Constructing Convex 3-Polytopes From Two Triangulations of a Polygon

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Constructing Convex 3-Polytopes From Two
Triangulations of a Polygon

• Benjamin MarlinDept. of Mathematics
  StatisticsMcGill University

Godfried ToussaintSchool of Computer
ScienceMcGill University
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Introduction
A configuration (P,T1,T2)

• A strictly convex polygon P (V, E) in the
  XY-plane.
• A set of diagonals T1 which triangulates P.
• A set of diagonals T2 which traingulates P and
  shares no diagonals with the set T1.

Example
P
P with T1
P with T2
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Introduction
Constructing Convex 3-Polytopes
Given a configuration (P,T1,T2), we are
interested in transforming it into a convex
3-polytope P by mapping the vertices of P into
3-space.
Unrestricted Mapping v(x,y) ? v(x,y,z)
For each vertex in the configuration new x, y and
z-coordinates are assigned.
Restricted Mapping v(x,y) ? v(x,y,z)
For each vertex in the configuration the x and
y-coordinates are preserved, and a new
z-coordinate is assigned.
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Introduction
Edge Mapping e(v1,v2) ? e(v1, v2)

• We consider edges and diagonals to be straight
  line segments joining two distinct verticies.
• When a configuration is mapped into 3-space we
  first assign each vertex to its new position, and
  then add edges between any two vertices that were
  connected by an edge or a diagonal in the
  starting configuration.

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Introduction
A Realization Example
A Configuration (P,T1,T2) is realized as a convex
3-polytope P by raising vertex A out of the
plane by 3 units. The complete mapping can be
seen below.
Configuration (P,T1,T2)
Convex 3-Polytope P
A(0,3)
B(4,0)
D(-4,0)
C(0,-3)
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Overview
Unrestricted Mapping

• We will show that any configuration can be
  mapped into a convex 3-polytope if there are no
  restrictions on the placement of the vertices in
  3-space.
• The proof of this statement involves the use of
  graph theory, particularly the theory of
  polyhedral graphs developed by Steinitz.

Steinitzs Theorem A graph G is isomorphic to
the edge graph of a convex 3-polytope P if and
only if G is planar and 3-connected.
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Overview
Restricted Mapping

• Leo Guibas conjectured that every configuration
  could be realized as convex 3-polytope if the
  verticies of the configuration were allowed to
  vary vertically only. Boris Dekster later proved
  this conjecture false. We will present an
  adaptation of Deksters Proof.
• Many configurations are realizable under the
  restricted mapping. In the final section we
  present a complete chrarcterization of the
  realizable configurations, as well as an
  alogorithm for deciding the realizabiltiy of a
  configuration based on linear programming
  techniques.

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Unrestricted Mapping
Theorem 1
Given a strictly convex polygon P in the XY-plane
along with two triangulations of it T1 and T2,
it is always possible to assign new x, y and z
coordinates to the vertices of P such that the
resulting 3-polytope P is convex.

Lemma 1 The edge graph G of (P U T1 U T2) is
planar.

• Let G1 be the edge graph of (P U T1) and G2 be
  the edge graph of (P U T2) .
• G1 and G2 are both plane graphs because by
  definition a triangulation of a convex polygon
  has no crossing edges.

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Unrestricted Mapping
Lemma 1 (Continued)

• Since G1 and G2 are plane graphs, each can be
  embeded on the surface of a sphere with the
  vertices along the equator and edges as
  noncrossing arcs of great circles restricted to
  one hemisphere.
• But then G (G1 U G2) can be mapped to the
  surface of a sphere with G1 in one hemisphere and
  G2 in the other.
• Taking a steriographic projection onto the
  xy-plane we obtain an embedding of G with no edge
  crossings so G must be a planar graph.

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Unrestricted Mapping
Lemma 2 The graph G of (P U T1 U T2) is
3-connected.

• If P has n vertices (PUT1UT2) has a total of
  3n-6 edges .
• Every planar graph with 3n-6 edges is a maximal
  planar graph (Whitney).
• All maximal planar graphs with n ? 4 are
  3-connected (Kuratowski).

Proof of Theorem 1

• By Lemma 1, Lemma 2, and Steinitzs Theorem, the
  edge graph G of (P U T1 U T2) is realizable as a
  convex 3-Polytope P.
• If P is a realization of G, then P is a
  realization of (P U T1 U T2) under the
  unrestricted mapping.

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Restricted Mapping
Guibas Conjecture Historical Aspects

• Proposed by Leo Guibas at the first CCCG held at
  McGill University in August 1989.
• Proved false by Boris Dekster of Mount Allison
  University in 1995.

Guibas Conjecture
Given a configuration (P,T1,T2) is it always
possible to assign height values to the vertices
of P such that the edges of the convex hull of
the spatial polygon P project back onto (P U T1
U T2).
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Deksters Counter Example
Dekster proves a general necessary condition for
realizability on the configuration (P,T1,T2). He
then produces a counter example for which the
condition fails, showing that Guibas conjecture
is false. The counter example is shown below
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Deksters Counter Example
Direct Proof Sketch

• Assume that the counter example configuration has
  an assignment of heights that satisfies Guibas
  conjecture.
• Proceed by geometric argument to show that there
  exist two points y1 and y2 where both y1 is
  strictly above y2, and y2 is strictly above y1
  for any height assignment.
• Conclude by contradiction that Guibas Conjecture
  must be false.

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Deksters Counter Example
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Deksters Counter Example
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Deksters Counter Example
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Deksters Counter Example
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Deciding Realizability
Overview

• Based on the characterization that a polyhedron
  is convex if and only if for each face, all the
  remaining vertices are on the same side of that
  face.
• Uses the signed volume of the simplex formed by
  each face and each of the remaining remaining
  vertices to construct a set of linear
  inequalities.
• Applies linear programming techniques to
  determine if a solution to the system exists.

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Deciding Realizability
Motivation

• As we have seen, the unrestricted mapping
  provides a link between Guibas conjecture, and
  Steinitzs theorem.
• The reduction to linear inequalities is inspired
  in part by similar methods used in quantitative
  treatments of Steinitzs Theorem (Onn and
  Sturmfels).
• A reduction to a system of linear equalities and
  inequalities is also used by Sugihara to
  characterize realizable line drawings, a related
  problem.

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Deciding Realizability
Constructing the Inequalities
Let fi(vi,1, vi,2, vi,3) be a face of P oriented
counter clockwise if fi is a face of the top of
P, and clockwise if fi is a face of the bottom of
P. Let vj be a vertex of P- fi. For the signed
volume to be positive we require that
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Deciding Realizability
Key Observation
The inequalities are linear in the Z coordinate
since the X and Y coordinates are given in the
problem instance. Cofactor expansion of the
determinant along the Z column gives a linear
inequality in Z
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Deciding Realizability
Theorem
A configuration (P,T1,T2) is realizable as a
convex 3-polytope if and only if the set of
linear inequalities just described has a solution.
Proof

• If the set of inequalities has a solution (z1,
  z2, ..., zn),then when these values are assigned
  to P, the resulting 3-polytope will be convex.
• If the set of inequalities has no solution, then
  for any assignment of values to the zjs at least
  one of the inequalities isn't satisfied. This
  means at least one vertex is on the wrong side of
  a face, and P is not realizable.

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Deciding Realizability
Algorithm
1. Given a configuration (P,T1,T2) compute its
combinatorial graph G. 2. Compute the set of
faces F of G with vertices listed in the
appropriate order with respect to top/bottom. 3.
Compute the set of linear inequalities as
described. 4. Apply linear programming techniques
to determine if the set of linear inequalities
has a solution. 5. If a solution exists the
configuration is realizable, otherwise it is not
realizable. Note To compute a realization we
return any solution found by the LP method, if a
solution exists.
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Computational Complexity

• A configuration (P,T1,T2) with n vertices
  generates an n dimensional linear programming
  instance with at most n(n-3) inequalities.
• The complexity of the proposed method is
  dominated by the complexity of the specific
  linear programming algorithm used to solve the
  system of linear inequalities.

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Directions for Further Study

• Determining which linear programming method
  gives the best results for the systems of
  inequalities produced (n dimensions, O(n2)
  inequalities, 4 variables per inequality).
• Finding a set of inequalities of size O(n) to
  replace the O(n2) set of inequalities described
  here. One possibility is to use a dihedral angle
  test for local convexity at each edge (suggested
  by Anna Lubiw).