Group pairings with equivalent rigidity properties for bar-joint and point-hyperplane frameworks Bernd Schulze (with Katie Clinch, Anthony Nixon and Walter Whiteley) Lancaster University June 13, 2019 Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 1 / 23
Outline Rigidity of Euclidean, spherical and point-hyperplane frameworks 1 Symmetric frameworks 2 Pairing symmetry groups in S d and R d 3 Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 2 / 23
Rigidity of Euclidean frameworks Framework in a space M : ( G , p ) , where G = ( V , E ) is a finite simple graph and p : V → M is a map. Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 3 / 23
Rigidity of Euclidean frameworks Framework in a space M : ( G , p ) , where G = ( V , E ) is a finite simple graph and p : V → M is a map. We first consider Euclidean frameworks ( G , p ) in R d . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 3 / 23
Rigidity of Euclidean frameworks Framework in a space M : ( G , p ) , where G = ( V , E ) is a finite simple graph and p : V → M is a map. We first consider Euclidean frameworks ( G , p ) in R d . Given ( G , p ) , is every framework ( G , q ) in an open neighborhood of p satisfying the same length constraints for the edges: � p i − p j � = const ( ij ∈ E ) , congruent to ( G , p ) ? Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 3 / 23
Rigidity of Euclidean frameworks Framework in a space M : ( G , p ) , where G = ( V , E ) is a finite simple graph and p : V → M is a map. We first consider Euclidean frameworks ( G , p ) in R d . Given ( G , p ) , is every framework ( G , q ) in an open neighborhood of p satisfying the same length constraints for the edges: � p i − p j � = const ( ij ∈ E ) , congruent to ( G , p ) ? If so, then ( G , p ) is called (locally) rigid. Otherwise ( G , p ) is called (locally) flexible. Figure: A flexible and a rigid framework in R 2 . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 3 / 23
Infinitesimal rigidity of Euclidean frameworks It is common to analyse rigidity by taking the derivative of the square of each length constraint, which leads to the linear system: � p i − p j , ˙ p i − ˙ p j � = 0 ( ij ∈ E ) . We then check the dimension of the solution space with variables ˙ p . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 4 / 23
Infinitesimal rigidity of Euclidean frameworks It is common to analyse rigidity by taking the derivative of the square of each length constraint, which leads to the linear system: � p i − p j , ˙ p i − ˙ p j � = 0 ( ij ∈ E ) . We then check the dimension of the solution space with variables ˙ p . p : V → R d is an infinitesimal motion of ( G , p ) if ˙ ˙ p satisfies the equations above. Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 4 / 23
Infinitesimal rigidity of Euclidean frameworks It is common to analyse rigidity by taking the derivative of the square of each length constraint, which leads to the linear system: � p i − p j , ˙ p i − ˙ p j � = 0 ( ij ∈ E ) . We then check the dimension of the solution space with variables ˙ p . p : V → R d is an infinitesimal motion of ( G , p ) if ˙ ˙ p satisfies the equations above. ( G , p ) is called infinitesimally rigid if the dimension of the space of � d + 1 � infinitesimal motions of ( G , p ) is equal to (assuming that the points 2 p ( V ) affinely span R d ). Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 4 / 23
Infinitesimal rigidity of Euclidean frameworks It is common to analyse rigidity by taking the derivative of the square of each length constraint, which leads to the linear system: � p i − p j , ˙ p i − ˙ p j � = 0 ( ij ∈ E ) . We then check the dimension of the solution space with variables ˙ p . p : V → R d is an infinitesimal motion of ( G , p ) if ˙ ˙ p satisfies the equations above. ( G , p ) is called infinitesimally rigid if the dimension of the space of � d + 1 � infinitesimal motions of ( G , p ) is equal to (assuming that the points 2 p ( V ) affinely span R d ). The matrix R ( G , p ) corresponding to this linear system above is the rigidity matrix. Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 4 / 23
Infinitesimal rigidity of Euclidean frameworks It is common to analyse rigidity by taking the derivative of the square of each length constraint, which leads to the linear system: � p i − p j , ˙ p i − ˙ p j � = 0 ( ij ∈ E ) . We then check the dimension of the solution space with variables ˙ p . p : V → R d is an infinitesimal motion of ( G , p ) if ˙ ˙ p satisfies the equations above. ( G , p ) is called infinitesimally rigid if the dimension of the space of � d + 1 � infinitesimal motions of ( G , p ) is equal to (assuming that the points 2 p ( V ) affinely span R d ). The matrix R ( G , p ) corresponding to this linear system above is the rigidity matrix. For regular configurations p (i.e., R ( G , p ) has maximum rank), rigidity is equivalent to infinitesimal rigidity. Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 4 / 23
Infinitesimal rigidity of spherical frameworks Next, we consider spherical frameworks ( G , p ) in S d . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 5 / 23
Infinitesimal rigidity of spherical frameworks Next, we consider spherical frameworks ( G , p ) in S d . In S d the ‘distance’ between two points is determined by their inner product. Hence we are interested in the solutions of: � p i , p j � = const ( ij ∈ E ) . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 5 / 23
Infinitesimal rigidity of spherical frameworks Next, we consider spherical frameworks ( G , p ) in S d . In S d the ‘distance’ between two points is determined by their inner product. Hence we are interested in the solutions of: � p i , p j � = const ( ij ∈ E ) . Since p i is constrained to be on S d , we also have � p i , p i � = 1 ( i ∈ V ) . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 5 / 23
Infinitesimal rigidity of spherical frameworks Next, we consider spherical frameworks ( G , p ) in S d . In S d the ‘distance’ between two points is determined by their inner product. Hence we are interested in the solutions of: � p i , p j � = const ( ij ∈ E ) . Since p i is constrained to be on S d , we also have � p i , p i � = 1 ( i ∈ V ) . Again, taking derivatives, we obtain the following system of first-order inner product constraints: � p i , ˙ p j � + � p j , ˙ p i � = 0 ( ij ∈ E ) � p i , ˙ p i � = 0 ( i ∈ V ) . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 5 / 23
Infinitesimal rigidity of spherical frameworks Next, we consider spherical frameworks ( G , p ) in S d . In S d the ‘distance’ between two points is determined by their inner product. Hence we are interested in the solutions of: � p i , p j � = const ( ij ∈ E ) . Since p i is constrained to be on S d , we also have � p i , p i � = 1 ( i ∈ V ) . Again, taking derivatives, we obtain the following system of first-order inner product constraints: � p i , ˙ p j � + � p j , ˙ p i � = 0 ( ij ∈ E ) � p i , ˙ p i � = 0 ( i ∈ V ) . p : V → R d + 1 is called an infinitesimal motion of ( G , p ) if it satisfies this ˙ system of linear constraints, and ( G , p ) is infinitesimally rigid if the � d + 1 � dimension of its space of infinitesimal motions is equal to 2 (assuming the points p ( V ) linearly span R d + 1 ). Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 5 / 23
> 0 and A d (or R d ) Transfer between S d Figure: The transfer of infinitesimal motions between S d > 0 and A d Theorem (S. and Whiteley, 2012) : A bar-joint framework ( G , p ) is infinitesimally rigid in A d if and only if ( G , φ ◦ p ) is infinitesimally rigid in S d > 0 , where φ is the central projection from the origin O . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 6 / 23
> 0 and A d (or R d ) Transfer between S d Figure: The transfer of infinitesimal motions between S d > 0 and A d Theorem (S. and Whiteley, 2012) : A bar-joint framework ( G , p ) is infinitesimally rigid in A d if and only if ( G , φ ◦ p ) is infinitesimally rigid in S d > 0 , where φ is the central projection from the origin O . Moreover, infinitesimal rigidity properties of ( G , p ) in S d remain unchanged if any subset of the joints are inverted in O . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 6 / 23
Point-line frameworks Jackson and Owen introduced the notion of a point-line framework in R 2 . Such a framework consists of points and lines in the plane which are linked by point-point distance constraints, point-line distance constraints, and line-line angle constraints. Figure: Point-line framework in R 2 . Bernd Schulze Group pairings with equivalent rigidity properties June 13, 2019 7 / 23
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