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)? Cake 4/24/2018 2

Riddle: Fair cake-cutting Cake with a hole! Still one cut Cake Cake 4/24/2018 3

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(n 2 ) What is the lower bound of the running time? Worst case: O(n 2 ) Is this optimal? 4/24/2018 7

Plane-sweep Algorithm 4/24/2018 8

Plane-sweep Simple Impl. Input 𝑀 = 𝑚 𝑗 𝑚 𝑗 = 𝑚 𝑗 . 𝑞 1 , 𝑚 𝑗 . 𝑞 2 Prepare a sorted list of all y coordinates S={} For each stop with a corresponding line 𝑚 𝑗 If top point Compare 𝑚 𝑗 to each 𝑡 ∈ 𝑇 Insert 𝑚 𝑗 to S If end point Remove 𝑚 𝑗 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 Sweep line state (in sorted order) 4/26/2018 28

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=S i ) checkIntersection(S i-1 , S i ) checkIntersection(S i , S i+1 ) Add the end point of p.l to P If p is the bottom point Remove p.l from S (p.l=S i before removal) checkIntersection(S i-1 ,S i ) If p is an interior point Report p as an intersection Find p.l in S (p.l=S i ) Swap(S i ,S i+1 ) checkIntersection(S i-1 , S i ) checkIntersection(S i , S i+1 ) 4/26/2018 30

Check Intersection(l 1 , l 2 ) If l 1 does not intersect l 2 then return Compute the intersection p i of l 1 and l 2 If p i .y is above the sweep line then return If 𝑞 𝑗 ∈ 𝑄 then return Insert p i 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 If two points have the same y-coordinate, sort l 1 them by the x-coordinate l 2 Starting point l 3 4/26/2018 33

Corner Case 2: Three Intersecting Lines l 1 l 2 l 3 Allow the event point to store a list of all intersecting line segments When processed, reverse the order of all the lines 4/26/2018 34

Rectangle Intersection Given a set of rectangles, find all intersections 5/3/2018 35

Example r1 r2 r3 r4 5/3/2018 36

Example r1 r2 r3 r4 5/3/2018 37

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(n 2 ) Is it optimal? 5/3/2018 39

Simple Plane-sweep Algo. r1 r2 r3 r4 5/3/2018 40

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 X-center 5/3/2018 43

A Simpler Interval Tree Store the intervals in a BST ordered by i.x min Augment the BST with the value x max which stores the maximum value of all the intervals in the subtree 5/3/2018 44

Augmented BST Credit: https://en.wikipedia.org/wiki/Interval_tree 5/3/2018 45

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 p6 A general polygon Corners might also contain p7 p4 holes, e.g., a grass area p5 with a lake inside For simplicity, we will only p3 deal with solid polygons p2 with no holes p8 p1 Edge or Segment 5/3/2018 47

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 overlapping 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: Polygon boundaries intersect 1. Polygons are disjoint 2. One polygon is contained in the other polygon 3. 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 overlap 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 P Q 5/3/2018 51

Case 2: Contained No intersection points Q If 𝑄 ⊂ 𝑅 , then any corner of P is ∈ 𝑅 If 𝑅 ⊂ 𝑄 , then any corner of Q is ∈ 𝑄 P The intersection is the contained polygon 5/3/2018 65

Case 3: Disjoint No intersection points Q P Neither 𝑄 ⊂ 𝑅 nor 𝑅 ⊂ 𝑄 The intersection is empty 5/3/2018 66

Special Case: Convex Polygons P Q 5/3/2018 67

Convex Polygon Overlaps Right-left Right-right Left-right left-left 5/3/2018 68

Left-right Split 5/3/2018 69

Left-right Overlap Test Compare the end point of one line segment to the other line segment using Half space L R If the end point of R is to the right of the segment in L ➔ Skip R. Why? 5/3/2018 70

Left-right Overlap Test Compare the end point of one line segment to the other line segment using Half space L R 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? 5/3/2018 71

Left-right Overlap Test Compare the end point of one line segment to the other line segment using Half space L R If the end point of L is to the left of the segment in R ➔ Skip L 5/3/2018 72

Left-right Overlap Test Compare the end point of one line segment to the other line segment using Half space L R Is the end point of L to the left of R? No! Do the line segments intersect? No! Switch to R 5/3/2018 73

Left-right Overlap Test Compare the end point of one line segment to the other line segment using Half space L R Is the end point of R to the right of L? Yes! Skip R 5/3/2018 74

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

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