09 computational geometry

8/9/2019 09 Computational Geometry

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CS 97SI: INTRODUCTION TOPROGRAMMING CONTESTS

Jaehyun Park


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Computational Geometry

Cross Product

Segment-Segment Intersection

Convex Hull Problem

Graham Scan

Sweep Line Algorithm

Intersecting Half-planes

A Useful Note on Binary/Ternary Search


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Cross Product

Arguably the most important operation in 2D

geometry

Well use it all the time

Applications:

Determining the (signed) area of a triangle

Testing if three points are collinear Determining the orientation of three points

Testing if two line segments intersect


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Cross Product

Define ccw , , =

ccw , , > 0

is a left turn

ccw , , < 0

is a right turn


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Segment-Segment Intersection Test

Given two segmentsand

Want to determine if they intersect properly: two

segments meet at a single point that are strictly

inside both segments

ProperNot Proper

Non-intersecting


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Assume that the segments intersect

Froms point of view, looking straight to , and

must lie on different sides

Holds true for the other segment as well

The intersection exists and is proper if:

ccw , , ccw , , < 0

AND ccw , , ccw , , < 0


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Determining non-proper intersections

We need more special cases to consider!

e.g. If ccw ,, , ccw ,, , ccw ,, ,

ccw ,, are all zeros, then two segments arecollinear

Very careful implementation is required


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Convex Hull Problem

Given points on the plane, find the smallest

convex polygon that contains all the given points

For simplicity, assume that no three points are collinear


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Simple algorithm

is an edge of the convex hull iff ccw ,,

have the same sign for all other given points

This gives us a simple algorithm

For eachand :

If ccw , , > 0for all , :

Record the edge Walk along the recorded edges to recover the

convex hull


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Faster Algorithm: Graham Scan

We know that the leftmost given point has to be in

the convex hull

We assume that there is a unique leftmost point

Make the leftmost point the origin So that all other points have positive coordinates

Sort the points in increasing order of /

Increasing order of angle, whatever you like to call it Incrementally construct the convex hull using a stack


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Incremental Construction

We maintain a convex chainof the given points

For each , we do the following:

Append point to the current chain

If the new point causes a concave corner, remove the

bad vertex from the chain that causes it

Repeat until the new chain becomes convex


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Example

Points are numbered in increasing order of /

new origin

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89


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2

Example

Add the first two points in the chain

new origin

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2

Example

Adding point 3 causes a concave corner 1-2-3

Remove 2

new origin

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Example

Thats better

new origin

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Example

Continue adding points

new origin

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Example


new origin

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Example


new origin

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Example

Bad corner!

new origin

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Example

Bad corner again!

new origin

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Example


new origin

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Example


new origin

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Example

Done!

new origin

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Pseudocode

Set the leftmost point 0,0 , and sort the rest of the

points in increasing order of /

Initialize stack

For = 1 :

Letbe the second topmost element of , be the

topmost element of , be the th point

If ccw , , < 0, pop and go back Push to

Points in form the convex hull


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A problem solving strategy for geometry problems

The main idea is to maintain a line (with some

auxiliary data structure) that sweeps through the

entire plane and solve the problem locally We cant simulate a continuous process, (e.g.

sweeping a line) so we define eventsthat causes

certain changes in our data structure And process the events in the order of occurrence

Well cover one sweep line algorithm


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Problem: Given axis-aligned rectangles, find the

area of the union of them

We will sweep the plane from left to right

Events: left and right edges of the rectangles

The main idea is to maintain the set of active

rectangles in order

It suffices to store the -coordinates of the rectangles


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Example

Sweep

line


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Example

Blue interval is added

to the data structure


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Example

green area = blue segment length

(distance that the sweep line traveled)


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Example


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Example


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Example


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Example


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Example


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Example


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Example

One of the segments is removed


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Example


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Pseudopseudocode

If the sweep line hits the left edge of a rectangle

Insert it to the data structure

Right edge?

Remove it

Move to the next event, and add the area(s) of the

green rectangle(s)

Finding the length of the union of the blue segments isthe hardest step

There is an easy method for this step


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Notes on Sweep Line Algorithms

Sweep line algorithm is a generic concept

Come up with the right set of events and data structures

for each problem

Exercise problems Finding the perimeter of the union of rectangles

Finding all intersections of line segments in

+ log time


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Representing a half-plane: + + 0

The intersection of half-planes is a convex area

If the intersection is bounded, it gives a convex polygon

Given half-planes, how do we compute theintersection of them?

i.e. Find vertices of the convex area

There is an easy algorithm and a hard log one

We will cover the easy one


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The intersection of half-planes can be unbounded

But usually, we are given limits on the min/max values

of the coordinates

Add four half-planes , , , (for large ) to ensure that the intersection is

bounded

Time complexity:

Pretty slow, but easy to code


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Ternary Search

Another useful method in many geometry problems

Finds the minimum point of a convex function

Not exactly convex, but lets use this word anyway

Initialize the search interval [,]

Until becomes small:

= + ( )/3, 2= ( )/3

If (2), then set to 2Otherwise, set to

09 computational geometry

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