Week 3 Polygon Triangulation

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Kellie Warner
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1 1 Week 3 Polygon Triangulation

2 2 Polygon A polygon is the region of a plane bounded by a fnite collection of line segments forming a simple closed curve. The intersection of adjacent segments is the shared end point Nonadjacent segments do not intersect

3 3 Polygon cover How many points do you need to cover a given polygon with n vertices? How many points are sufcient to cover any polygon with n vertices?

4 4 Fisk's Sufciency Triangulate, 3-color, guards at one color 4 3 5

5 5 Triangulation Theorem Every polygon with n vertices may be partitioned into triangles by the addition of (zero or more) diagonals

6 6 Number of Diagonals & Triangles Every triangulation of a polygon of n vertices has n-3 diagonals and n-2 triangles Holds for n = 3 For n 4, Vertices Diagonals Triangles n1 n1-3 n -2 ( 0 diagonals, 1 triangle ) + n2 +1+ n2-3 + n -2 =n+2 =n-3 =n-2

7 7 Sum of Internal Angles The sum of the internal angles of a polygon of n vertices is (n-2)π π π π π π

8 8 Triangulation Dual The dual T of a triangulation is a tree, with each node of degree at most 3

9 9 Two Ears Theorem Three consecutive vertices a,b,c of a polygon form an ear if ac is a diagonal Theorem: Every polygon of n 4 vertices has at least two nonoverlapping ears. Proof: The triangulation dual has at least two leaves! Each leaf is an ear

10 10 3-coloring a Triangulation Graph The triangulation graph of a polygon P of n vertices can be 3-colored Proof by induction A triangle is 3-colorable Assume every triangulation graph of n-1 vertices is 3-colorable By Two Ears Theorem, there is an ear abc Remove b Rest is 3-colorable Color b according to a and c

11 11 Area of a Polygon Area of a triangle (base * height) / 2 Why? height base

12 12 Area of a Triangle What if you are only given the coordinates of vertices? (x3,y3) (x1,y1) (x2,y2)

13 13 Cross Product of Two Vectors Cross product of vectors A and B gives the area of the paralelogram determined by A and B. B Area = AxB A

14 14 Area of a Triangle A(abc) = A x B / 2 length of product = B=c-a a (a0,a1,a2) A0 A1 A2 c (c0,c1,c2) A=b-a b (b0,b1,b2) B0 X B1 B2 = A1B2 - A2B1 A2B0 - A0B2 A0B1 - A1B0

15 15 Area of a Triangle In 2D, A0 A1 0 B0 X B = A0B1 - A1B0 = A0B1 - A1B0 Then, area of the triangle becomes (A0B1 A1B0) / 2

16 16 Area of a Triangle (A0B1 A1B0) / 2 Replace A = b a, B = c a, A0 = b0 a0 A1 = b1 a1 B0 = c0 a0 B1 = c1 a1 B=c-a a (a0,a1,a2) c (c0,c1,c2) A=b-a ½ * [(b0 a0)(c1 a1) (b1 a1)(c0 a0)] b (b0,b1,b2)

17 17 Example c (6,5) a (3,1) A(abc) b (8,-1) = [(8 3)(5 1) (6 3)(-1 1)] / 2 = [5.4 3.(-2)] / 2 = 26 / 2 = 13

18 18 Area of a Convex Polygon Area =?

19 19 Area of a Convex Polygon Triangulate!

20 20 Area of a Convex Quadrilateral c d a b A(Q) = A(a,b,c) + A(a,c,d) or A(Q) = A(d,a,b) + A(d,b,c)

21 21 Area of Convex Quadrilateral A(Q) = A(a,b,c) + A(a,c,d) = ½ * [(b0 a0)(c1 a1) (b1 a1)(c0 a0)] + ½ * [(c0 a0)(d1 a1) (c1 a1)(d0 a0)] = ½ * [b0c1 b0a1 a0c1 + a0a1 b1c0 + b1a0 + a1c0 a0a1 + c0d1 c0a1 a0d1 + a0a1 c1d0 + c1a0 + a1d0 a0a1] = ½ * [a0b1 b0a1 + b0c1 b1c0 + c0d1 d0c1 + d0a1 a0d1]

22 22 Area of a Convex Polygon ½ * [a0b1 b0a1 + b0c1 b1c0 + c0d1 d0c1 + d0a1 a0d1] n 1 1 A P = xi yi 1 yi xi 1 2 i =0 (x4,y4) (x6,y6) (x7,y7) (x3,y3) (x8,y8) (x0,y0) (x2,y2) (x1,y1)

23 23 Area of a Nonconvex Quadrilateral c c d + b - + a d + a Area = A(abd) + A(bcd) Area = A(abc) + A(cda) b

24 24 Area From an Arbitrary Center An alternative way of computing the area Area from an arbitrary center Why do we need this? Convex polygons are easily triangulated as a fan Not true for nonconvex polygons! Fan triangulation does not work in nonconvex We need a new approach to compute the area of nonconvex polygons

25 25 Area From an Arbitrary Center c a T b p CLAIM: A(T) = A(p,a,b) + A(p,b,c) + A(p,c,a)

26 26 Area From an Arbitrary Center c a A1 T A2 A4 A3 b p A(T) = A(p,a,b) + A(p,b,c) + A(p,c,a) A(T) = -A3 + -A4 + A2 + A3 + A1 + A4 = A1 + A2

27 27 Area From an Arbitrary Center c a T Another example A1 b A A2 3 p A(T) = A(p,b,c) + A(p,c,a) + A(p,a,b) A(T) = -A3 + A1 + A2 + A3 + -A2 = A1

28 28 LEMMA If T(abc) is a triangle with vertices ordered counterclockwise, and p is an arbitrary point in the plane, then A T = A p, a, b A p, b, c A p, c, a

29 29 THEOREM: Area of Polygon Let a polygon P have vertices v0, v1, v2,, vn-1 labeled counterclockwise, and let p be an arbitrary point in the plane. Then, A P = A p, v0, v1 A p, v1, v 2 A p, v2, v3... A p, vn 2, v n 1 A p, vn 1, v0 n 1 A P = 1 xi yi 1 yi xi 1 2 i =0 n 1 1 A P = xi xi 1 yi 1 yi 2 i=0

30 30 Area of Polygon PROOF: by induction Base case: triangle (already shown) Assume truth for all polygons with n-1 vertices Let P be a polygon with n vertices By Two Ears Theorem, P has at least two ears W.l.o.g., let E=(vn-2, vn-1, v0) be one ear Let Pn-1 = P E A(Pn-1) = A(p,v0,v1) + + A(p,vn-3,vn-2) + A(p,vn-2,v0) A(E) = A(p,vn-2,vn-1) + A(p,vn-1,v0) + A(p,v0,vn-2) A(P) = A(Pn-1) + A(E)

31 31 Computing a Triangulation So far, we have seen properties of triangulations where triangulations are used But, how can we compute a triangulation?

polygon P Try all possible line segments

all edges e, not adjacent to a or b e and

32 32 Computing a Triangulation Algorithm: Triangulation Find a diagonal d of polygon P Try all possible line segments ab we know that there is at least one For all edges e, not adjacent to a or b e and ab do not intersect ab is internal to P Divide the polygon into 2 smaller polygons p1 and p2 at d Triangulate p1 and p2 recursively

33 33 Segment Intersection Given two line segments ab and cd, do they intersect? b c a d

34 34 Segment Intersection Given two line segments ab and cd, do they intersect? Algorithm: Segment Intersection Compute the line equations Solve the corresponding linear equation If the intersection is on both segments Then the segments do intersect Not very pleasing to implement Vertical lines division by zero Parallel lines division by zero

35 35 Segment Intersection First a helper algorithm: Left Given a point and a line segment, is the point to the left of the segment? c b a Algorithm: Left ( Point a, Point b, Point c ) If A(abc) > 0, then return TRUE else return FALSE

36 36 Segment Intersection Algorithm: Better Segment Intersection Given line segments ab and cd Handle special cases Return ( Left(a,b,c) XOR Left(a,b,d) ) && ( Left(c,d,a) XOR Left(c,d,b) ) b c a d

37 37 Computing a Triangulation Algorithm: Triangulation Find a diagonal d of polygon P Try all possible line segments ab For all edges e, not adjacent to a or b Better Segment Intersection(ab, e) ab is internal to P Divide the polygon into 2 smaller polygons p1 and p2 at d Triangulate p1 and p2 recursively

38 38 Internality Given segment ab, is it internal to P? Claim: ab is internal, if it is in the cone a- a a+ b- b b + a- a b a+

39 39 Internality If a is convex, a- is to the LEFT of ab a+ is to the RIGHT of ab a- a b a+

40 40 Internality If a is refex, Consider the cone a+ a a- If ab is not in a+ a a-, then it is in a- a a+ b aa a+

41 41 Computing a Triangulation Algorithm: Triangulation Find a diagonal d of polygon P Try all possible line segments ab For all edges e, not adjacent to a or b Better Segment Intersection(ab, e) In Cone(a, b) && InCone(b, a) Divide the polygon into 2 smaller polygons p1 and p2 at d Triangulate p1 and p2 recursively

42 42 Computing a Triangulation O(n2) diagonal candidates each requires O(n) checks to fnd whether it is a diagonal or not we need to fnd (n-3) diagonals overall running time: O(n4)!!! very slow!

43 43 Triangulation by Ear Removal First improvement: We know that there is an internal diagonal Moreover, we know that there is an ear And it is seperated by a diagonal There are O(n) ear candidates Overall running time: O(n3) Faster But still very slow!

44 44 Triangulation by Ear Removal Second Improvement Initialize the ear tip status of each vertex Requires O(n2) time While n>3 Locate an ear tip v2 Output diagonal v1v3 Delete v2 Update the ear tip status of v1 and v3 Requires O(n) time for each ear O(n2) overall

45 45 How It Works v1 v2 ear v0 v3 v4 v5

46 46 QUIZ TIME Prepare for the quiz

Week 2 Polygon Triangulation

Week 2 Polygon Triangulation What is a polygon? Last week A polygonal chain is a connected series of line segments A closed polygonal chain is a polygonal chain, such that there is also a line segment