CS133 Computational Geometry Intersection Problems 4/24/2018 1 - - PowerPoint PPT Presentation

CS133

Computational Geometry

Intersection Problems

4/24/2018 1

Riddle: Fair Cake-cutting

Using only one cut, how to split the cake into two equal pieces (by area)?

4/24/2018 2

Cake

Riddle: Fair cake-cutting

Cake with a hole! Still one cut

4/24/2018 3

Cake Cake

Line Segment Intersections

Given a set of line segments, each defined by two end points, find all intersecting line segments

4/24/2018 4

Line Segment Intersection

4/24/2018 5

Line Segment Intersection

4/24/2018 6

Naïve Algorithm

Enumerate all possible pairs of lines Test for intersection Running time O(n2) What is the lower bound of the running time? Worst case: O(n2) Is this optimal?

4/24/2018 7

Plane-sweep Algorithm

4/24/2018 8

Plane-sweep Algorithm

4/24/2018 9

Plane-sweep Algorithm

4/24/2018 10

Plane-sweep Algorithm

4/24/2018 11

Plane-sweep Algorithm

4/24/2018 12

Plane-sweep Algorithm

4/24/2018 13

Plane-sweep Algorithm

4/24/2018 14

Plane-sweep Algorithm

4/24/2018 15

Plane-sweep Algorithm

4/24/2018 16

Plane-sweep Algorithm

4/24/2018 17

Plane-sweep Algorithm

4/24/2018 18

Plane-sweep Algorithm

4/24/2018 19

Plane-sweep Algorithm

4/24/2018 20

Plane-sweep Algorithm

4/24/2018 21

Plane-sweep Algorithm

4/24/2018 22

Plane-sweep Algorithm

4/24/2018 23

Plane-sweep Algorithm

4/24/2018 24

Plane-sweep Algorithm

4/24/2018 25

Plane-sweep Simple Impl.

Input 𝑀 = 𝑚𝑗 𝑚𝑗 = 𝑚𝑗. 𝑞1, 𝑚𝑗. 𝑞2 Prepare a sorted list of all y coordinates S={} For each stop with a corresponding line 𝑚𝑗

If top point

Compare 𝑚𝑗 to each 𝑡 ∈ 𝑇 Insert 𝑚𝑗 to S

If end point

Remove 𝑚𝑗 from S

4/24/2018 26

Bentley-Ottmann Algorithm

An improved scan-line algorithm Maintains the state S in a sorted order to speed up checking a line segment against segments in S

4/26/2018 27

Example

4/26/2018 28

Sweep line state (in sorted order)

Algorithm Pseudo code

Create a list of event points P

P is always sorted by the y coordinate

Initialize the state S to an empty list Initialize P with the first point (top point) of each line segment While P is not empty

p  P.pop processEvent(p)

4/26/2018 29

Process Event Point (p)

If p is the top point Add p.l to S at the order p.x (p.l=Si) checkIntersection(Si-1, Si) checkIntersection(Si, Si+1) Add the end point of p.l to P If p is the bottom point Remove p.l from S (p.l=Si before removal) checkIntersection(Si-1,Si) If p is an interior point

Report p as an intersection Find p.l in S (p.l=Si) Swap(Si,Si+1) checkIntersection(Si-1, Si) checkIntersection(Si, Si+1)

4/26/2018 30

Check Intersection(l1, l2)

If l1 does not intersect l2 then return Compute the intersection pi of l1 and l2 If pi.y is above the sweep line then return If 𝑞𝑗 ∈ 𝑄 then return Insert pi into P

4/26/2018 31

Analysis

Initial sort of starting points O(n log n) Number of processed event points (2n+k) For each event point

Remove from P: O(log |P|) Insert or remove from S: O(log |S|) Check intersection with at most two lines: O(1) Insert a new event points: O(log |P|)

Upper limit of |P| = 2n Upper limit of |S| = n Overall running time O((n+k)log n)

4/24/2018 32

Corner Case 1: Horizontal Line

4/26/2018 33

l1 l2 l3 If two points have the same y-coordinate, sort them by the x-coordinate Starting point

Corner Case 2: Three Intersecting Lines

4/26/2018 34

l1 l2 l3 Allow the event point to store a list of all intersecting line segments When processed, reverse the

• rder of all the lines

Rectangle Intersection

Given a set of rectangles, find all intersections

5/3/2018 35

Example

5/3/2018 36

r1 r2 r3 r4

Example

5/3/2018 37

r1 r2 r3 r4

Rectangle Primitives

A rectangle is represented by its two corner points, lower and upper Test if two rectangles overlap

Two rectangles overlap if both their x intervals and y-intervals overlap Intervals overlap [x1,x2], [x3,x4]: x4>=x1 and x2>=x3

R1(x1,y1,x2,y2)×R2(x3,y3,x4,y4) ➔ R3(Max(x1,x3), Max(y1,y3), Min(x2,x4), Min(y2,y4))

5/3/2018 38

Naïve Algorithm

Test all pairs of rectangles and report the intersections Running time O(n2) Is it optimal?

5/3/2018 39

Simple Plane-sweep Algo.

5/3/2018 40

r1 r2 r3 r4

Simple Plane-sweep Algo.

What is the state of the sweep line? What is an event? What processing should be done at each event?

5/3/2018 41

Improved Plane-sweep Algo.

Keep the sweep line state sorted

But how?

Interval tree A variation of BST Stores intervals Supports two operation

Find all intervals that overlap a query point 𝑞 Find all intervals that overlap a query interval 𝑟

5/3/2018 42

Interval Tree Structure

5/3/2018 43

X-center

A Simpler Interval Tree

Store the intervals in a BST ordered by i.xmin Augment the BST with the value xmax which stores the maximum value of all the intervals in the subtree

5/3/2018 44

Augmented BST

5/3/2018 45

Credit: https://en.wikipedia.org/wiki/Interval_tree

Polygon Intersection

Given a set of polygons, find all intersecting polygons

5/3/2018 46

Polygon Representation

A polygon is represented as a sequence of points that form its boundary A general polygon might also contain holes, e.g., a grass area with a lake inside For simplicity, we will only deal with solid polygons with no holes

5/3/2018 47

p1 p2 p3 p4 p5 p6 p7 p8 Corners Edge or Segment

Filter-and-refine Approach

Convert all polygons to rectangles

For each polygon, compute its minimum bounding rectangle (MBR)

Filter: Find all overlapping rectangles Refine: Test the polygons that have

• verlapping MBBs

5/3/2018 48

Filter-and-refine Approach

Filter step: Already studied Refine: How to find the intersection of two polygons? For any two polygons, there are three cases:

1.

Polygon boundaries intersect

2.

Polygons are disjoint

3.

One polygon is contained in the other polygon

5/3/2018 49

Test Case 1: Intersecting

If the boundaries of the two polygons overlap, then there is at least two polygon edges that

• verlap

Naïve solution: Compute all intersections between every pair of edges 𝑃 𝑛 ⋅ 𝑜

Where 𝑛 and 𝑜 are the sizes of the two polygons

We can also use the line-segment sweep-line algorithm

Run in 𝑃 𝑙 + 𝐽 log 𝑙 where 𝑙 = 𝑛 + 𝑜 and 𝐽 is the number of intersections If we only need to test, we can stop at the first intersection

5/3/2018 50

Computing the Intersection

5/3/2018 51

P Q

Computing the Intersection

5/3/2018 52

P Q

Computing the Intersection

5/3/2018 53

P Q

Computing the Intersection

5/3/2018 54

P Q

Computing the Intersection

5/3/2018 55

P Q

Computing the Intersection

5/3/2018 56

P Q

Computing the Intersection

5/3/2018 57

P Q

Computing the Intersection

5/3/2018 58

P Q

Computing the Intersection

5/3/2018 59

P Q

Computing the Intersection

5/3/2018 60

P Q

Computing the Intersection

5/3/2018 61

P Q

Computing the Intersection

5/3/2018 62

P Q

Computing the Intersection

5/3/2018 63

P Q

Computing the Intersection

5/3/2018 64

P Q

Case 2: Contained

5/3/2018 65

P Q No intersection points If 𝑄 ⊂ 𝑅, then any corner

• f P is ∈ 𝑅

If 𝑅 ⊂ 𝑄, then any corner

• f Q is ∈ 𝑄

The intersection is the contained polygon

Case 3: Disjoint

5/3/2018 66

P Q No intersection points Neither 𝑄 ⊂ 𝑅 nor 𝑅 ⊂ 𝑄 The intersection is empty

Special Case: Convex Polygons

5/3/2018 67

P Q

Convex Polygon Overlaps

5/3/2018 68

Right-left Right-right left-left Left-right

Left-right Split

5/3/2018 69

Left-right Overlap Test

5/3/2018 70

Compare the end point of

• ne line segment to the
• ther line segment using

Half space R L If the end point of R is to the right of the segment in L ➔ Skip R. Why?

Left-right Overlap Test

5/3/2018 71

Compare the end point of

• ne line segment to the
• ther line segment using

Half space R L Is the end point of R to the right of the segment in L? No! Do the line segments intersect? No! Switch to L Why?

Left-right Overlap Test

5/3/2018 72

Compare the end point of

• ne line segment to the
• ther line segment using

Half space R L If the end point of L is to the left of the segment in R ➔ Skip L

Left-right Overlap Test

5/3/2018 73

Compare the end point of

• ne line segment to the
• ther line segment using

Half space R L Is the end point of L to the left of R? No! Do the line segments intersect? No! Switch to R

Left-right Overlap Test

5/3/2018 74

Compare the end point of

• ne line segment to the
• ther line segment using

Half space R L Is the end point of R to the right of L? Yes! Skip R

Left-right Overlap Test

5/3/2018 75

Compare the end point of

• ne line segment to the
• ther line segment using

Half space R L Is the end point of R to the right of L? No! Do the line segments intersect? Yes!

Convex Polygon Intersection

If only testing is required, the algorithm can terminate as soon as the first intersection is found If the polygon intersection is needed, the algorithm reports all intersections The algorithm terminates when all segments are inspected Running time: 𝑃 𝑛 + 𝑜 , each iteration skips

• ne segment

5/3/2018 76