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Recursive regularity and the finest regular coarsening of a - - PowerPoint PPT Presentation

Apr 06, 2023 421 likes •1.6k views

Recursive regularity and the finest regular coarsening of a polyhedral subdivision Rafel Jaume joint work with G unter Rote Overview Basic definition and tools Recursive regularity and finest regular coarsening Properties of recursively

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Recursive regularity and the finest regular coarsening

  • f a polyhedral subdivision

joint work with G¨ unter Rote Rafel Jaume

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Basic definition and tools Recursive regularity and finest regular coarsening Applications Properties of recursively regular subdivisions Overview

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Recursive regularity and the finest regular coarsening

  • f a polyhedral subdivision
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Let A = {p1, ...pn} be a set of points in Rd

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Let A = {p1, ...pn} be a set of points in Rd A polyhedral subdivision S of A is

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Let A = {p1, ...pn} be a set of points in Rd A polyhedral subdivision S of A is a polytopal complex

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Let A = {p1, ...pn} be a set of points in Rd A polyhedral subdivision S of A is a polytopal complex covering conv(A)

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Let A = {p1, ...pn} be a set of points in Rd A polyhedral subdivision S of A is a polytopal complex covering conv(A) whose set of vertices is contained in A

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Let A = {p1, ...pn} be a set of points in Rd A polyhedral subdivision S of A is Full-dimensional polytopes of the complex are called cells a polytopal complex covering conv(A) whose set of vertices is contained in A

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Recursive regularity and the finest regular coarsening

  • f a polyhedral subdivision
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S

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A coarsening of a subdivision S is a polyhedral subdivision S′ such that every face f ∈ S is contained in some face κ(f) ∈ S′ S

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A coarsening of a subdivision S is a polyhedral subdivision S′ such that every face f ∈ S is contained in some face κ(f) ∈ S′ S

′

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A coarsening of a subdivision S is a polyhedral subdivision S′ such that every face f ∈ S is contained in some face κ(f) ∈ S′ S

′

Reciprocally, we say that S is a refinement of S′

  • r that S is finer than S′
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Recursive regularity and the finest regular coarsening

  • f a polyhedral subdivision
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S is regular if it is the orthogonal projection of a (d + 1)-dimensional polytope P Rd

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Rd+1

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Rd+1 pi ωi pi

ωi

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Rd+1 P = conv( pi

ωi

  • )
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Rd+1

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A non-regular subdivision

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A non-regular subdivision

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A regular coarsening

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Recursive regularity and the finest regular coarsening

  • f a polyhedral subdivision
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Whether S is regular can be checked by a LP

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wall C D

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pb1 pb2 pb3 q affine basis of C point in D \ C C D

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pb1 pb2 pb3 q C D pb1

ωb1

  • pb3

ωb3

  • pb2

ωb2

  • q

ωq

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pb1 pb2 pb3 q C D

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  • 1

... 1 pb1 ... pbd+1

  • 1

... 1 1 pb1 ... pbd+1 q wb1 ... wbd+1 wq

  • > 0

pb1 pb2 pb3 q C D Local folding condition

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pc1 pc2 pcd+1 C

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pc1 pc2 pcd+1 q1 C

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pc1 pc2 pcd+1 q1 C

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pc1 pc2 pcd+1 C q2

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pc1 pc2 pcd+1 Coplanarity conditions

  • 1

... 1 1 pc1 ... pcd+1 qi wc1 ... wcd+1 wi

  • = 0

C q2

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  • 1

... 1 pb1 ... pbd+1

  • 1

... 1 1 pb1 ... pbd+1 q wb1 ... wbd+1 wq

  • > 0

Coplanarity conditions Local folding conditions

  • 1

... 1 1 pc1 ... pcd+1 qi wc1 ... wcd+1 wi

  • = 0
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  • 1

... 1 pb1 ... pbd+1

  • 1

... 1 1 pb1 ... pbd+1 q wb1 ... wbd+1 wq

  • > 0

Coplanarity conditions Local folding conditions s1 · ω > 0 sm · ω > 0 e1 · ω = 0 el · ω = 0 . . . . . . ω =    ω1 . . . ωn   

S

  • 1

... 1 1 pc1 ... pcd+1 qi wc1 ... wcd+1 wi

  • = 0
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  • 1

... 1 pb1 ... pbd+1

  • 1

... 1 1 pb1 ... pbd+1 q wb1 ... wbd+1 wq

  • > 0

Coplanarity conditions Local folding conditions s1 · ω > 0 sm · ω > 0 e1 · ω = 0 el · ω = 0 . . . . . . ω =    ω1 . . . ωn    Lemma (De Loera, Rambau, Santos) S is regular ⇐ ⇒ S is compatible

S

  • 1

... 1 1 pc1 ... pcd+1 qi wc1 ... wcd+1 wi

  • = 0
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The finest regular coarsening is well defined Theorem

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C D

Local folding condition >

Proof The finest regular coarsening is well defined Theorem

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C D

Local folding condition Coplanarity condition > =

Proof The finest regular coarsening is well defined Theorem

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Proof The finest regular coarsening is well defined Theorem s1 · ω > 0 sm · ω > 0 e1 · ω = 0 el · ω = 0

S

. . . . . .

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Proof The finest regular coarsening is well defined Theorem s1 · ω > 0 sm · ω > 0 e1 · ω = 0 el · ω = 0

S

s1 · ω ≥ 0 sm · ω ≥ 0 e1 · ω = 0 el · ω = 0

S′

. . . . . . . . . . . .

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Proof The finest regular coarsening is well defined Theorem s1 · ω > 0 sm · ω > 0 e1 · ω = 0 el · ω = 0

S

s1 · ω ≥ 0 sm · ω ≥ 0 e1 · ω = 0 el · ω = 0

S′

. . . . . . . . . . . . Is a closed cone. A point in its relatives interior satisfies in an strict way the maximum number of inequalities.

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Recursive regularity and the finest regular coarsening

  • f a polyhedral subdivision
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A polyhedral subdivision S is recursively regular if: S is regular or There exists a regular coarsening S′ of S with coarsening function κ such that κ−1(C) is recursively regular, for each cell C ∈ S′

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A polyhedral subdivision S is recursively regular if: S is regular or There exists a regular coarsening S′ of S with coarsening function κ such that κ−1(C) is recursively regular, for each cell C ∈ S′

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A polyhedral subdivision S is recursively regular if: S is regular or There exists a regular coarsening S′ of S with coarsening function κ such that κ−1(C) is recursively regular, for each cell C ∈ S′

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A polyhedral subdivision S is recursively regular if: S is regular or There exists a regular coarsening S′ of S with coarsening function κ such that κ−1(C) is recursively regular, for each cell C ∈ S′ Has it applications? How different from regularity is? Is flip-connected? How can we identify a recursively regular subdivision ?

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Regularity tree

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Regularity tree

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Regularity tree

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Regularity tree FRC

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Regularity tree FRC

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Regularity tree FRC

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Regularity tree FRC FRC

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Regularity tree FRC FRC

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Regularity tree FRC FRC

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S is recursively regular

  • its regularity tree has regular leaves

Proposition

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S is recursively regular

  • its regularity tree has regular leaves

Proposition Corollary It can be efficiently decided if a subdivision is recursively regular

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S is recursively regular

  • its regularity tree has regular leaves

Proposition Proof Regularity tree with regular leaves ⇒ recursively regular (def)

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S is recursively regular

  • its regularity tree has regular leaves

Proposition Proof Regularity tree with regular leaves ⇒ recursively regular (def) Recursively regular ⇒ regular leaves:

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S is recursively regular

  • its regularity tree has regular leaves

Proposition Proof Regularity tree with regular leaves ⇒ recursively regular (def) Recursively regular ⇒ regular leaves: There exists a “regularity tree” certifying that S is recursively

  • regular. The first coarsening must be coarser than the FRC.

Then, each cell of the FRC is recursively regular, since regularity is preserved by intersection with convex sets.

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Regular Cyclic Regular subdivisions are acyclic Acyclicity theorem (Edelsbrunner) Subdivisions

1 2 3 4

Cyclic

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Regular Cyclic

  • Rec. regular

Regular subdivisions are acyclic Acyclicity theorem (Edelsbrunner) Subdivisions Recursively regular subdivisions are a superset of the regular subdivisions and a subset of the acyclic ones Proposition

1 2 3 4

Cyclic

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Recursively regular subdivisions are a superset of the regular subdivisions and a subset of the acyclic ones Proposition Proof Assume that it exists a cycle in a recursively regular subdivision.

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Recursively regular subdivisions are a superset of the regular subdivisions and a subset of the acyclic ones Proposition Proof We can assume that the cycle belongs to more than one cell of the FRC and apply recursion otherwise. Assume that it exists a cycle in a recursively regular subdivision.

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Recursively regular subdivisions are a superset of the regular subdivisions and a subset of the acyclic ones Proposition Proof We can assume that the cycle belongs to more than one cell of the FRC and apply recursion otherwise. Then, the FRC cannot be regular, since a cycle cannot be destroyed by means of merging cells. Assume that it exists a cycle in a recursively regular subdivision.

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There is a 5-dimensional point set with at least 12 recursively regular triangulations pairwise disconnected and disconnected from any regular triangulation in the graph of flips Proposition

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Applications Floodlights Homothecies Graph embeddings Spider webs

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Floodlights polyhedral fan with n cells

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Floodlights polyhedral fan with n cells n points

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Floodlights polyhedral fan with n cells n points

  • ne to one
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Floodlights polyhedral fan with n cells n points

  • ne to one

covering assignment

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Covering fans A polyhedral fan is universally covering if for any point set there exists a covering assignment

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Covering fans A polyhedral fan is universally covering if for any point set there exists a covering assignment Regular fans are universally covering Theorem (Galperin & Galperin, 1981)

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Covering fans A polyhedral fan is universally covering if for any point set there exists a covering assignment Regular fans are universally covering Theorem (Galperin & Galperin, 1981) Recursively regular fans are universally covering Cyclic fans are not universally covering Theorem

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Regular Cyclic

  • Rec. regular

Fans

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Regular Cyclic

  • Rec. regular

Universally covering Fans

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Regular Cyclic

  • Rec. regular

Universally covering Fans

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Regular Cyclic

  • Rec. regular

Universally covering Fans

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Regular Cyclic

  • Rec. regular

Universally covering Fans

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Overlapping condition

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Overlapping condition

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Overlapping condition

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V = {1, ..., n} Directional graph embeddings − → E ⊂ V × V

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V = {1, ..., n} P = {p1, ...pn} ⊂ Rd Directional graph embeddings − → E ⊂ V × V

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N : − → E − → Rd V = {1, ..., n} P = {p1, ...pn} ⊂ Rd Directional graph embeddings − → E ⊂ V × V

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N : − → E − → Rd V = {1, ..., n} P = {p1, ...pn} ⊂ Rd Is there a σ : V ↔ P N(i, j) · pi ≥ N(i, j) · pj ∀(i, j) ∈ − → E ? proper embedding Directional graph embeddings − → E ⊂ V × V

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N : − → E − → Rd V = {1, ..., n} P = {p1, ...pn} ⊂ Rd Is there a σ : V ↔ P N(i, j) · pi ≥ N(i, j) · pj ∀(i, j) ∈ − → E ? A path is always embeddable proper embedding Directional graph embeddings − → E ⊂ V × V

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N : − → E − → Rd V = {1, ..., n} P = {p1, ...pn} ⊂ Rd Is there a σ : V ↔ P N(i, j) · pi ≥ N(i, j) · pj ∀(i, j) ∈ − → E ? A path is always embeddable A cycle is not always embeddable proper embedding Directional graph embeddings − → E ⊂ V × V

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N : − → E − → Rd V = {1, ..., n} P = {p1, ...pn} ⊂ Rd Is there a σ : V ↔ P N(i, j) · pi ≥ N(i, j) · pj ∀(i, j) ∈ − → E ? A path is always embeddable A cycle is not always embeddable A tree? proper embedding Directional graph embeddings − → E ⊂ V × V

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Proposition If the graph is the normal graph (or a projection of a subgraph) of a fan and there is a covering assignment forP, there is a proper graph embedding for P. If there is no assignment satisfying OC for... , there is no proper embedding

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Proposition If the graph is the normal graph (or a projection of a subgraph) of a fan and there is a covering assignment forP, there is a proper graph embedding for P. If there is no assignment satisfying OC for... , there is no proper embedding Corollary The graph of a polytope is always embeddable.

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Homothecies 1,3 1,5 0,9 1,1 0,6 1,2

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Homothecies 1,3 1,5 0,9 1,1 0,6 1,2

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Homothecies

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Homothecies

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Homothecies

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Homothecies

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Homothecies

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Homothecies Proposition If there is a covering assignment for the fan and the points corresponding to the transformations, there is a transformation assignment If there is no assignment satisfying OC for a fan and a pointset, there is a set of transformations for which there is no transformation assignment

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Spiderwebs

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Spiderwebs (Positive) equilibrium stress ↔ (convex) lifting (for non-crossing planar frameworks) Maxwell-Cremona correspondence Theorem (Connelly) If a tensegrity framework is infinitesimally rigid, then it has an equilibrium stress that is nonzero

  • n all struts and cables

Theorem (Crapo, Whiteley) If a spider web has a positive equilibriuem stress, then it is rigid

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Spiderwebs

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Spiderwebs

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Spiderwebs

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Spiderwebs

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Spiderwebs

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Spiderwebs A recursively regular spider web is rigid. A recursively regular non-regular spider web is infinitesimally flexible. Theorem

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Conclusions The finest regular coarsening is well defined

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Conclusions The finest regular coarsening is well defined Recursively regular subdivisions are Superset of regular subdivisions, subset of acyclic Easy to identify Not connected by flips

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Conclusions The finest regular coarsening is well defined Recursively regular subdivisions are Superset of regular subdivisions, subset of acyclic Easy to identify Not connected by flips They are related to the problems Covering by floodlights Covering by homothecies Directional graph embedding Spider webs

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Conclusions The finest regular coarsening is well defined Recursively regular subdivisions are Superset of regular subdivisions, subset of acyclic Easy to identify Not connected by flips They are related to the problems Covering by floodlights Covering by homothecies Directional graph embedding Universality, counterexamples Spider webs

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Conclusions The finest regular coarsening is well defined Recursively regular subdivisions are Superset of regular subdivisions, subset of acyclic Easy to identify Not connected by flips They are related to the problems Covering by floodlights Covering by homothecies Directional graph embedding Universality, counterexamples Spider webs Open: Decision problem, universality limits

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Conclusions The finest regular coarsening is well defined Recursively regular subdivisions are Superset of regular subdivisions, subset of acyclic Easy to identify Not connected by flips They are related to the problems Covering by floodlights Covering by homothecies Directional graph embedding Universality, counterexamples Thank you! Spider webs Open: Decision problem, universality limits