Download CONVEX POLYTOPES

CONVEX POLYTOPES
Gleb Kodinets
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GALE TRANSFORM
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‫‪GALE TRANSFORM‬‬
‫• בדומה לטרנספורם דואלי טרנספורם גאיל מעביר‬
‫קונפיגורציה גאומטרית אחד לקונפיגורציה גאומטרית‬
‫אחרת‪ .‬המציאו אותו כדי ללמוד יותר את פאונים‬
‫הקמורים ממימד גבוה יותר טוב‪.‬‬
‫• טרנספורם גאיל קשור גם לדואליות של תיכנות ליניארית‪,‬‬
‫אבל לא נדבר על זה היום‪.‬‬
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Gale Transform
• The Gale transform assigns to a sequence
𝑎=(𝑎1 , 𝑎2 , … , 𝑎𝑛 ) of n ≥ d+1 points in 𝑅𝑑
another sequence of n points. The result
points 𝑔=(𝑔1 ,𝑔2 , … , 𝑔𝑛 ) live in a different
dimension, namely in 𝑅𝑛−𝑑−1 .
For example, n points in the plane are
transformed to n vectors in 𝑅𝑛−3 .
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• Gale transform operates on sequences, not
individual points: We cannot say what 𝑔1 is
without knowing all of 𝑎1 , 𝑎2 , … , 𝑎𝑛 .
• We also require that the affine hull of the 𝑎𝑖 be
the whole 𝑅𝑑 ; otherwise, the Gale transform is
not defined.
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How to
• In order to obtain the Gale transform of 𝑎, we
first convert the 𝑎𝑖 into (d+1)-dimensional
vectors: 𝑎𝑖 ∈ 𝑅𝑑+1 is obtained from 𝑎𝑖 by
appending a (d+1)st coordinate equal to 1.
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• Let A be the (d+1) x n matrix with: 𝑎𝑖 as the ith
column. Since we assume that there are d+1
affinely independent points in a, the matrix A
has rank d+1.
• And so the vector space V generated by the
rows of A is a (d+1)-dimensional subspace of
𝑅𝑛 .
• We let 𝑉 ⊥ be the orthogonal complement of V in
𝑅𝑛 ; that is, 𝑉 ⊥ = {w ∈ 𝑅𝑛 :<v,w>= 0 for all v ∈ V}.
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• We have dim(𝑉 ⊥ ) = n-d-1. Let us choose some
basis (𝑏1 , 𝑏2 , … , 𝑏𝑛−𝑑−1 ) of 𝑉 ⊥ , and let В be the
(n-d-1) x n matrix with 𝑏𝑗 as the jth row. Finally,
we let 𝑔𝑖 ∈ 𝑅𝑛−𝑑−1 be the ith column of B. The
sequence 𝑔 =( 𝑔1 , 𝑔2 , … , 𝑔𝑛 ) is the Gale
transform of a.
•
Here is а pictorial summary:
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Observation.
• (i) The Gale transform is determined up to
linear isomorphism
In the construction of g, we can choose an arbitrary
basis of 𝑉 ⊥ . Choosing a different basis corresponds
to multiplying the .matrix В from the left by a
nonsingular (n-d-1) x (n-d-1) matrix T and this means
transforming ( 𝑔1 , 𝑔2 , … , 𝑔𝑛 ) by a linear
isomorphism of 𝑅𝑛−𝑑−1 .
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Observation.
• (ii) A sequence 𝑔 in 𝑅𝑛−𝑑−1 is the Gale
transform of some 𝑎 if and only if it spans
𝑅𝑛−𝑑−1 and has 0 as the center of gravity:
•
𝑛
𝑖=1 𝑔𝑖
=0
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Observation
• (iii) Let us consider a sequence 𝑔 in 𝑅𝑛−𝑑−1
satisfying the condition in (ii). If we apply the
Gale transform to it, and apply the Gale
transform the second time, we recover the
original 𝑔, up to linear isomorphism.
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Two ways of probing a
configuration
• For a sequence 𝑎 = (𝑎1 , 𝑎2 , … , 𝑎𝑛 ) of vectors in
𝑅𝑑+1 , we define two vector subspaces of 𝑅𝑛 :
• Linear function – linear combination without
constant(free coefficient)
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Two ways of probing a
configuration
• For a point sequence 𝑎 = (𝑎1 , 𝑎2 , … , 𝑎𝑛 ) , we
then let AffVal(𝑎) = LinVal(𝑎) and AffDep(𝑎) =
LinDep(𝑎), where 𝑎 is obtained from 𝑎 as
above, by appending 1's. Another description is
• Affine function – linear combination with free
coefficient
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• The knowledge of LinVal(𝑎) tells us a lot about
𝑎, and we only have to learn to decode the
information. As usual, we assume that 𝑎 linearly
spans all of 𝑅𝑑+1
• Each nonzero linear function 𝑓: 𝑅𝑑+1 → 𝑅
determines the hyperplane passing through 0.
•
ℎ𝑓 = {x ∈ 𝑅𝑑+1 : 𝑓 𝑥 = 0}
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• ℎ𝑓 = {x ∈ 𝑅𝑑+1 : 𝑓 𝑥 = 0} , 𝑓: 𝑅𝑑+1 → 𝑅
• This ℎ𝑓 is oriented (one of its half-spaces is
positive and the other negative), and the sign of
𝑓(𝑎𝑖 ) determines whether (𝑎𝑖 lies on ℎ𝑓 , on its
positive side, or on its negative side.
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• Let’s choose our favorite index set I⊆ {1, 2,..., n},
and ask whether the points of the subsequence
𝑎𝐼 = (𝑎𝑖 : i ∈ I ) span a linear hyperplane.
• First, we observe that they lie in a common linear
hyperplane if and only if there is a nonzero φ ∈
LinVal(𝑎) such that φ𝑖 = 0 for all i ∈ I.
• It could still happen that all of 𝑎𝐼 lies in a lowerdimensional linear subspace.
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• Using the assumption that 𝑎 spans 𝑅𝑑+1 , it is
not difficult to see that 𝑎𝐼 spans a linear
hyperplane if and only if all φ ∈ LinVal(𝑎) that
vanish on 𝑎𝐼 have identical zero sets; that is,
the set {i: φ𝑖 =0} is the same for all such φ. If we
know that 𝑎𝐼 spans a linear hyperplane, we can
also see how the other vectors in 𝑎 are
distributed with respect to this linear
hyperplane.
• (𝑎1 , 𝒂𝟐 , 𝑎3 , 𝒂𝟒 , 𝒂𝟓 ) I={2,4,5} 𝜑𝑖 = 𝑓 𝑎𝑖
𝑓 𝑎2 = 𝑓 𝑎4 = 𝑓 𝑎5 = 0
𝑓 𝑎1 = 0
𝑎1 = 𝑐2 𝑎2 + 𝑐4 𝑎4 + 𝑐5 𝑎5
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• Similar information can be inferred from
AffDep(𝑎) (exactly the same information, in fact,
since AffDep(𝑎) = AffVal(𝑎) ⊥ ). For an α ϵ
AffDep(𝑎) let 𝐼+ (α) = {i ∈ {1,2, ...,n}: 𝛼𝑖 > 0} and
𝐼− (α) = {i ∈ {1,2, ...,n}: 𝛼𝑖 < 0} . As we learned in
the proof of Radon's lemma, 𝐼+ = 𝐼+ (α) and 𝐼−
= 𝐼− (α) correspond to Radon partitions of 𝑎.
• AffDep(𝒂) = AffVal(𝒂) ⊥ :
• < 𝑓 𝑎1 , … , 𝑓 𝑎𝑛 >×< 𝛼1 , … , 𝛼𝑛 >
𝛼1 𝑎1 + ⋯ + 𝛼𝑛 𝑎𝑛 = 0 , 𝑓 − 𝑙𝑖𝑛𝑒𝑎𝑟 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
• 𝛼1 𝑓 𝑎1 + ⋯ + 𝛼𝑛 𝑓 𝛼𝑛 = 𝑓 𝛼1 𝑎1 + ⋯ + 𝛼𝑛 𝑎𝑛
=𝑓 0 =0
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• Namely, 𝑖∈𝐼+ 𝛼𝑖 𝑎𝑖 = 𝑖∈𝐼− (−𝛼𝑖 )𝑎𝑖 and dividing
𝑖∈𝐼+ 𝛼𝑖 = 𝑖∈𝐼− (−𝛼𝑖 ) , we have convex
combinations on both sides, and so conv(𝑎𝐼+ ) ∩
conv(𝑎𝐼− ) ≠∅. For example, 𝑎𝑖 is a vertex of
conv(𝑎) if and only if there is no α ∈ AffDep(𝑎)
with 𝐼+ (𝛼) ={i}.
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Lemma
• Let 𝑎 be a sequence of 𝑛 points in 𝑅𝑑 whose
points affinely span 𝑅𝑑 , and let 𝑔 be its Gale
transform. Then LinVal( 𝑔 ) = AffDep(a) and
LinDep( 𝑔 ) = AffVal(a).
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Proof
• LinVal( 𝑔 ) = AffDep(𝑎)
• ∀𝛼. 𝛼1 𝑎1 + ⋯ + 𝛼𝑛 𝑎𝑛 = 0 , ∃𝑓
< 𝑓 𝑔1 , … , 𝑓 𝑔𝑛 > = < 𝛼1 , … , 𝛼𝑛 >
|
|
𝑐1 , … , 𝑐𝑛−𝑑−1 × 𝑔1 … 𝑔𝑛 =< 𝛼1 , … , 𝛼𝑛 >
|
|
− 𝑎1 −
⋮
𝛼1 , … , 𝛼𝑛 ×
=0
− 𝑎𝑛 −
− 𝑎1 −
|
|
⋮
𝑐1 , … , 𝑐𝑛−𝑑−1 × 𝑔1 … 𝑔𝑛 ×
|
|
− 𝑎𝑛 −
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Proof
• LinVal( 𝑔 ) = AffDep(𝑎)
−
|
|
𝑐1 , … , 𝑐𝑛−𝑑−1 × 𝑔1 … 𝑔𝑛 ×
|
|
−
𝑐1 , … , 𝑐𝑛−𝑑−1 × 0 = 0
𝑎1
⋮
𝑎𝑛
−
−
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Dictionary of the Gale transform
• (i)Lying in a common hyperplane
For every (d+1)-point index set I ⊆ {1,2,..., n}, the
points 𝑎𝑖 with i∈I lie in a common hyperplane if and
only if all the vectors 𝑔𝑗 with j ∉ I lie in a common
linear hyperplane.
• (ii)General position
In particular, the points of 𝑎 are in general position
(no d+1 on a common hyperplane) if and only if
every n-d-1 vectors among 𝑔1 , … , 𝑔𝑛 span
𝑅𝑛−𝑑−1 (which is a natural condition of general
position for vectors).
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Dictionary of the Gale transform.
• (iii)Faces of the convex hull
The points 𝑎𝑖 with i∈I are contained in a common
facet of P =conv(𝑎) if and only if 0 ∈ conv{ 𝑔𝑗 : j ∉ I}. In
particular, if P is a simplicial polytope, then its k-faces
exactly correspond to complements of the (n-k-1)element subsets of 𝑔 containing 0 in the convex hull.
• (iv)Convex independence
The 𝑎𝑖 form a convex independent set if and only if
there is no oriented linear hyperplane with exactly
one of the 𝑔𝑗 on the positive side.
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• Here is a picture of a 3-dimensional convex
polytope with 6 vertices and the (planar) Gale
transform of its vertex set.
• For example, the facet 𝑎1 𝑎2 𝑎5 𝑎6 is reflected by
the complementary pair 𝑔3 , 𝑔4 of parallel
oppositely oriented vectors, and so on.
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Signs suffice.
• As was noted above, in order to find out whether
some 𝑎𝑖 is a vertex of conv(𝑎), we ask whether
there is an α ∈ AffDep(𝑎) with 𝐼+ (α) = {i}.
• Only the signs of the vectors in AffDep(𝑎) are
important here, and this is the case with all the
combinatorial-geometric information about point
sequences or vector sequences.
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• For such purposes, the knowledge of
sgn(AffDep(𝑎)) = {(sgn(𝛼𝑖 ),... ,sgn(𝛼𝑛 )): α ∈
AffDep(𝑎)} is as good as the knowledge of
AffDep(𝑎).
• We can thus declare two sequences 𝑎 and 𝑏
combinatorially isomorphic if sgn(AffDep(𝑎)) =
sgn(AffDep(𝑏)) and sgn(AffVal(𝑎)) =
sgn(AffVal(𝑏)).
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• Here we need only one very special case: If
𝑔 = ( 𝑔1 ,..., 𝑔𝑛 ) is a sequence of vectors,
𝑡1 , … , 𝑡𝑛 > 0 are positive real numbers, and
𝑔′ = ( 𝑡1 𝑔1 ,...,𝑡𝑛 𝑔𝑛 ), then clearly,
• and so 𝑔 and 𝑔′are combinatorially
isomorphic vector configurations.
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Affine Gale diagrams.
• We have seen a certain asymmetry of the Gale
transform: While the sequence 𝑎 is interpreted
affinely, as a point sequence, its Gale transform
needs to be interpreted linearly, as a sequence
of vectors (with 0 playing a special role). Could
one reduce the dimension of 𝑔 by 1 and pass
to an "affine version" of the Gale transform?
This is indeed possible, but one has to
distinguish "positive" and "negative" points in
the affine version.
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• Let 𝑔 be the Gale transform of some 𝑎,
𝑔1 , … , 𝑔𝑛 ∈𝑅𝑛−𝑑−1 . Let us assume for simplicity
that all the 𝑔𝑖 are nonzero. We choose a
hyperplane h not parallel to any of the 𝑔𝑖 and
not passing through 0, and we project the
𝑔𝑖 centrally from 0 into h, obtaining points 𝑔1 ,...
, 𝑔𝑛 ∈ h ≅ 𝑅𝑛−𝑑−1 . If 𝑔𝑖 lies on the same side of
0 as 𝑔𝑖 , i.e., if 𝑔𝑖 =𝑡𝑖 𝑔𝑖 with 𝑡𝑖 > 0, we set 𝜎𝑖 = +1,
and call 𝑔𝑖 a positive point.
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• For 𝑔𝑖 lying on the other side of 0 than 𝑔𝑖 we
let 𝜎𝑖 = −1 and we call 𝑔𝑖 a negative point.
Here is an example with the 2-dimensional Gale
transform from the previous drawing:
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• The positive 𝑔𝑖 are marked by full circles, the
negative ones by empty circles, and we have
borrowed the (incomplete) yin-yang symbol for
marking the positions shared by one positive
and one negative point. This sequence 𝑔 of
positive and negative points in 𝑅𝑛−𝑑−1 , or more
formally the pair (𝑔,σ), is called an affine Gale
diagram of 𝑎.
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• It conveys the same combinatorial information
as 𝑔 , although we cannot reconstruct 𝑎 from it
up to linear isomorphism, as was the case with
𝑔 . (For this reason, we speak of Gale diagram
rather than Gale transform.) One has to get
used to interpreting the positive and negative
points properly. If we put
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• Easy to check:
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Proposition (Dictionary of affine
Gale diagrams)
• Let 𝑎 be a sequence of n points in 𝑅𝑑 , let 𝑔 be
the Gale transform of 𝑎, and assume that all the
𝑔𝑖 are nonzero. Let (𝑔, σ) be an affine Gale
diagram of 𝑎 in 𝑅𝑛−𝑑−2 :
• (i) A subsequence 𝑎𝐼 lies in a common facet of
conv(𝑎) if and only if conv({𝑔𝑗 : j∉I, 𝜎𝑖 = 1}) ∩ ({𝑔𝑗 :
j∉I, 𝜎𝑖 = -1}) ≠∅.
• (ii) The points of 𝑎 are in convex position if and
only if for every oriented hyperplane in 𝑅𝑛−𝑑−2 ,
the number of positive points of 𝑔 on its positive
side plus the number of negative points of 𝑔 on
its negative side is at least 2.
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• So far we have assumed that 𝑔𝑖 ≠ 0 for all 𝑖.
This need not hold in general, and points 𝑔𝑖 =
0 need a special treatment in the affine Gale
diagram: They are called the special points,
and for a full specification of the affine Gale
diagram, we draw the positive and negative
points and give the number of special points.
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A nonrational polytope.
• Configurations of k+4 points in 𝑅𝑘 have planar
affine Gale diagrams. This leads to many
interesting constructions of k-dimensional
convex polytopes with k+4 vertices.
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Example
• 8-dimensional polytope with 12 vertices that
cannot be realized with rational coordinates;
that is, no polytope with isomorphic face lattice
has all vertex coordinates rational. First one has
to become convinced that if 9 distinct points are
placed in R2 so that they are not all collinear
and there are collinear triples and 4-tuples as is
marked by segments in the left drawing below,
then not all coordinates of the points can be
rational.
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• We declare some points negative, some
positive, and some both positive and negative,
as in the right drawing, obtaining 12 points.
These points have a chance of being an affine
Gale diagram of the vertex set of an 8dimensional convex polytope, since condition is
satisfied.
• How do we construct such a polytope? For 𝑔𝑖 =
(𝑥𝑖 , 𝑦𝑖 ), we put 𝑔𝑖 = (𝑡𝑖 𝑥𝑖, 𝑡𝑖 𝑦𝑖 , 𝑡𝑖 ) ∈ 𝑅3 , choosing
𝑡𝑖 > 0 for positive 𝑔𝑖 and 𝑡𝑖 < 0 for negative 𝑔𝑖 ,
in such a way that 12
𝑖=1 𝑔𝑖 = 0. Then the Gale
transform of 𝑔 is the vertex set of the desired
convex polytope P.
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• Let P' be some convex polytope with an
isomorphic face lattice and let (g’, σ') be an
affine Gale diagram of its vertex set a'. We
have, for example, 𝑔′7 = 𝑔′10 because {𝑎′𝑖 . i =
7,10} form a facet of P', and similarly for the
other point coincidences. The triple 𝑔′1 , 𝑔′12 ,
𝑔′8 (where 𝑔′8 is positive) is coUinear, because
{𝑎′𝑖 . i ≠ 1,8,12} is a facet. In this way, we see
that the point coincidences and collinearities
are preserved, and so no affine Gale diagram of
P' can have all coordinates rational. At the
same time, by checking the definition, we see
that a point sequence with rational coordinates
has at least one affine Gale diagram with
rational coordinates. Thus, P cannot be realized
with rational coordinates.
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Voronoi Diagrams
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• Consider a finite set P ⊂ 𝑅𝑑 . For each point p ∈
P, we define a region reg(p), which is the
"sphere of influence" of the point p: It consists
of the points x ∈ 𝑅𝑑 for which p is the closest
point among the points of P. dist(x, y) denotes
the Euclidean distance of the points x and y.
• The Voronoi diagram of P is the set of all
regions reg(p) for p ∈ P.
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• Here is an example of the Voronoi diagram of 2
points in the plane:
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• Here is an example of the Voronoi diagram of 3
points in the plane:
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• Here is an example of the Voronoi diagram of a
point set in the plane:
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Observation
• Each region reg(p) is a convex polyhedron with
at most |P|-1 facets.
• Indeed, reg(p) is an intersection of |P| - 1 halfspaces.
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• For d = 2, a Voronoi diagram of n points is a
subdivision of the plane into n convex polygons
(some of them are unbounded).
•
It can be regarded as a drawing of a planar
graph (with one vertex at the infinity, say), and
hence it has a linear combinatorial complexity:
n regions, O(n) vertices, and O(n) edges.
• Euler’s formula: v+f=2+e
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Examples of applications.
• Voronoi diagrams have been reinvented and
used in various branches of science.
Sometimes the connections are surprising.
• For instance, in archaeology, Voronoi diagrams
help study cultural influences.
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Examples of applications: "Post
office problem" or nearest
neighbor searching
• Given a point set P in the plane, we want to
construct a data structure that finds the point of
P nearest to a given query point x as quickly as
possible. This problem arises directly in some
practical situations or, more significantly, as a
subroutine in more complicated problems. The
query can be answered by determining the
region of the Voronoi diagram of P containing x.
For this problem (point location in a subdivision
of the plane), efficient data structures are
known.
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Robot motion planning
• Consider a disk-shaped robot in the plane. It
should pass among a set P of point obstacles,
getting from a given start position to a given
target position and touching none of the
obstacles.
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Robot motion planning
• If such a passage is possible at all, the robot
can always walk along the edges of the Voronoi
diagram of P, except for the initial and final
segments of the tour. This allows one to reduce
the robot motion problem to a graph search
problem: We define a subgraph of the Voronoi
diagram consisting of the edges that are
passable for the robot.
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A nice triangulation: the
Delaunay triangulation
• Let P ⊂ 𝑅2 be a finite point set. In many
applications one needs to construct a
triangulation of P (that is, to subdivide conv(P)
into triangles with vertices at the points of P) in
such a way that the triangles are not too skinny.
Of course, for some sets, some skinny triangles
are necessary, but we want to avoid them as
much as possible.
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A nice triangulation: the
Delaunay triangulation
• One particular triangulation that is usually very
good, and provably optimal with respect to
several natural criteria, is obtained as the dual
graph to the Voronoi diagram of P. Two points
of P are connected by an edge if and only if
their Voronoi regions share an edge.
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A nice triangulation: the
Delaunay triangulation
• If no 4 points of P lie on a common circle then
this indeed defines a triangulation, called the
Delaunay triangulation of P.
• The definition extends to points sets in Rd in a
straightforward manner.
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Interpolation
• Suppose that f: 𝑅2 → 𝑅 is some smooth function
whose values are known to us only at the points
of a finite set P ⊂ 𝑅2 . We would like to
interpolate f over the whole polygon conv(P).
• We don’t know how f looks like outside P, but
still we want a reasonable interpolation rule that
provides a nice smooth function with the given
values at P.
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Interpolation
• Multidimensional interpolation is an extensive
semiempirical discipline, which we do not
seriously consider here; we explain only one
elegant method based on Voronoi diagrams. To
compute the interpolated value at a point 𝑥
∈ conv(𝑃), we construct the Voronoi diagram of
P, and we overlay it with the Voronoi diagram of
P U {x}.
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Interpolation
• The region of the new point x cuts off portions
of the regions of some of the old points. Let 𝜔𝑝
be the area of the part of reg(p) in the Voronoi
diagram of P that belongs to reg(x) after
inserting x. The interpolated value f(x) is
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Relation of Voronoi diagrams to
convex polyhedra.
• We now show that Voronoi diagrams in Rd
correspond to certain convex polyhedra in
𝑅𝑑+1 .
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• First we define the unit paraboloid in 𝑅𝑑+1 :
• For d = 1, U is a parabola in the plane.
• In the sequel, let us imagine the space 𝑅𝑑 as
the hyperplane 𝑥𝑑+1 = 0 in 𝑅𝑑+1 . For a point 𝑝 =
(𝑝1 , … , 𝑝𝑑 ) ∈𝑅𝑑 , let e(𝑝) denote the hyperplane
in 𝑅𝑑+1 with equation
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• Geometrically, e(𝑝) is the hyperplane tangent to
the paraboloid U at the point u(p) = (𝑝1 , … , 𝑝𝑑 ,
𝑝12 + … + 𝑝𝑑2 ) lying vertically above p.
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Proposition
• Let p, x ∈ Rd be points and let u(x) be the point
of U vertically above x. Then u(x) lies above the
hyperplane e(p) or on it, and the vertical
distance of u(x) to e(p) is 𝛿 2 , where 𝛿 =
dist(x,p).
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Proof:
• We just substitute into the equations of U and of
e(p). The 𝑥𝑑+1 -coordinate of u(x) is 𝑥12 + … +𝑥𝑑2 ,
while the 𝑥𝑑+1 -coordinate of the point of e(p)
above x is 2𝑝1 𝑥1 + • • • + 2𝑝𝑑 𝑥𝑑 - 𝑝12 - … - 𝑝𝑑2
The difference is (𝑥1 −𝑝1 ) 2 + ...+(𝑥𝑑 −𝑝𝑑 ) 2 = δ2
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• Let ε(𝑝) denote the half-space lying above the
hyperplane e(𝑝). Consider an n-point set 𝑃
⊂ 𝑅𝑑 . As we saw x ϵ reg(𝑝) holds if and only if
e(𝑝) is vertically closest to U at x among all
e(𝑞), 𝑞 ∈ 𝑃.
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Corollary.
• The Voronoi diagram of P is the vertical
projection of the facets of the polyhedron
𝑝∈𝑃 𝜀(𝑃) onto the hyperplane 𝑥𝑑+1 =0.
• Here is an illustration for a planar Voronoi
diagram:
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The farthest-point Voronoi
diagram.
• The projection of the H-polyhedron 𝑝∈𝑃 𝜀(𝑝)𝑜𝑝 ,
where 𝛾 𝑜𝑝 denotes the half-space opposite to γ,
forms the farthest-neighbor Voronoi diagram, in
which each point 𝑝 ∈ P is assigned the regions
of points for which it is the farthest point. It can
be shown that all nonempty regions of this
diagram are unbounded and they correspond
precisely to the points appearing on the surface
of conv(P).
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