Computational Geometry. Geometry Cross Product Convex Hull Problem Sweep Line Algorithm

Transcription

1 GEOMETRY COMP 321 McGill University These slides are mainly compiled from the following resources. - Professor Jaehyun Park slides CS 97SI - Top-coder tutorials. - Programming Challenges books.

2 Computational Geometry Geometry Cross Product Convex Hull Problem Sweep Line Algorithm

3 Geometry Basics - Line A Line can be described with mathematical equation: y = mx + c or ax + bx + c = 0. The y = mx + c equation involves gradient / slope m. Note: Be careful with vertical lines with infinite slope. Usually, we treat the vertical lines separately in the solution code (example of the special cases in geometry problems). A Line Segment is a line with two end points with finite length.

4 Geometry Basics - Circles

5 Geometry Basics - Circles In a 2-D Cartesian coordinate system, the Circle centered at (a, b) with radius r is the set of all points (x, y) such that (x a) 2 + (y b) 2 = r 2. The constant π is the Ratio of any circle s circumference to its diameter in the Euclidean space. To avoid precision error, the safest value for programming contest is pi = 2 X acos(0.0), unless if this constant is defined in the problem description! The Circumference c of a circle with a Diameter d is c = π X d where d = 2 X r. The length of an Arc of a circle with a circumference c and an angle α (in degrees) is (α/360.0) x c

6 Geometry Basics - Circles The length of a Chord of a circle with a radius r and an angle α (in degrees) can be obtained with the Law of Cosines: 2r 2 X(1 cos(α)) The Area A of a circle with a radius r is A = π X r 2 The area of a Sector of a circle with an area A and an angle α (in degrees) is (α /360.0) X A The area of a Segment of a circle can be found by subtracting the area of the corresponding Sector with the area of an Isosceles Triangle with sides: r, r, and Chordlength.

7 Geometry Basics - Triangles

8 Geometry Basics - Triangles A Triangle is a polygon with three vertices and three edges. Equilateral Triangle, all three edges have the same length and all inside/interior angles are 60 degrees; Isosceles Triangle, two edges have the same length; Scalene Triangle, no edges have the same length; Right Triangle, one of its interior angle is 90 degrees (or a right angle). The Area A of triangle with base b and height h is A = 0.5 X b X h The Perimeter p of a triangle with 3 sides: a, b, and c is p = a + b + c. The Heron s Formula: The area A of a triangle with 3 sides: a, b, c, is A = sqrt(s X (s a) X (s b) X (s c)), where s = 0.5 X p (the Semi-Perimeter of the triangle).

9 Geometry Basics - Triangles The radius r of the Triangle s Inner Circle with area A and the semi-perimeter s is r = A/s. The radius R of the Triangle s Outer Circle with 3 sides: a, b, c and area A is R =a X b X c/(4 X A). Also please check Law of Cosines. Law of Sines Pythagorean Theorem. Pythagorean Triple.

10 Geometry Basics - Quadrilateral It is a polygon with four edges (and four vertices).

11 Geometry Basics - Rectangles A Rectangle is a polygon with four edges, four vertices, and four right angles. The Area A of a rectangle with width w and height h is A = w X h. The Perimeter p of a rectangle with width w and height h is p = 2 X (w + h). A Square is a special case of rectangle where w = h.

12 Geometry Basics - Polygons A Polygon is a plane figure that is bounded by a closed path or circuit composed of a finite sequence of straight line segments. A polygon is said to be Convex if any line segment drawn inside the polygon does not intersect any edge of the polygon. Otherwise, the polygon is called Concave. The area A of an n-sided polygon (either convex or concave) with n pairs of vertex coordinates given in some order (clockwise or counter-clockwise) is:

13 Geometry Basics - Polygons Testing if a polygon is convex (or concave) is easy with a quite robust geometric predicate test called CCW (Counter Clockwise). This test takes in 3 points p, q, r in a plane and determine if the sequence p q r is a left turn. Or in other words: p q r is counter-clockwise

14 CCW We ll 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

15 Counterclockwise Predicate Define ccw(a,b,c) = (B A) x (C A) = (b x a x )(c y a y ) (b y a y )(c x a x ).

16 Segment-Segment Intersection Test Given two segments AB and CD Want to determine if they intersect properly: two segments meet at a single point that are strictly inside both segments

17 Segment-Segment Intersection Test Assume that the segments intersect From A s point of view, looking straight to B, C and D must lie on different sides. Holds true for the other segment as well The intersection exists and is proper if: ccw(a,b,c) X ccw(a,b,d) < 0 and ccw(c,d,a) X ccw(c,d,b) < 0

18 Non-proper Intersections We need more special cases to consider! e.g., If ccw(a,b,c), ccw(a,b,d), ccw(c,d,a), ccw(c,d,b) are all zeros, then two segments are collinear Very careful implementation is required

19 Convex Hull Problem Given n points on the plane, find the smallest convex polygon that contains all the given points For simplicity, assume that no three points are collinear

20 Convex Hull Problem: Simple Algorithm AB is an edge of the convex hull iff ccw(a,b,c) have the same sign for all other points C This gives us a simple algorithm For each A and B: If ccw(a,b,c) > 0 for all C A,B: Record the edge A -> B Walk along the recorded edges to recover the convex hull

21 Convex Hull Problem: 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 x coordinates Sort the points in increasing order of y/x Increasing order of angle, whatever you like to call it Incrementally construct the convex hull using a stack

22 Convex Hull Problem: Graham Scan Points are numbered in increasing order of y/x

23 Convex Hull Problem: Graham Scan Add the first two points in the chain

24 Convex Hull Problem: Graham Scan Adding point 3 causes a concave corner 1-2-3: remove 2

25 Convex Hull Problem: Graham Scan That s better...

26 Convex Hull Problem: Graham Scan Adding point 4 to the chain causes a problem: remove 3

27 Convex Hull Problem: Graham Scan Continue adding points...

28 Convex Hull Problem: Graham Scan Continue adding points...

29 Convex Hull Problem: Graham Scan Continue adding points...

30 Convex Hull Problem: Graham Scan Bad corner!

31 Convex Hull Problem: Graham Scan Bad corner again!

32 Convex Hull Problem: Graham Scan Continue adding points...

33 Convex Hull Problem: Graham Scan Continue adding points...

34 Convex Hull Problem: Graham Scan Continue adding points...

35 Convex Hull Problem: Graham Scan Done!

36 Convex Hull Problem: Graham Scan Set the leftmost point as (0, 0), and sort the rest of the points in increasing order of y/x Initialize stack S For i = 1,..., n: Let A be the second topmost element of S, B be the topmost element of S, and C be the i-th point If ccw(a,b,c) < 0, pop S and go back Push C to S Points in S form the convex hull

37 Sweep Line Algorithm 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 can t simulate a continuous process, (e.g. sweeping a line) so we define events that causes certain changes in our data structure And process the events in the order of occurrence We ll cover one sweep line algorithm

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49 Sweep Line Algorithm 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 is the hardest step There is an easy O(n) method for this step

50 Sweep Line Algorithm Sweep line algorithm is a generic concept Come up with the right set of events and data structures for each problem