EE602 Algorithms GEOMETRIC INTERSECTION CHAPTER 27

Transcription

1 EE602 Algorithms GEOMETRIC INTERSECTION CHAPTER 27

2 The Problem Given a set of N objects, do any two intersect? Objects could be lines, rectangles, circles, polygons, or other geometric objects Simple to evaluate physical objects but takes some effort for a computer Work with only line intersection as base case Consider simple case first and expand implementation

3 Methods Brute force Horizontal and Vertical Lines General Line Intersection

4 Brute Force Check each pair of objects to see if they intersect using ccw Time proportional to O(N 2 ) Not uncommon to deal with very large number of objects

5 Simple case Constrain lines to be horizontal or vertical Apply general strategy for horizontal-vertical case to lines with no constraints Focus on counting problem Algorithms can be extended to return all intersecting pairs

6 Horizontal and Vertical Lines A F B H I C D G E Given a set of N horizontal and vertical lines, do any two intersect?

7 Horizontal and Vertical Lines Properties of line segment defined by P1(x1, y1) and P2(x2, y2): Line segment is vertical if x1 == x2 Line segment is horizontal if y1 == y2 All lines have one of these properties Idea: Sweep scan line from bottom to top Scan line acts like time Line segments must occupy same space in same time to intersect Project Vertical and Horizontal line segments onto scan line Vertical line segments appear as single point Horizontal line segments appear as an interval Intersection found if vertical line segment projection (single point) lies in horizontal line segment (interval) on scan line Reduces problem into one-dimensional range search Does interval contain point?

8 Horizontal and Vertical Lines Build binary search tree (named x-tree) to represent relationships between vertical line segments Node stores lines x coordinates For vertical line segments: If bottom endpoint encountered, add x-coordinate into x- tree If a vertical line is left to the root line, then add it to the left leaf, otherwise add it to right leaf If top endpoint encountered, delete from x-tree For horizontal line segments: Do interval range search using min and max x-coordinates

9 Scan Process F A C H B G D E range I Sort line endpoints by their y-coordinates Each vertical line appears twice in list Each horizontal line appears once For vertical lines Bottom endpoint encountered = insert into tree Top endpoint encountered = delete from tree For horizontal lines Endpoints (L, R) encountered = range search delete delete range delete delete delete delete delete CEDG IBFCHBA IEDHF insert insert insert insert insert insert insert

10 Sweep Scanline CD F HE A F C D B G E H I DB HB HI

11 Y-Tree instead of line sweep Instead of presorting on y-coordinates to get scan order, use binary search tree for initial y sort (called y-tree) y-tree contains all the line endpoints Scan is inorder traversal of y-tree A B D C B E F D C I H E G F H I Figure 27.5 Sorting for scan using the y-tree

12 Generic Program Procedure Intersection (y-tree) Set x-tree null and count 0 Loop If y-tree completes traverse, then return count Set focus Suitable-pick (y-tree) If it is a horizontal line, then RangeSearch x-tree and count the intersection numbers else Update x-tree End-loop Count: record the intersection number Suitable-pick: pick one line s endpoint from y-tree with left priority RangeSearch: take a range search in x-tree for intersection Update: add line to x-tree when encountering the bottom endpoint; delete line from x- tree when encountering the top endpoint

13 Analysis Property 27.1: All intersections among N horizontal and vertical lines can be found in time proportional to N log N + I, where I is the number of intersections Tree manipulations on average take time proportional to log N Range searching depends on total number of intersections Bad-case: cross-hatch pattern Number of intersections proportional to N 2 If I is known to be large, use brute force (N 2 )/2 instead of (N log N + N 2 )

14 General Line Intersection F A C D B E H I G Given a set of N lines, do any two intersect?

15 General Line Intersection Changes Interval range test no longer sufficient, necessary to explicitly test using the line intersect function Sort on (min) y-coordinate to divide space into strips Use binary search tree (BST) to store order of lines Node stores line Proceed through sorted list [same idea as scan] If bottom endpoint encountered, add line to BST If top endpoint encountered, delete line from BST Building tree not that simple

16 Building The Tree Use a more general ordering relationship Line x is to the right of line y if both endpoints of x are on the same side of y as a point infinitely far to the right Or if y is to the left of x Line x is to the left of line x If both endpoints of x are on the same size of y as a point infinitely far to the left Or if y is to the right of x Line comparison can be implemented using ccw procedure

17 CCW Review Slope Comparison Collinear Cases

18 Line Ordering Example B is to the right of F F is to the left of B H is to the right of B B is to the left of H A E I F C D B H G

19 Building the Tree inserting a line Line X Defined by P0(x0, y0) and P1(x1, y1) Already in tree Line Y Defined by P2(x2, y2) and P3(x3, y3) Inserting into tree Comparison Test If ccw(x.p0, X.P1, Y.P2) == ccw(x.p0, X.P1, Y.P3) == counterclockwise Insert Y as left child of X If ccw(x.p0, X.P1, Y.P2) == ccw(x.p0, X.P1, Y.P3) == clockwise Insert Y as right child of X If ccw(x.p0, X.P1, Y.P2)!= ccw(x.p0, X.P1, Y.P3) Lines must intersect

20 Building the tree comparison operation The ordering relationship between lines for the binary tree is more complicated Comparison operation is not transitive E I A F is the left of B, B is to the left of D, F D but F is not to the left of D C B H G

21 Building the tree tree manipulation Insertion: Starting at root, compare line Each comparison performed during the tree manipulation procedures is a line-intersection test Deletion: Explicitly test the 2 children of the line being deleted, because comparison is not transitive We must test explicitly that comparisons are valid each time we change the tree structure. Default BST assumes transitivity

22 General Line Intersection CD B E I F H A DB G D F G C B H

23 What happens after intersection? A E I The y values is likely to be different between the intersection point and line endpoints F C D B H G D and F should swap places after the point of intersection

24 After Intersection Problem: After a point of intersection, the two lines should swap places Solution 1: use priority queue instead of binary tree for sort on y-coordinate Work the scan line by taking the smallest y-coordinate When intersection is found, add new entries for each line using the intersection point as the lower endpoints Solution 2: remove one of the intersecting lines when intersection found Appropriate if not too many intersections expected All intersecting pairs must involve removed line After scan is complete, use brute-force to enumerate all intersections

25 Analysis All intersections among N lines can be found in time proportional to (N+I) log N I is the number of intersections log N for tree manipulations If there is an intersection, we need to add two new lines endpoint into the priority queue, so we have N+I lines.

26 General geometric shapes Rectangles Defined by four lines Define to the left of and to the right of X is left of Y is right edge of X is left of the left edge of Y Circles Circle defined as center coordinate and radius Use center coordinate to order and compare distance between centers and distance between radii Brute force