plane: F = ( G, P ) R 2 : e �→ P ( e ) = ( x e , y e ) P : E → I Choose host graph G 2 = ( V, E 2 ) . pin merging in planar body frameworks – p. 14/25

The rigidity matrix I Realization of hypergraph G = ( V, E ) as body and pin framework in the plane: F = ( G, P ) R 2 : e �→ P ( e ) = ( x e , y e ) P : E → I Choose host graph G 2 = ( V, E 2 ) . For each edge ij ∈ E 2 with { i, j } ⊂ e ∈ E : = (0 , . . . , 0 , 1 , 0 . . . , 0 , − 1 , 0 , . . . , 0 | 0 , . . . , 0 | 0 , . . . , 0 , − x e , 0 , . . . , 0 , x e , 0 , . . . , 0) JX ij (0 , . . . , 0 | 0 , . . . , 0 , 1 , 0 . . . , 0 , − 1 , 0 , . . . , 0 | 0 , . . . , 0 , − y e , 0 , . . . , 0 , y e , 0 , . . . , 0) = JY ij (non-zero entries in positions i and j in subsequences of length | V | ) pin merging in planar body frameworks – p. 14/25

The rigidity matrix I Realization of hypergraph G = ( V, E ) as body and pin framework in the plane: F = ( G, P ) R 2 : e �→ P ( e ) = ( x e , y e ) P : E → I Choose host graph G 2 = ( V, E 2 ) . For each edge ij ∈ E 2 with { i, j } ⊂ e ∈ E : = (0 , . . . , 0 , 1 , 0 . . . , 0 , − 1 , 0 , . . . , 0 | 0 , . . . , 0 | 0 , . . . , 0 , − x e , 0 , . . . , 0 , x e , 0 , . . . , 0) JX ij (0 , . . . , 0 | 0 , . . . , 0 , 1 , 0 . . . , 0 , − 1 , 0 , . . . , 0 | 0 , . . . , 0 , − y e , 0 , . . . , 0 , y e , 0 , . . . , 0) = JY ij (non-zero entries in positions i and j in subsequences of length | V | ) ⇒ 2 w ( E ) × | V | matrix M ( G 2 , P ) . pin merging in planar body frameworks – p. 14/25

The rigidity matrix II Given hypergraph G = ( V, E ) with realization F = ( G, P ) pin merging in planar body frameworks – p. 15/25

The rigidity matrix II Given hypergraph G = ( V, E ) with realization F = ( G, P ) Let C i = ( a i , b i , c i ) with i = 1 , . . . , v = | V | R 3 v and put γ = ( a 1 , . . . , a v , b 1 , . . . , b v , c 1 , . . . , c v ) ∈ I pin merging in planar body frameworks – p. 15/25

The rigidity matrix II Given hypergraph G = ( V, E ) with realization F = ( G, P ) Let C i = ( a i , b i , c i ) with i = 1 , . . . , v = | V | R 3 v and put γ = ( a 1 , . . . , a v , b 1 , . . . , b v , c 1 , . . . , c v ) ∈ I Property: The C i are centers of motion for bodies of F ⇐ ⇒ M ( G 2 , P ) · γ T = 0 for any host G 2 ⇒ M ( G 2 , P ) · γ T = 0 for every host G 2 ⇐ pin merging in planar body frameworks – p. 15/25

The rigidity matrix II Given hypergraph G = ( V, E ) with realization F = ( G, P ) Let C i = ( a i , b i , c i ) with i = 1 , . . . , v = | V | R 3 v and put γ = ( a 1 , . . . , a v , b 1 , . . . , b v , c 1 , . . . , c v ) ∈ I Property: The C i are centers of motion for bodies of F ⇐ ⇒ M ( G 2 , P ) · γ T = 0 for any host G 2 ⇒ M ( G 2 , P ) · γ T = 0 for every host G 2 ⇐ Remarks: rank M ( G 2 , P ) independent from host F inf. rigid ⇐ ⇒ rank M ( G 2 , P ) = 3 | V | − 3 F isostatic ⇐ ⇒ M ( G 2 , P ) has independent rows and 2 · w ( E ) = 3 | V | − 3 pin merging in planar body frameworks – p. 15/25

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

Independent hypergraphs Definition: Hypergraph G = ( V, E ) is 2-independent iff. there is a realization P such that for some (hence for every) host G 2 the rows of M ( G 2 , P ) are linearly independent. pin merging in planar body frameworks – p. 16/25

Independent hypergraphs Definition: Hypergraph G = ( V, E ) is 2-independent iff. there is a realization P such that for some (hence for every) host G 2 the rows of M ( G 2 , 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

Independent hypergraphs Definition: Hypergraph G = ( V, E ) is 2-independent iff. there is a realization P such that for some (hence for every) host G 2 the rows of M ( G 2 , 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

Independent hypergraphs Definition: Hypergraph G = ( V, E ) is 2-independent iff. there is a realization P such that for some (hence for every) host G 2 the rows of M ( G 2 , 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

Results and conjectures: overview G = ( V, E ) : hypergraph with no isolated vertices pin merging in planar body frameworks – p. 17/25

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

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

Results and conjectures: overview G = ( V, E ) : hypergraph with no isolated vertices G realizable as inf. rigid planar body-pin framework � ⇑ no hyperleaves G contains 3 2 HT-decomposition and G contains ( 3 2 , 3 2 ) -hypertight ⇐ subgraph pin merging in planar body frameworks – p. 17/25

Results and conjectures: overview G = ( V, E ) : hypergraph with no isolated vertices G realizable as inf. rigid planar body-pin framework � ⇑ ⇓ no hyperleaves G contains 3 2 HT-decomposition and G contains ( 3 2 , 3 2 ) -hypertight ⇒ ⇐ subgraph (conjectured) pin merging in planar body frameworks – p. 17/25

Matrix proof Proposition: G is 3 2 HT ⇒ G is independent. pin merging in planar body frameworks – p. 18/25

Matrix proof Proposition: G is 3 2 HT ⇒ G is independent. Proof: hypergraph G → host graph G 2 → M ( G 2 , X , Y ) variables ( X , Y ) = ( X e , Y e , . . . ) for each hyperedge e pin merging in planar body frameworks – p. 18/25

Matrix proof Proposition: G is 3 2 HT ⇒ G is independent. Proof: hypergraph G → host graph G 2 → M ( G 2 , X , Y ) 3 2 HT ⇒ 2 G 2 = T 1 ∪ T 2 ∪ T 3 (doubled) edges hosting the same hyperedge belong to the same trees pin merging in planar body frameworks – p. 18/25

Matrix proof Proposition: G is 3 2 HT ⇒ G is independent. Proof: hypergraph G → host graph G 2 → M ( G 2 , X , Y ) 3 2 HT ⇒ 2 G 2 = T 1 ∪ T 2 ∪ T 3 Rearrange rows of M ( G 2 , X , Y ) : ( T 2 = T 2 x ∪ T 2 y )   I ( T 1 ) 0( T 1 ) X ( T 1 )   I ( T 2 x ) 0( T 2 x ) X ( T 2 x )       0( T 2 y ) I ( T 2 y ) Y ( T 2 y )     0( T 3 ) I ( T 3 ) Y ( T 3 ) pin merging in planar body frameworks – p. 18/25

Matrix proof (continued) Specialization ( X , Y ) : e covered by T 1 ⇒ X e = 0 e not covered by T 1 ⇒ X e = 1 e covered by T 3 ⇒ Y e = 0 e not covered by T 1 ⇒ Y e = 1 pin merging in planar body frameworks – p. 19/25

Matrix proof (continued) Specialization ( X , Y ) : e covered by T 1 ⇒ X e = 0 e not covered by T 1 ⇒ X e = 1 e covered by T 3 ⇒ Y e = 0 e not covered by T 1 ⇒ Y e = 1 ⇒ M ( G 2 , X , Y ) becomes:   I ( T 1 ) 0( F 1 ) 0( F 1 ) − I ( T 2 x ) I ( T 2 x ) 0( T 2 x )   M =   − I ( T 2 y )  0( T 2 y ) I ( T 2 y )    0( T 3 ) I ( T 3 ) 0( T 3 ) pin merging in planar body frameworks – p. 19/25

Matrix proof (continued) Specialization ( X , Y ) : e covered by T 1 ⇒ X e = 0 e not covered by T 1 ⇒ X e = 1 e covered by T 3 ⇒ Y e = 0 e not covered by T 1 ⇒ Y e = 1 ⇒ M ( G 2 , X , Y ) becomes:   I ( T 1 ) 0( F 1 ) 0( F 1 ) − I ( T 2 x ) I ( T 2 x ) 0( T 2 x )   M =   − I ( T 2 y )  0( T 2 y ) I ( T 2 y )    0( T 3 ) I ( T 3 ) 0( T 3 ) Observe: rows M lin. independent. pin merging in planar body frameworks – p. 19/25

Matrix proof (continued) Specialization ( X , Y ) : e covered by T 1 ⇒ X e = 0 e not covered by T 1 ⇒ X e = 1 e covered by T 3 ⇒ Y e = 0 e not covered by T 1 ⇒ Y e = 1 ⇒ M ( G 2 , X , Y ) becomes:   I ( T 1 ) 0( F 1 ) 0( F 1 ) − I ( T 2 x ) I ( T 2 x ) 0( T 2 x )   M =   − I ( T 2 y )  0( T 2 y ) I ( T 2 y )    0( T 3 ) I ( T 3 ) 0( T 3 ) Observe: rows M lin. independent. Q.E.D. pin merging in planar body frameworks – p. 19/25

Generalization to higher dimensions d : dimension workspace pin merging in planar body frameworks – p. 20/25

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

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 +1 � D = : dimension space of hinges d − 1 D Theorem: G contains a D − 1 HT-decomposition ⇒ realizable as inf. rigid body-hinge framework in d -space. pin merging in planar body frameworks – p. 20/25

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 +1 � D = : dimension space of hinges d − 1 D Theorem: G contains a D − 1 HT-decomposition ⇒ realizable as inf. rigid body-hinge framework in d -space. Proof: cf. d = 2 . pin merging in planar body frameworks – p. 20/25

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 +1 � D = : dimension space of hinges d − 1 D Theorem: G contains a D − 1 HT-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

Generating 3 2 T-decompositions Given a 3 2 T-graph G = ( V, E ) : pin merging in planar body frameworks – p. 21/25

Generating 3 2 T-decompositions Given a 3 2 T-graph G = ( V, E ) : E = F R ∪ F Y ∪ F G with covering trees T RY = F R ∪ F Y , T RG = F R ∪ F G and T Y G = F Y ∪ F G . 1 2 4 3 5 6 7 pin merging in planar body frameworks – p. 21/25

Generating 3 2 T-decompositions Given a 3 2 T-graph G = ( V, E ) : E = F R ∪ F Y ∪ F G with covering trees T RY = F R ∪ F Y , T RG = F R ∪ F G and T Y G = F Y ∪ F G . 1 2 4 3 5 6 7 Question: How do we find other decompositions? pin merging in planar body frameworks – p. 21/25

Generating 3 2 T-decompositions Given a 3 2 T-graph G = ( V, E ) : E = F R ∪ F Y ∪ F G with covering trees T RY = F R ∪ F Y , T RG = F R ∪ F G and T Y G = F Y ∪ F G . 1 1 2 2 4 4 3 3 5 5 6 6 7 7 Question: How do we find other decompositions? pin merging in planar body frameworks – p. 21/25

Generating 3 2 T-decompositions II colour swap: an edge for an edge! pin merging in planar body frameworks – p. 22/25

Generating 3 2 T-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