Arrangements and Duality
Sanjay Sthapit
Comp290
10/6/98
Contents
• Duality
• Arrangements
• Applications
Duality
• Duality is the
association of a point
on the primal plane
with a line on the dual
plane, and vice versa.
Duality Transforms
•
–
–
–
Some duality transforms :Slope:
y = mx + b <=> p:(m, b)
Polar:
ax + by = 1 <=> p:(a, b)
Parabolic:
y = 2ax - b <=> p:(a, b)
Duality Properties
• * is its own inverse: (x*)* = x
• Point p lies on line L iff point L* lies on line
p*
• Lines L1 and L2 intersect at a point p iff line
p* passes thru L1* and L2*
• If point p lies above line L, then line p* lies
below point L* and vice-versa
Geometric Interpretation of Duality
• Let U = x2/2
• If p lies on U, then
line p* is the tangent to
U at p
• If q lies directly above
or below p, then q* is
the line thru q and
parallel to p
Duality for Other Objects
• For line segments, the
dual of a line segment
s between points p and
q is the double wedge
between lines p* and
q* on the dual plane.
Why Study Duality?
• Looking at things on the dual plane
provides new perspectives.
• For problems dealing with points, the
arrangement of their lines on the dual plane
(whose structure is more apparent) can be
studied to provide solutions.
Arrangements
• An arrangement is the
subdivision of a plane
(and the consequent
vertices, edges, and
faces induced) by a set
of lines on it.
Complexity of an Arrangement
• The complexity of an
arrangement is defined as
the total number of
vertices, edges, and faces
of the arrangement.
• An arrangement is simple
if it does not contain
parallel lines, and 3 or
more lines with a common
intersection point
Properties of Arrangements
• For a set L of n lines on a plane and their
arrangement A(L),
– number of vertices in A(L) <= n(n - 1)/2
– number of edges in A(L) <= n2
– number of faces in A(L) <= n2/2 + n/2 + 1
(Equality holds for the above iff A(L) is
simple)
Properties of Arrangements
• For a set L of n lines on a plane and their
arrangement A(L),
– number of vertices in A(L) <= n(n - 1)/2
• Proof :
Any pair of lines gives a maximum of 1
vertex.
Properties of Arrangements
• For a set L of n lines on a plane and their
arrangement A(L),
– number of edges in A(L) <= n2
• Proof :
Max number of edges for each line is (n - 1)
+ 1. Therefore max number of edges is n2.
Properties of Arrangements
• Proof :
Add an extra vertex at infinity
and join all line open-ended
edges to this vertex.
Let nv = # of vertices, ne = # of
edges, and nf = # of faces.
Then by Euler’s formula for
connected graphs,
nv - ne + nf = 2
=> nf = 2 - (nv + 1) + ne
= 2 - (n(n-1)/2 + 1) + n2
= n2/2 + n/2 + 1
Properties of Arrangements
• Zone theorem: The
complexity of the zone
of a line in an
arrangement of n lines
is O(n).
(The zone of a line l in
an arrangement is the
set of faces whose
closure intersects l)
Properties of Arrangements
• Zone theorem proof for n
lines:
Proof by induction.
Trivial for n = 1.
Assume true for n-1
lines.
Adding another line adds
at most 4 left edges.
QED
Constructing an Arrangement
• Doubly-connected edge lists are well suited
to represent arrangements.
Doubly-Connected Edge Lists
• For a subdivision of a plane by a set of lines
on it,
A doubly-connected edge list consists of
three collections of records - for one for
faces, one for half-edges, and one for
vertices.
Doubly-Connected Edge Lists
• Half-edges : An edge
E connecting p and q
is thought of as 2
distinct half-edges one from p to q (e),
and the other its twin
from q to p (e`). Thus
a half edge bounds
only 1 face.
Doubly-Connected Edge Lists
• Vertex record :
for a vertex v, it stores
– coordinates
– a pointer to an
arbitrary half-edge
with v as its origin
Doubly-Connected Edge Lists
• Face record :
for a face f, it stores
– a pointer to some edge
on its boundary
Doubly-Connected Edge Lists
• Half-edge record :
for an edge e, it stores
pointers to
– its origin
– its twin
– the face it bounds
– next edge in face
– previous edge in face
Constructing an Arrangement
• Construct a bounding box B that
contains all vertices of An.
• Incrementally add each line li to
Ai-1 :
– find leftmost intersection e with B
– split face bounded by e
– move on to next intersected face
Constructing an Arrangement
• Splitting a face f intersected by l
with leftmost intersected edge e:
– find the edge e` where l leaves f
and its twin
– create 2 new records for new faces
f` and f`` created by l
– create a new vertex record for
vertex v` where l intersects e`
– create 2 new records for half edges
created by v`
– delete records for e` and f
– move to face bounded by twin(e`)
Constructing an Arrangement
ConstructArrangement(L) :
1. Compute bounding box B(L) that contains all vertices of A(L) in its
interior.
2. Construct a doubly-connected edge list for subdivision induced by
B(L)
3. For i = 1 to n do
4. Find edge on B(L) that contains the leftmost intersection of li and
Ai-1
5. f = bounded face incident to e
6. While f is inside B(L) do
7.
Split f, and set f to be next intersected face
Constructing an Arrangement
Running time for ConstructArrangement(L) :
1. Compute bounding box B(L) that contains all vertices of A(L) in its
interior.
Compute all intersections of pairs of lines and
choose leftmost, rightmost, top and bottom
intersection points for bounding box.
Step 1 can be done in O(n2) time.
Constructing an Arrangement
Running time for ConstructArrangement(L) :
2. Construct a doubly-connected edge list for subdivision induced by
B(L)
Step 2 takes constant time.
Constructing an Arrangement
Running time for ConstructArrangement(L) :
4.
Find edge on B(L) that contains the leftmost intersection of li and
Ai-1
Step 4 takes O(n) time.
Constructing an Arrangement
Running time for ConstructArrangement(L) :
6.
7.
While f is inside B(L) do
Split f, and set f to be next intersected face
Splitting a face f is linear in complexity f.
By the Zone theorem, the number of faces a line
intersects is O(n).
Step 6 and 7 can be done in O(n) time.
Constructing an Arrangement
Running time for ConstructArrangement(L) :
O(n2) + O(1) + n ( O(n) + O(n) )
= O(n2)
Constructing a 1D Vornoi Diagram
For each point x, construct an
arrangement of parabolic
duals for (x, x2).
Project the intersections of
these duals onto the x-axis.
This is the Vornoi diagram.
( The process generalizes to
higher order diagrams)
Computing Discrepancy
Discrepancy is the difference
between percentage of
actual area covered and
percentage of sample hits
in a pixel.
Construct an arrangement A
of the duals of the sample
points and the dividing
edge e in the pixel.
Count the number of lines
above and below e*.
Other Applications
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Computing minimum area triangles
Ray-shooting
Visibility graphs
Hidden surface removal
Ham-sandwich cuts
Motion planning
Etc.