On the Classification of Motions of Laman Graphs Georg Grasegger, - PowerPoint PPT Presentation

On the Classification of Motions of Laman Graphs Georg Grasegger, Jan Legersk´ y, Josef Schicho RISC JKU Linz, Austria Geometric constraint systems: rigidity, flexibility and applications Lancaster, UK, June 12, 2019 A R C A D E S

Dixon (1899), Walter and Husty (2007) 1

Dixon (1899), Walter and Husty (2007) Q 1 1

Proper flexible labelings An edge labeling λ : E → R + of a graph G = ( V , E ) is called flexible if there are infinitely many non-congruent realizations ρ : V → R 2 such that � ρ ( u ) − ρ ( v ) � = λ ( uv ) for all edges uv in E . ( x ¯ u , y ¯ u ) = (0 , 0) ( x ¯ v , y ¯ v ) = ( λ (¯ u ¯ v ) , 0) ( x u − x v ) 2 + ( y u − y v ) 2 = λ ( uv ) 2 , ∀ uv ∈ E An irreducible component of the solution set is called an algebraic motion . 2

Proper flexible labelings An edge labeling λ : E → R + of a graph G = ( V , E ) is called proper flexible if there are infinitely many non-congruent injective realizations ρ : V → R 2 such that � ρ ( u ) − ρ ( v ) � = λ ( uv ) for all edges uv in E . ( x ¯ u , y ¯ u ) = (0 , 0) ( x ¯ v , y ¯ v ) = ( λ (¯ u ¯ v ) , 0) ( x u − x v ) 2 + ( y u − y v ) 2 = λ ( uv ) 2 , ∀ uv ∈ E ( x u − x v ) 2 + ( y u − y v ) 2 � = 0 , ∀ uv / ∈ E . 2

NAC-colorings Definition A coloring of edges δ : E → { blue, red } is called a NAC-coloring , if it is surjective and for every cycle in G , either all edges in the cycle have the same color, or there are at least two blue and two red edges in the cycle. 5 4 5 4 6 3 6 3 1 2 1 2 ω 1 ǫ 56 3

NAC-colorings Definition A coloring of edges δ : E → { blue, red } is called a NAC-coloring , if it is surjective and for every cycle in G , either all edges in the cycle have the same color, or there are at least two blue and two red edges in the cycle. 5 4 5 4 6 3 6 3 1 2 1 2 ω 1 ǫ 56 Theorem (GLS) A connected graph with at least one edge has a flexible labeling if and only if it has a NAC-coloring. 3

Active NAC-colorings uv = ( x v − x u ) 2 + ( y v − y u ) 2 λ 2 = (( x v − x u ) + i ( y v − y u )) (( x v − x u ) − i ( y v − y u )) � �� � � �� � W u , v Z u , v Lemma (GLS) Let λ be a flexible labeling of a graph G. Let C be an algebraic motion of ( G , λ ) . If α ∈ Q and ν is a valuation of the complex function field of C such that there exists edges ¯ u ¯ v , ˆ u ˆ v with ν ( W ¯ v ) = α and ν ( W ˆ v ) > α , then δ : E G → { red , blue } given by u , ¯ u , ˆ δ ( uv ) = red ⇐ ⇒ ν ( W u , v ) > α , δ ( uv ) = blue ⇐ ⇒ ν ( W u , v ) ≤ α . is a NAC-coloring, called active. 4

Active NAC-colorings of quadrilaterals Quadrilateral Motion Active NAC-colorings parallel Rhombus degenerate resp. Parallelogram Antiparallelogram nondegenerate Deltoid degenerate General 5

Three-prism 6

Leading coefficient system Assume a valuation that gives only one active NAC-coloring = ⇒ Laurent series parametrization. For every cycle C = ( u 1 , . . . , u n ): � � ( w u i u i +1 t + h.o.t. ) + ( w u i u i +1 + h.o.t. ) = 0 . � �� � � �� � i ∈{ 1 ,..., n } i ∈{ 1 ,..., n } W ui , ui +1 W ui , ui +1 δ ( u i u i +1 )=red δ ( u i u i +1 )=blue 7

Leading coefficient system Assume a valuation that gives only one active NAC-coloring = ⇒ Laurent series parametrization. For every cycle C = ( u 1 , . . . , u n ): � w u i u i +1 = 0 . i ∈{ 1 ,..., n } δ ( u i u i +1 )=blue 7

Leading coefficient system Assume a valuation that gives only one active NAC-coloring = ⇒ Laurent series parametrization. For every cycle C = ( u 1 , . . . , u n ): � w u i u i +1 = 0 . i ∈{ 1 ,..., n } δ ( u i u i +1 )=blue For all uv ∈ E G : w uv z uv = λ 2 uv . = ⇒ elimination using Gr¨ obner basis provides an equation in λ uv ’s. 7

Singleton NAC-colorings If a valuation yields two active NAC-colorings δ, δ ′ , then the set { ( δ ( e ) , δ ′ ( e )): e ∈ E G } has 3 elements. 8

Triangle in Q 1 3 4 1 7 2 5 6 57 r 2 + λ 2 67 s 2 + � � ⇒ λ 2 λ 2 56 − λ 2 57 − λ 2 = rs = 0 , 67 r = λ 2 24 − λ 2 23 , s = λ 2 14 − λ 2 13 9

Triangle in Q 1 3 4 1 7 2 5 6 57 r 2 + λ 2 67 s 2 + � � ⇒ λ 2 λ 2 56 − λ 2 57 − λ 2 = rs = 0 , 67 r = λ 2 24 − λ 2 23 , s = λ 2 14 − λ 2 13 Considering the equation as a polynomial in r , the discriminant is ( λ 56 + λ 57 + λ 67 )( λ 56 + λ 57 − λ 67 )( λ 56 − λ 57 + λ 67 )( λ 56 − λ 57 − λ 67 ) s 2 . 9

Triangle in Q 1 3 4 1 7 2 5 6 57 r 2 + λ 2 67 s 2 + � � ⇒ λ 2 λ 2 56 − λ 2 57 − λ 2 = rs = 0 , 67 r = λ 2 24 − λ 2 23 , s = λ 2 14 − λ 2 13 Considering the equation as a polynomial in r , the discriminant is ( λ 56 + λ 57 + λ 67 )( λ 56 + λ 57 − λ 67 )( λ 56 − λ 57 + λ 67 )( λ 56 − λ 57 − λ 67 ) s 2 . Theorem (GLS) The vertices 5, 6 and 7 are collinear for every proper flexible labeling of Q 1 . 9

Orthogonal diagonals Lemma (GLS) If there is an active NAC-coloring δ of an algebraic motion of ( G , λ ) such that a 4-cycle (1 , 2 , 3 , 4) is blue and there are red paths from 1 to 3 and from 2 to 4 , then λ 2 12 + λ 2 34 = λ 2 23 + λ 2 14 , namely, the 4-cycle (1 , 2 , 3 , 4) has orthogonal diagonals. 10

Ramification formula Theorem (GLS) Let C be an algebraic motion of ( G , λ ) with the set of active NAC-colorings N. There exists µ δ ∈ Z ≥ 0 for all NAC-colorings δ of G such that: 1. µ δ � = 0 if and only if δ ∈ N, and 2. for every 4-cycle ( V i , E i ) of G, there exists a positive integer d i such that � for all δ ′ ∈ { δ | E i : δ ∈ N } . µ δ = d i δ ∈ NAC G δ | Ei = δ ′ 11

Ramification formula Theorem (GLS) Let C be an algebraic motion of ( G , λ ) with the set of active NAC-colorings N. There exists µ δ ∈ Z ≥ 0 for all NAC-colorings δ of G such that: 1. µ δ � = 0 if and only if δ ∈ N, and 2. for every 4-cycle ( V i , E i ) of G, there exists a positive integer d i such that � for all δ ′ ∈ { δ | E i : δ ∈ N } . µ δ = d i δ ∈ NAC G δ | Ei = δ ′ � � � � � � p = , o = , , g = , , , � � � � a = , , e = , . 11

Example ǫ 13 ǫ 14 ǫ 23 ǫ 24 γ 1 γ 2 η ψ 1 ψ 2 φ 3 φ 4 ζ 12

Example ǫ 13 ǫ 14 ǫ 23 ǫ 24 γ 1 γ 2 η ψ 1 ψ 2 φ 3 φ 4 ζ � � Antiparallelogram , = ⇒ 12

Example ǫ 13 ǫ 14 ǫ 23 ǫ 24 γ 1 γ 2 η ψ 1 ψ 2 φ 3 φ 4 ζ � � Antiparallelogram , = ⇒ µ ǫ 13 = µ γ 1 = µ η = 0 12

Example ǫ 13 ǫ 14 ǫ 23 ǫ 24 γ 1 γ 2 η ψ 1 ψ 2 φ 3 φ 4 ζ � � Antiparallelogram , = ⇒ µ ǫ 13 = µ γ 1 = µ η = 0 µ ǫ 14 + µ ψ 1 12

Example ǫ 13 ǫ 14 ǫ 23 ǫ 24 γ 1 γ 2 η ψ 1 ψ 2 φ 3 φ 4 ζ � � Antiparallelogram , = ⇒ µ ǫ 13 = µ γ 1 = µ η = 0 µ ǫ 14 + µ ψ 1 = µ ǫ 23 + µ γ 2 + µ φ 3 + µ ζ 12

Classification of motions • Find all possible types of motions of quadrilaterals with consistent µ δ ’s 13

Classification of motions • Find all possible types of motions of quadrilaterals with consistent µ δ ’s • Remove combinations with coinciding vertices (due to edge lengths, perpendicular diagonals) 13

Classification of motions • Find all possible types of motions of quadrilaterals with consistent µ δ ’s • Remove combinations with coinciding vertices (due to edge lengths, perpendicular diagonals) • Identify symmetric cases 13

Classification of motions • Find all possible types of motions of quadrilaterals with consistent µ δ ’s • Remove combinations with coinciding vertices (due to edge lengths, perpendicular diagonals) • Identify symmetric cases • Compute necessary conditions for λ uv ’s using leading coefficient systems 13

Classification of motions • Find all possible types of motions of quadrilaterals with consistent µ δ ’s • Remove combinations with coinciding vertices (due to edge lengths, perpendicular diagonals) • Identify symmetric cases • Compute necessary conditions for λ uv ’s using leading coefficient systems • Check if there is a proper flexible labeling satisfying the necessary conditions 13

Classification of motions • Find all possible types of motions of quadrilaterals with consistent µ δ ’s • Remove combinations with coinciding vertices (due to edge lengths, perpendicular diagonals) • Identify symmetric cases • Compute necessary conditions for λ uv ’s using leading coefficient systems • Check if there is a proper flexible labeling satisfying the necessary conditions Implementation – SageMath package FlexRiLoG ( https://github.com/Legersky/flexrilog ) 13

2 2 2 1 3 5 1 3 5 1 3 5 4 4 4 6 6 6 4-cycles active NAC-colorings # NAC K 3 , 3 1 ggggggggg { ǫ 12 , ǫ 23 , ǫ 34 , ǫ 14 , ǫ 16 , ǫ 36 , ω 1 , ω 3 } 6 Dixon I ooogggggg { ǫ 12 , ǫ 23 , ǫ 34 , ǫ 14 } 9 pooggogge { ǫ 12 , ǫ 34 , ω 5 , ω 6 } 18 Dixon II pgggaggag 1 4 2 6 5 3 14