Epsilon local rigidity and numerical algebraic geometry Andrew - PowerPoint PPT Presentation

Epsilon local rigidity and numerical algebraic geometry Andrew Frohmader 1 Alexander Heaton 2 1 University of Wisconsin Milwaukee 2 Technische Universit¨ at Berlin and Max Planck Institute for Mathematics in the Sciences April 2, 2020 Frohmader, Heaton Epsilon local rigidity

Connected graph ( V, E ) with n nodes and m edges. Embedded in R d by the map p : [ n ] → R d We say p = ( p ik ) ∈ R nd . A deformation of p is a continuous map p ( t ) : [0 , 1] → R nd . A rigid motion is a deformation preserving � d � � ( p ik ( t ) − p jk ( t )) 2 ij ∈ ( [ n ] k =1 2 ) � d +1 � . Euclidean group of rigid motions has dimension 2 Frohmader, Heaton Epsilon local rigidity

� [ n ] � , | E | = m . Consider p : [ n ] → R d , and E ⊂ 2 [6] = { 1 , 2 , 3 , 4 , 5 , 6 } E = { 12 , 13 , 14 , 15 , 23 , 25 , 26 , 34 , 36 , 45 , 46 , 56 }     1 0 0 p 11 p 12 p 13 √   − 1 3     0 p 21 p 22 p 23     2 2 √     − 1 3 p 31 p 32 p 33 − 0     2 2 p = = .     √  p 41 p 42 p 43   3 − 1  − 3     2 2 √     p 51 p 52 p 53 3 − 1  3  2 2 p 61 p 62 p 63 0 1 3 We say p ∈ R nd . Frohmader, Heaton Epsilon local rigidity

g : C nd → C m p ∈ R nd � [ n ] � = ⇒ g ( x ) = 0 E ⊂ 2 | E | = m V ( g ) , V R ( g ) � � x ∈ C nd : g ( x ) = 0 V ( g ) := V R ( g ) := V ( g ) ∩ R nd . Frohmader, Heaton Epsilon local rigidity

g : C nd → C m p ∈ R nd � [ n ] � = ⇒ g ( x ) = 0 E ⊂ 2 V ( g ) , V R ( g ) | E | = m   ( x 11 − x 21 ) 2 + ( x 12 − x 22 ) 2 + ( x 13 − x 23 ) 2 − 3 ( x 11 − x 31 ) 2 + ( x 12 − x 32 ) 2 + ( x 13 − x 33 ) 2 − 3 � √ 3 + 2 � 2     ( x 11 − x 41 ) 2 + ( x 12 − x 42 ) 2 + ( x 13 − x 43 ) 2 − 1   − 37 0 � √ 3 − 2 � 2  4 4  0  ( x 11 − x 51 ) 2 + ( x 12 − x 52 ) 2 + ( x 13 − x 53 ) 2 − 1    − 37 0  4 4    ( x 21 − x 31 ) 2 + ( x 22 − x 32 ) 2 + ( x 23 − x 33 ) 2 − 3 0     � √ 3 + 1 � 2    0  ( x 21 − x 51 ) 2 + ( x 22 − x 52 ) 2 + ( x 23 − x 53 ) 2 − 1 − 9     � √ 3 − 2 � 2 0 g ( x ) = = . 2     0 ( x 21 − x 61 ) 2 + ( x 22 − x 62 ) 2 + ( x 23 − x 63 ) 2 − 1 − 37     � √ 3 − 1 � 2 4 4 0     ( x 31 − x 41 ) 2 + ( x 32 − x 42 ) 2 + ( x 33 − x 43 ) 2 − 1 0  − 9    � √ 3 + 2 � 2 2 0   ( x 31 − x 61 ) 2 + ( x 32 − x 62 ) 2 + ( x 33 − x 63 ) 2 − 1  − 37  0 4 4   0 ( x 41 − x 51 ) 2 + ( x 42 − x 52 ) 2 + ( x 43 − x 53 ) 2 − 3 ( x 41 − x 61 ) 2 + ( x 42 − x 62 ) 2 + ( x 43 − x 63 ) 2 − 3 ( x 51 − x 61 ) 2 + ( x 52 − x 62 ) 2 + ( x 53 − x 63 ) 2 − 3 Frohmader, Heaton Epsilon local rigidity

Definition A flex of p is a deformation p ( t ) : [0 , 1] → R nd such that g ( p ( t )) = 0 for all t ∈ [0 , 1] and which is not a rigid motion. Definition The configuration p is called locally rigid if no flex exists. In the 2009 paper by Timothy Abbott, Reid Barton, and Eric Demaine, “Generalizations of Kempe’s Universality Theorem” [ABD09] deciding local rigidity was shown to be Co-NP hard. Frohmader, Heaton Epsilon local rigidity

g : C nd → C m p ∈ R nd � [ n ] � = ⇒ g ( x ) = 0 E ⊂ 2 V ( g ) , V R ( g ) | E | = m The Jacobian dg : C nd → C m is a linear map, a matrix. Frohmader, Heaton Epsilon local rigidity

Jacobian dg : C nd → C m is a linear map, a matrix. Frohmader, Heaton Epsilon local rigidity

  f 1 = − x 3 + xy   f 2 = − x 4 + xz  : C 3 → C 3 f ( x ) =  f 3 = x 7 − x 5 y − x 4 z + x 2 yz Frohmader, Heaton Epsilon local rigidity

� � − 3 x 2 + y 0 x − 4 x 3 + z d f = 0 x 7 x 6 − 5 x 4 y − 4 x 3 z + 2 xyz − x 5 + x 2 z − x 4 + x 2 y f | (0 , 0 , 0) f | (0 , 5 , 3) f | (1 , 1 , 1) d d d � 0 � � 5 � � − 2 � 0 0 0 0 1 0 0 0 0 3 0 0 − 3 0 1 0 0 0 0 0 0 0 0 0 Frohmader, Heaton Epsilon local rigidity

Jacobian dg : C nd → C m has a generic rank . Choosing q = ( q ik ) ∈ C nd randomly, dg | q becomes a matrix with scalar entries. Calculate its rank (Gaussian elimination if exact computation, or SVD for floating point calculations). Frohmader, Heaton Epsilon local rigidity

By the way, 3 / 4 of triangles are obtuse. From Edelman and Strang: Frohmader, Heaton Epsilon local rigidity

Theorem A configuration p with n nodes embedded in R d is infinitesimally rigid if � � d + 1 rank ( dg | p ) = nd − , 2 � � d + 1 corank ( dg | p ) = . 2 Theorem Infinitesimal rigidity implies local rigidity. Why does this work? Frohmader, Heaton Epsilon local rigidity

Theorem Infinitesimal rigidity implies local rigidity. Proof. Since the Euclidean group acts we have a lower bound � � d + 1 ≤ dim p V R ( g ) . 2 But � � d + 1 dim p V R ( g ) ≤ dim p V ( g ) ≤ corank ( dg | p ) = . 2 Frohmader, Heaton Epsilon local rigidity

  1 0 0 √ 3 − 1  0  2 2   √ 3 − 1 0 p = −   2 2 √  3 − 1  3 − 2 2 √ 3 − 1 3 2 2 0 1 3 Frohmader, Heaton Epsilon local rigidity

Flexible? Frohmader, Heaton Epsilon local rigidity

Flexible? Frohmader, Heaton Epsilon local rigidity

Flexible? Frohmader, Heaton Epsilon local rigidity

...and now for something completely different. Frohmader, Heaton Epsilon local rigidity

We know the roots of q ( x ) = x 3 − 1 . x 3 − 7 x 2 + 17 x − 15 x 3 − 5 x 2 − 7 x + 51 ( x − 3)( x − 2 − i )( x − 2 + i ) ( x + 3)( x − 4 + i )( x − 4 − i ) Frohmader, Heaton Epsilon local rigidity

    x d 1 1 − 1 f 1 ( x )     x d 2 f 2 ( x ) 2 − 1     h ( z, t ) = (1 − t )  + γt  ,     . . . .   . . x d N f N ( x ) N − 1 h ( x, t ) = (1 − t ) f + γtq h ( x ( t ) , t ) = 0 Frohmader, Heaton Epsilon local rigidity

h ( x, t ) = (1 − t ) f + γtq h ( x ( t ) , t ) = 0 ∂h dx dt + ∂h ∂t = 0 . ∂x Frohmader, Heaton Epsilon local rigidity

Say that f : C 7 → C 4 . Its irreducible components X can have possible dimensions dim X ∈ { 3 , 4 , 5 , 6 } . To find 4 -dimensional components, create a square system of equations C 7 → C 7 :       f 1 1 0 0 c 11     f 2         0 1 0 c 21     f 3 0   0 0 1 c 31     f 4 0       � �     x 1 0     Af         = x 2 = 0       Lx L 11 L 12 · · · L 17       x 3 0         L 21 L 22 · · · L 27         x 4 0       L 31 L 32 · · · L 37     x 5 0     L 41 L 42 · · · L 47     x 6 x 7 Frohmader, Heaton Epsilon local rigidity

The numerical irreducible decomposition uses witness sets and the following Theorem (Bertini’s Theorem, Theorem 9.3 of [BSHW13]) Given a polynomial system f : C N → C n , there is a Zariski-open, dense set U ⊂ C k × n of matrices A such that V ( Af ) \ V ( f ) is either empty or consists of exactly C f ∈ Z > 0 irreducible components, each smooth (and hence disjoint) and of dimension N − k . The number C f of these extraneous components is independent of A . Frohmader, Heaton Epsilon local rigidity

But now back to epsilon local rigidity... Frohmader, Heaton Epsilon local rigidity

Using a moving frame [Olv], we can change coordinates in a useful way, creating zeros.     1 0 0 0 . 0 0 . 0 0 . 0 √ − 1 3 0   2 2 1 . 7320508075688772 0 . 0 0 . 0     √ − 1 3 0 0 . 8660254037844388 − 1 . 5 0 . 0 −  �→  .    2 2 √ 1 . 3660254037844386 − 1 . 3660254037844386 3 . 0  3 − 1 3 − 2 2 − 0 . 1339745962155613 − 0 . 5 3 . 0 √ 3 − 1 3 1 . 3660254037844388 0 . 3660254037844386 3 . 0 2 2 0 1 3 Frohmader, Heaton Epsilon local rigidity

Real parameter homotopy Start with equations g ( x ) = 0 but then adjoin one additional equation: ℓ v := v T x − v T p = 0 . Here, we can choose v ∈ R N randomly, or we could choose v from some infinitesimal flex. Then perturb this equation to ℓ v,ǫ := v T x − v T p − ǫ = 0 . for some small real 0 < ǫ ∈ R . In the computer, we use a real parameter homotopy (without γ ) as in � � � � g g h ( x, t ) = (1 − t ) + t . ℓ v,ǫ ℓ v Frohmader, Heaton Epsilon local rigidity

Frohmader, Heaton Epsilon local rigidity

Flexible? Frohmader, Heaton Epsilon local rigidity