COT 5520 Computational Geometry

1
Definitions
Coordinate systems and dimensions The objects
considered in Computational Geometry are
points, lines, line segments, polygons,
polyhedron, hyper-rctacgles etc. A coordinate
system provides a means to specify positions or
points in space. The Cartesian coordinate system
labels a d-dimensional space with d mutually
perpendicular (orthogonal) coordinate axes, one
per dimension. d-dimensional space
(d-space) Notation d defined as number of
dimensions of space or a geometric object. As a
prefix on the name of an object, d- denotes the
number of dimensions of the object, e.g.
d-rectangle or 2-rectangle. We will most often
work in d 2 (plane), which is the
default, sometimes in d 1 or d gt 2.
d 2
d 3 Right-handed coordinate system
y
x
0
z
2
Definitions
Point Object with d dimensions and 0
extent. Location in d-space. Given as an ordered
sequence of d coordinates. d 1 (x) or x d
2 (x, y) d 3 (x, y, z) d ? 4 (x1, x2, ..., xd)
or (x0, x1, ..., xd-1)
Line Infinite straight 1-dimensional set of
points, determined by two points p0, p1 ? p0 ?
p1.
3
Definitions
Segment Finite 1-dimensional subset of a
line, determined by two endpoints p0, p1.
Ray Infinite 1-dimensional subset of a
line determined by two points p0, p1 ? p0 ?
p1, where one point is denoted as the endpoint.
4
Preliminaries
Point-Line classification We now consider the
geometric primitive operation of classifying a
point w.r.t. a line (both in the plane). A
directed line segment partitions the plane into
7 non-overlapping regions. The possibilities are
shown below. The problem, given p0, p1, and p2,
is to determine which region p2 lies in.
beyond
p1
left
terminus
between
p0
origin
right
behind
5
Preliminaries
Parametric equation of a line We use the
following equation of a line line ?(p0) (1
- ??(p1) , where ????? (real numbers) where p0
and p1 as usual are the points determining the
line. p0 (x0, y0) p1 (x1, y1) Substituting
gives ?(x0, y0) (1 - ??(x1, y1) Multiplying
through gives the coordinates ?x0 (1 - ??x1,
?y0 (1 - ??y1 Work out an example with points
(4,3) and (7,5) as the two end points with
values of ? as 0, 1, 0.5, 2 and -3. For example,
when ? equal to 2 , (x,y)(2 x0 x1 , 2 y0 -
y1 ) (1,1).
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Line Segment A line segment is a closed subset
of a line contained between two points which
are called the end points. The subset is closed
in the sense that it includes the end points.
The equation of the line segment is the same as
the parametric equation of a line with
the restriction that ? has the value
0?????1. This is also called the convex
combination of the two end points.
7
Explicit Form of Line Equation y mx c
mslopetan? where ? is the angle made by the
line with positive x-axis cintercept of the
line with the y-axis. Vertical line with xk
cannot be represented since these lines have
infinite slopes. Expressed in terms of the
coordinates of two points on the line (x1,y1)
and (x2,y2), we can write y(y2-y1)/(x2-x1) x
(y1x2 -y2x1)/(x2-x1) Or (x2-x1)(y-y1)
(y2-y1)(x-x1) This is called an implicit form of
an equation of a line. Now, x2x1 and y2?y1 means
a vertical line with equation xx1.
y
?? mtan?
c
x
8
In general, a line equation in a plane can be
specified as Ax By C 0 where A, B and C
are constants. A vertical line is simply a line
with B0. Note the coefficients are not unique
for a given constant k, kA, kB and kC will give
the same line. In general, in a d-dimension,
given a set of k points p1,p2,..,pk, the set of
points p ?1 p1 ?2 p2. ?k pk such that
the ?-coefficients are real and their sum equals
1, is called an affine combination of the given
set of k points . For k2, this set defines a
straight line through two points for k3, it is
a plane and for higher k value it is called a
hyperplane.
9
Definitions
Plane Infinite 2-dimensional subset of
space, determined by three points p0, p1, p2, ?
p0 ? p1 ? p2 ? p0.
Interval Pair of coordinate values. Often treated
like a segment on a coordinate axis. l,
r closed x ? l ? x ? r is within interval (l,
r) open x ? l lt x lt r is within interval l,
r) half open x ? l ? x lt r. (l, r half open x
? l lt x ? r is within interval
r
l
closed
0
r
l
open
0
10
Definitions
Rectangle Quadrilateral with opposite sides
parallel and only right angles.
Rectilinear or axis-parallel rectangle Cartesian
product of d intervals.
2-rectangle or simply rectangle
11
Definitions
Dimensional prefixes on geometric object names No
prefix d- means usual or expected number of
dimensions for the object. rectangle
2-rectangle cube 3-cube Prefix hyper- means
d is unspecified and it may be more than the
usual number of dimensions for that
object. hyper-rectangle d-rectangle, d
unspecified d-dimensional rectilinear
hyper-rectangle
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Definition ---------------------------------------
------------------ Polygon In two dimensions,
it is defined by a finite set of line segments
such that every point (vertex) is shared by two
line segments (edges) and no subset of the
segments has the same property. Example
Individually each collection of line segments is
a Polygon, but taken together it is not a polygon
since it violates the subset property. Simple
Polygon A polygon is simple if there are no
points between non-consecutive line segments,
that is, vertices are the only intersection
points. In the above diagram, the first two are
simple polygons But the third one is not. The
following is not even a Polygon
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Note a non-simple polygon may be drawn
alternately as a planar graph but as drawn it
is still not a simple polygon
4
1
2
3
3
2
1
5
4
5
Non-simple polygon
Corresponding planar graph
In computational geometry, the relative
geometrical positions and dimensions matter. The
edges do not represent an abstract relation
as in graph theory.
14
Definitions
Polygons ORourke, pp. 1-2 A polygon is the
region of a plane bounded by a finite set
of segments forming a simple closed curve. (Note
that we are working in d 2 by definition.) Let
v0, v1, ..., vN-1 be N points in the plane the
points are called vertices. Let e0 v0v1, e1
v1v2, ..., eN-1 vN-1v0 be N segments connecting
the points the segments are called edges. The
edges bound a polygon iff the intersection of
each pair of edges adjacent in the ordering is
the single vertex shared between them ei ? ei1
vi1 for i 0, N - 1
v4
v5
v3
N 8
v6
v2
v7
v0
v1
Vertices are numbered in counterclockwise
sequence by convention.
15
Definitions
The segments are connected end-to-end in a sort
of a curve, they form a cycle and hence it is
closed, and the closed curve is simple since
non-adjacent segments do not intersect. We also
have this famous theorem Interior and
exterior Jordan curve theorem. Every simple
closed plane curve divides the plane into two
parts.
Exterior
Polygon interior ? boundary If we are
interested in just the interior or just the
boundary, they will be referred to as such. (Same
as true for other similar objects, e.g.,
rectangle.). The two parts are called the
exterior (unbounded) and the interior (bounded).
Thus the polygon P is the region of a plane
bounded by a finite collection of line segments
forming a simple closed curve. Sometimes a
polygon is considered to be just the segments
bounding the region and not the region itself.
This is defined as ?P. Thus, ?P ? P.
16
Definitions
Polygon
Not vertices
Simple polygon A polygon is simple iff
non-adjacent edges do not intersect. ei ? ej ?
for all 0 ? j, i ? N - 1 and j ? i 1.
17
Definitions
Convex polygon A polygon is convex if and only
if for any two points in the polygon (interior ?
boundary) the segment connecting the points
is entirely within the polygon.
convex
not convex
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CONVEX SET
Let p and q be two arbitrary points in a
d-dimensional Euclidean space belonging to a set
of points C. Then C is said to be convex if for
all pairs (p,q) in C, the set of points
?p (1- ?)q ? C for 0lt ? lt1 That is, if
two pints p and q belong to C, then the set of
points on the line segments connecting p and q
also belong to C. When d2, the points belong to
a convex polgon.
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Planar Graph
A graph G(V,E) is palnar if it can be embedded in
a plane without crossings. ( Kuratowskis
Theorem no subgraph or homomorphic to the
subgraphs
A straight line planar embedding of a
planar graph determines a partition of the plane
called planar subdivisions or a map. Let v
number of vertices, e number of edges and
fnumber of faces. Theorem ( Euler) v - e
f 2 Proof A smple polygon has always ve
and f2 ( interior and exterior).
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If the interior is further subdivided by a
chord, it creates one extra face. V remains same,
e becomes e1 and f becomes f1. So the equation
remains valid. If a chain is used with t new
vertices and necessarily with t1 edges, we have
v becomes vt, e becomes et1 and f becomes
f1. So, the Eulers formula still remains valid.
chain
chord
It can also be shown that for any planar graph,
e lt 3v -6. Using Eulers formula, we then have
flt 2/3 e and f lt 2v-4, giving the upper bounds
on f in terms of e and v, respectively. Furtherm
ore, if for each vertex vgt 3 (
polyhedron) then, 3vlt2e which yields elt 3f-6
and vlt2f-4. All these relations are linear.
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POLYHEDRON
In 3-d Euclidean space, a polyhedron is
defined to be a finite set of planar polygons
such that every edge of the polygon is shared by
exactly one other neighboring polygon and no
subset of polygons has the same property ( to
avoid union of polygons). Edges and vertices
have usual meaning. The polygons are called the
facets of the polyhedron. A polyhedron is
simple if there is no pair of non-adjacent
facets sharing a point. A simple polyhedron
partitions the 3-d space into two disjoint
domains - the interior and the exterior. A simple
polyhedron is convex if its interior is
convex. The surface of a polyhedron is
isomorphic to a planar subdivision ( on a
sphere). Thus the numbers v, e and f of a simple
polyhedron obey Eulers formula.
22
Definitions
Vertices A polygon vertex is convex if its
interior angle ??????????? It is reflex if its
interior angle gt ?????????
reflex
convex
In a convex polygon, all the vertices are convex.
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Computational Model ------------------------------
----------------------------------- Human brains
perform complex geometrical computations using a
model not yet understood. We certainly dont use
floating point numbers! Neural networks attempt
to mimic very poorly the actions of the brain
cells but in the process again introduces
numbers as weights! Nature computes with
molecules and atoms, not numbers. The great
enigma is that human brain invented mathematics
which is supposed to be independent of nature.
Ironically, mathematics was invented to explain
nature! We are stuck with the RAM ( Random
Access Memory) model in our computation but we
modify the model by assuming that each cell in
the memory can hold a number of infinite
precision. This is a very powerful model since it
isolates geometry from the details of
computation. Thus, we can say that the
computation of intersection of two points, which
may not be identified by any rational numbers in
some cases, takes constant amount of time.
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--------------------------------------------------
--------- In practice, we do indeed take
constant time for such a computation by
introducing computational errors, which may
multiply, propagate, proliferate ultimately
creating a non-sense computation. For example
Draw a complete graph on 5 vertices as on a
plane such that the boundary is a convex 5-gon.
You will see an inner pentagon. Repeat this
process. If computers have to carry on this
process, after several iterations, all the points
will become fuzzy and inaccurate. The study of
errors in computational geometry and solid
modeling and its impact on the models that you
create for simulation is a research topic that
has attracted many reserachers.
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Preliminaries
Example problem RANGE SEARCHINGGiven N points
in the plane, how many lie in a given rectangle
with sides parallel to the coordinate axes? That
is, how many points (x, y) satisfy lx ? x ? rx,
ly ? y ? ry, for given lx? rx, ly? ry?
ry
ly
lx
rx
The range searching problem is (informally)
defined above. In this example, the instance of
the range searching problem is a particular set
of points and a particular range. In an instance
like this, the points are the data set.
26
Preliminaries
Algorithms and models of computation We are
interested in efficient algorithm for geometric
problems. Efficiency is evaluated in terms of
computational cost, given as a function of the
size of the instance of the problem. By
convention, notation N denotes input instance
size. To determine the computational cost of an
algorithm, we must know what primitive operations
are available and what they cost. This is a
model of computation. Turing machine too
primitive C language on Unix workstation too
specific We will use a highly abstract model, the
familiar random-access machine from Aho,1974.
We use a slightly modified real RAM model. These
operations are available at unit cost 1.
arithmetic , -, , / 2. comparisons lt, ?, ,
?, ?, gt 3. memory access 4. analytic functions
root, trig, exp, log Numbers are assumed to be
real, with infinite precision. This is
considerably abstracted in a real machine 1.
different operations vary widely in cost 2.
floating point precision is limited You must deal
with the latter in your code. The abstract model
of computation allows us to focus on
how computation cost changes w.r.t. input
instance size.
27
Preliminaries
Order notation We are interested in the amount of
time and memory used by algorithms, as a function
of the input instance size N. Worst case or
Upper bound O(f(N)) denotes the set of all
functions g(N) such that there exist positive
constants C and N0 with g(N) ? Cf(N) for all N
??N0. Best case or Lower Bound ?(f(N)) denotes
the set of all functions g(N) such that there
exist positive constants C and N0 with g(N) ?
Cf(N) for all N ??N0. Optimal case Or Optimal
Bound ?(f(N)) denotes the set of all functions
g(N) such that there exist positive constants C1,
C2, and N0 with C1f(N) ? g(N) ??C2f(N) for all N
??N0. Notes These notations denote sets f(N)
? O(log N), not f(N) O(log N). Larger terms
dominate the order, e.g., (4N 20 log N 100) ?
O(N) We will use the notation for both time and
memory. Most of our attention will be towards
worst case.
28
Preliminaries
Example analysis SEGMENT INTERSECTION
COUNTING INSTANCE Set S s1, s2, ..., sN of
line segments in the plane. QUESTION Count the
number of intersections of segments in S.
29
Preliminaries
Example analysis SEGMENT INTERSECTION
COUNTING INSTANCE Set S s1, s2, ..., sN of
line segments in the plane. QUESTION Count the
number of intersections of segments in
S. 1 procedure SegmentIntersectionCounting(S) 2 b
egin 3 count 0 3 for i 1 to N 4 for j 1
to N 5 if i ? j and si ? sj ? ? 6 count
count 1 7 endif 8 endfor 9 endfor 10 pr
int count 11 end Storage O(N), for set S Time
O(N2), for nested loops Notice the assumed
primitive operation segment intersection. What
geometric operations are primitive?
30
Preliminaries
Algorithmic complexity measures Preprocessing.
Time spent organizing the data set, usually
into some data structure. Less important than
query and storage. Query. Time spent producing
the answer for a query relative to the data
set. Storage. Memory required for static and
dynamic data structures used by the query
algorithm. Single shot vs. repetitive-mode Sing
le shot. Given a single data set and a single
query, produce ananswer one time. Almost always
best handled by scan of data set no
preprocessing, query O(N), storage
O(N). Repetitive-mode. Given a single data set
and a sequence of queries, produce the answer for
each query relative to the data set. Here we are
willing to spend time preprocessing to enable
query time better than O(N).
31
Preliminaries
Counting vs. reporting Counting. Determine the
number of objects in the data set that satisfy
the query. Reporting. Report (list, identify)
the objects that satisfy the query. For example,
consider the standard range search
problem RANGE SEARCHING. INSTANCE Set S
p1, p2, ..., pN, pi (xi, yi) of points in
the plane, and rectangle R lx, rx ? ly, ry
in the plane. QUESTION (counting) How many
points of S are within R? QUESTION (reporting)
Which points of S are within R?
32
Preliminaries
Output sensitive or report-mode algorithms The
time complexity of algorithms is often expressed
as a function of input data set size, e.g. O(N
log N). Reportingproblems can have query time
complexity that is output sensitive. Output-sensi
tive example INTERVAL ENCLOSURE INSTANCE Set S
x1, x2, ..., xN of points on the number line
(x-axis), and an interval Q l, r. QUESTION
Which points of S are within Q, i.e. l ?
xi???r? Naive repetitive mode algorithm and
analysis. Preprocessing 1. Sort S into an array
A. O(N log N) Query 1. Binary search A for xi
? l. O(log N) 2. Binary search A for xj???r.
O(log N) 3. Report points from xi to xj in A.
? There can be O(N) points from l to r ? step
3 is O(N) ???query is O(N). Without reporting,
query is O(log N), with reporting O(N). Time
complexity of this type is usually written O(log
N K), where N is input size and K is output
size.
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Average Complexity observed complexity
in practice. Space or Storage Complexity. Pre-pr
ocessing Cost trade-off between space and time
complexity with or without pre-processing
. Amortized Cost average over expensive
and inexpensive operations. Normalization It
will sometimes be useful to have available
normalized values for coordinates. For a
coordinate value x, its normalized x coordinate
is in 1, N, assigned in order of increasing x
coordinate, relative to the set from which the
coordinates are/will be drawn. Normalization
usually implies an O(N log N) sort in
preprocessing and O(log N) normalization search
in query.
34
Preliminaries
Segment Tree A segment tree is a rooted binary
tree that stores data intervals onthe real line
whose extremes (endpoints) belong to a fixed set
of N abscissae (x-values). It is the set of
x-values from which the endpoints are chosen that
is fixed, not the intervals themselves. The tree
structure in which the intervals are stored is
defined for a scope interval l, r. For a given
l, r there is exactly one segment tree
structure. The data intervals are stored within
the fixed tree. We will assume WLOG that the
data interval endpoints have been normalized to
1, N and the tree has been built for scope
interval 1, N. T(l, r) Segment tree over
scope interval l, r. Each node v of T(l, r) is
associated with a scope interval ? l, r. A
node v has these parameters B(v) Beginning of
scope interval associated with this
node. E(v) End of scope interval associated with
this node. Lchild(v) Left subtree T(B(v),
?(B(v) E(v)) / 2?) Rchild(v) Right subtree
T(?(B(v) E(v)) / 2?? E(v)) A(v) List of data
intervals stored (allocated) to this
node. B(v), E(v)) is the scope interval
associated with node v. It is closed on the left
and open on the right (except nodes on
the rightmost path of T, which are closed on both
ends).
35
Preliminaries
Example The structure of the segment tree
T(1,12) is shown. Each node is labeled with its
associated interval B(v), E(v)).
10,12
1,2)
6,7)
7,9)
9,10)
4,6)
3,4)
2,3)
5,6)
4,5)
The set of intervals B(v), E(v)) or B(v),
E(v) ? v a node of T(l, r)are the standard
intervals of T(l, r). Standard intervals which
are also leaves are elementary intervals. These
are simply i, i 1) for l ? i lt r and r - 1,
r. T(l, r) is balanced, with depth ? log(r - l)
? ? O(log N).
36
Preliminaries
Insertion The segment tree structure supports
insertions and deletions of intervals with
endpoints ? l, l 1, l 2, ..., r, in O(log
N) time per operation. For r - l gt 3, an
arbitrary interval b, e inserted into T(l, r)
willbe partitioned into and allocated as a
collection of standard intervalsof T(l, r).
There will be at most ?log (r - 1) ? ?log (r -
1) - 2???O(log N) standard intervals in the
partition. To insert interval b, e into
segment tree T InsertSegmentTree(b, e,
root(T)) procedure InsertSegmentTree(b, e,
v) begin if (b ? B(v) and E(v) ? e) then add
b, e to A(v) else if (b lt ?(B(v) E(v)) /
2? ) then InsertSegmentTree(b, e,
Lchild(v)) if (?(B(v) E(v)) / 2? lt e)
then InsertSegmentTree(b, e,
Rchild(v)) end end
37
Preliminaries
Example Insertion of 2, 8) into T(1,12)
?(B(v) E(v)) / 2?
6
3
9
7
4
2
10,12
1,2)
7,9)
9,10)
3,4)
8
6
5
8,9)
7
5,6)
4,5)
Interval 2, 8) allocated to nodes 2, 3), 3,
6), 6, 7), 7,8). An underlying assumption here
is that all the intervals inserted and deleted is
closed on the left and open on the right. Thus
if we have to represent 2,8 we will have to
start with 2,9).
38
Preliminaries
Deletion Deletion is symmetric with
insertion. To delete interval b, e from
segment tree T DeleteSegmentTree(b, e,
root(T)) procedure DeleteSegmentTree(b, e,
v) begin if (b ? B(v) and E(v) ? e)
then remove b, e from A(v) else if (b lt
?(B(v) E(v)) / 2? ) then DeleteSegmentTree(b,
e, Lchild(v)) if (?(B(v) E(v)) / 2? lt e)
then DeleteSegmentTree(b, e,
Rchild(v)) end end