Lec 1 Introduction and Convex Hull

8/3/2019 Lec 1 Introduction and Convex Hull

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IntroductionConvex hulls

Computational Geometry

Lecture 1: Introduction and convex hulls

Computational Geometry Lecture 1: Introduction and Convex Hulls


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Geometric objectsGeometric relationsCombinatorial complexityComputational geometry

Geometry: points, lines, ...

Plane (two-dimensional), R2

Space (three-dimensional), R3

Space (higher-dimensional), Rd

A point in the plane, 3-dimensional space, higher-dimensionalspace.p = (px,py), p = (px,py,pz), p = (p1,p2, . . . ,pd)

A line in the plane: y = m x+ c; representation by m and c

A half-plane in the plane: y m x+ c or ym x+ c

Represent vertical lines? Not by m and c . . .



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Geometry: line segments

A line segment pq is defined by itstwo endpoints p and q:( px + (1) qx, py + (1)

qy)where 0 1

Line segments are assumed to beclosed = with endpoints, not open

Two line segments intersect if theyhave some point in common. It is aproper intersection if it is exactly oneinterior point of each line segment



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Polygons: simple or not

A polygon is a connected region ofthe plane bounded by a sequence ofline segments

simple polygon

polygon with holes

convex polygon

non-simple polygon

The line segments of a polygon arecalled its edges, the endpoints ofthose edges are the vertices

Some abuse: polygon is only

boundary, or interior plus boundary

interior

exterior


G i bj


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Other shapes: rectangles, circles, disks

A circle is only the boundary, a diskis the boundary plus the interior

Rectangles, squares, quadrants,slabs, half-lines, wedges, . . .


G t i bj t


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The distance between two geometric objects other than points

usually refers to the minimum distance between two points thatare part of these objects

Question: How can the distance between two line segments berealized?


Geometric objects


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The intersection of two geometricobjects is the set of points (part ofthe plane, space) they have in

common

Question 1: How many intersectionpoints can a line and a circle have?

Question 2: What are the possibleoutcomes of the intersection of arectangle and a quadrant?


Geometric objects


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Question 3: What is the maximumnumber of intersection points of a

line and a simple polygon with 10vertices (trick question)?

Question 4: What is the maximumnumber of intersection points of aline and a simple polygon boundarywith 10 vertices (still a trickquestion)?


Geometric objects


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Description size

A point in the plane can berepresented using two reals

A line in he plane can be representedusing two reals and a Boolean (forexample)

A line segment can be represented bytwo points, so four reals

A circle (or disk) requires three realsto store it (center, radius)

A rectangle requires four reals tostore it

false, m , c

true, . . , c

y = m

x + c

x = c


Geometric objects


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Description size

A simple polygon in the plane can be represented using 2n reals ifit has n vertices (and necessarily, n edges)

A set ofn points requires 2n reals

A set ofn line segments requires 4n reals

A point, line, circle, . . . requires O(1), or constant, storage.

A simple polygon with n vertices requires O(n), or linear, storage


Geometric objects


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jGeometric relationsCombinatorial complexityComputational geometry

Computation time

Any computation (distance, intersection) on two objects ofO(1)description size takes O(1) time!

Question: Suppose that a simple polygon with n vertices is given;the vertices are given in counterclockwise order along theboundary. Give an efficient algorithm to determine all edges thatare intersected by a given line.

How efficient is your algorithm? Why is your algorithm efficient?


Geometric objects


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Geometric relationsCombinatorial complexityComputational geometry

Algorithms, efficiency

Recall from your algorithms and data structures course:

A set ofn real numbers can be sorted in O(n logn) time

A set ofn real numbers can be stored in a data structure that usesO(n) storage and that allows searching, insertion, and deletion inO(logn) time per operation

These are fundamental results in 1-dimensional computationalgeometry!


Geometric objects


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Computational geometry scope

In computational geometry, problems on input with more thanconstant description size are the ones of interest

Computational geometry (theory): Study of geometric problems ongeometric data, and how efficient geometric algorithms that solvethem can be

Computational geometry (practice): Study of geometric problems

that arise in various applications and how geometric algorithms canhelp to solve well-defined versions of such problems


I d iGeometric objectsG i l i


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Computational geometry theory

Computational geometry (theory):Classify abstract geometric problemsinto classes depending on howefficiently they can be solved

smallest enclosing circle

closest pair

any intersection?

find all intersections


I t d tiGeometric objectsG t i l ti


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Computational geometry practice

Application areas that require geometric algorithms are computergraphics, motion planning and robotics, geographic informationsystems, CAD/CAM, statistics, physics simulations, databases,

games, multimedia retrieval, . . .

Computing shadows from virtual light sources

Spatial interpolation from groundwater polution measurements

Computing a collision-free path between obstacles

Computing similarity of two shapes for shape databaseretrieval


IntroductionGeometric objectsGeometric relations


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Computational geometry history

Early 70s: First attention for geometric problems from algorithmsresearchers

1976: First PhD thesis in computational geometry (Michael

Shamos)1985: First Annual ACM Symposium on Computational Geometry.Also: first textbook

1996: CGAL: first serious implementation effort for robust

geometric algorithms

1997: First handbook on computational geometry (second one in2000)


IntroductionConvexityConvex hull


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Convex hullAlgorithm developmentAlgorithm analysis

Convexity

A shape or set is convex if for any

two points that are part of theshape, the whole connecting linesegment is also part of the shape

Question: Which of the following

shapes are convex? Point, linesegment, line, circle, disk, quadrant?




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Convex hull

For any subset of the plane (set ofpoints, rectangle, simple polygon),its convex hull is the smallest convexset that contains that subset




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Convex hull problem

Give an algorithm that computes theconvex hull of any given set ofn

points in the plane efficiently

The input has 2n coordinates, soO(n) size

Question: Why cant we expect todo any better than O(n) time?




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Convex hulls Algorithm developmentAlgorithm analysis

Convex hull problem

Assume the n points are distinct

The output has at least 4 and at most 2n coordinates, so it hasbetween O(1) and O(n) size

The output is a convex polygon so it should be returned as a sortedsequence of the points, counterclockwise along the boundary

Question: Is there any hope of finding an O(n) time algorithm?




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Developing an algorithm

To develop an algorithm, find usefulproperties, make variousobservations, draw many sketches togain insight

Property: The vertices of the convexhull are always points from the input

Consequently, the edges of theconvex hull connect two points of

the input

Property: The supporting line of anyconvex hull edge has all input pointsto one side

p

q

all points lie left of the di-rected line from p to q, if

the edge from p to q is a

CCW convex hull edge




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Algorithm SlowConvexHull(P)

Input. A set P of points in the plane.Output. A list L containing the vertices ofCH(P) in clockwise

order.

1. E /0.2. for all ordered pairs (p,q) PP with p not equal to q3. do valid true4. for all points r P not equal to p or q5. do if r lies left of the directed line from p to q

6. then valid false7. if valid then Add the directed edge pq to E8. From the set E of edges construct a list L of vertices of

CH(P), sorted in clockwise order.




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Question: How must line 5 be interpreted to make the algorithmcorrect?

Question: How efficient is the algorithm?


IntroductionC h ll

ConvexityConvex hullAl i h d l


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Another approach: incremental, from left to right

Lets first compute the upper boundary of the convex hull this way(property: on the upper hull, points appear in x-order)

Main idea: Sort the points from left to right (= by x-coordinate).Then insert the points in this order, and maintain the upper hull so

far


IntroductionC h ll

ConvexityConvex hullAl ith d l t


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Observation: from left toright, there are only right turnson the upper hull



ConvexityConvex hullAlgorithm development


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Initialize by inserting theleftmost two points





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If we add the third point therewill be a right turn at theprevious point, so we add it





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If we add the fourth point weget a left turn at the thirdpoint





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. . . so we remove the thirdpoint from the upper hullwhen we add the fourth





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g pAlgorithm analysis


If we add the fifth point we geta left turn at the fourth point





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g pAlgorithm analysis


. . . so we remove the fourthpoint when we add the fifth





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Algorithm analysis


If we add the sixth point weget a right turn at the fifthpoint, so we just add it





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Algorithm analysis


We also just add the seventhpoint



ConvexityConvex hullAlgorithm developmentAl ith l i


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Algorithm analysis


When adding the eight point. . . we must remove theseventh point



ConvexityConvex hullAlgorithm developmentAlgorithm anal sis


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Algorithm analysis


. . . we must remove theseventh point



ConvexityConvex hullAlgorithm developmentAlgorithm analysis


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Algorithm analysis


. . . and also the sixth point





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Algorithm analysis


. . . and also the fifth point





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Algorithm analysis


After two more steps we get:





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Algorithm analysis

The pseudo-code

Algorithm ConvexHull(P)Input. A set P of points in the plane.Output. A list containing the vertices ofCH(P) in clockwise order.1. Sort the points by x-coordinate, resulting in a sequence

p1, . . . ,pn.

2. Put the points p1 and p2 in a list Lupper, with p1 as the firstpoint.

3. for i 3 to n4. do Append pi to Lupper.

5. while Lupper contains more than two points and thelast three points in Lupper do notmake a right turn

6. do Delete the middle of the last three points fromLupper.





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g y

The pseudo-code

Then we do the same for thelower convex hull, from right

to leftWe remove the first and lastpoints of the lower convex hull

. . . and concatenate the two

lists into one

1, 2, 10, 13, 14

p14, p12, p8, p4, p1





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Algorithm analysis

Algorithm analysis generally has two components:

proof of correctness

efficiency analysis, proof of running time





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Correctness

Are the general observations on which the algorithm is basedcorrect?

Does the algorithm handle degenerate cases correctly?

Here:

Does the sorted order matter if two or more points have thesame x-coordinate?

What happens if there are three or more collinear points, inparticular on the convex hull?





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Efficiency

Identify of each line of pseudo-code how much time it takes, if it isexecuted once (note: operations on a constant number ofconstant-size objects take constant time)

Consider the loop-structure and examine how often each line ofpseudo-code is executed

Sometimes there are global arguments why an algorithm is moreefficient than it seems, at first





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The pseudo-code

Algorithm ConvexHull(P)Input. A set P of points in the plane.Output. A list containing the vertices ofCH(P) in clockwise order.1. Sort the points by x-coordinate, resulting in a sequence

p1, . . . ,pn.

2. Put the points p1 and p2 in a list Lupper, with p1 as the firstpoint.

3. for i 3 to n4. do Append pi to Lupper.5. while L

uppercontains more than two points and the

last three points in Lupper do notmake a right turn

6. do Delete the middle of the last three points fromLupper.





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Efficiency

The sorting step takesO

(n logn

) timeAdding a point takes O(1) time for the adding-part. Removingpoints takes constant time for each removed point. If due to anaddition, k points are removed, the step takes O(1+ k) time

Total time:O(n logn) +

n

i=3

O(1+ ki)

if ki points are removed when adding pi

Since ki = O(n), we get

O(n logn) +n

i=3

O(n) = O(n2)





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Efficiency

Global argument: each point can be removed only once from theupper hull

This gives us the fact:

n

i=3

ki n

Hence,

O(n logn) +n

i=3

O(1+ ki) = O(n logn) +O(n) = O(n logn)





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Final result

The convex hull of a set ofn points in the plane can be computedin O(n logn) time, and this is optimal


Lec 1 Introduction and Convex Hull

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