Computational Geometry [csci 3250]
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1 Computational Geometry [csci 3250] Laura Toma Bowdoin College
2 Polygon Triangulation
3 Polygon Triangulation The problem: Triangulate a given polygon. (output a set of diagonals that partition the polygon into triangles).
4 Polygon Triangulation The problem: Triangulate a given polygon. (output a set of diagonals that partition the polygon into triangles).
5 Definitions Given a simple polygon P A diagonal is a segment between 2 non-adjacent vertices that lies entirely within the interior of the polygon. A ear with tip vi is a set of 3 consecutive vertices vi-1, vi, vi+1 if vi-1vi+1 is a diagonal. Put differently, vi is an ear tip if the vertex right before it and the vertex right after it are visible to each other
6 Known Results Theorem: Any simple polygon must have a convex vertex. Theorem: Any simple polygon with n>3 vertices contains (at least) a diagonal. Theorem: Any polygon can be triangulated by adding diagonals. Theorem: Any simple polygon has at least two ears. All triangulations of a polygon of n vertices have n-2 triangles and n-3 diagonals.
7 Algorithms Is pipj a diagonal of P? Find a diagonal of P Is vi the tip of an ear? Find all ears ==> Triangulate by finding ears
8 Polygon triangulation: First steps Algorithm 1: Triangulation by identifying ears Idea: Find an ear, output the diagonal, delete the ear tip, repeat. Analysis: checking whether a vertex is ear tip or not: O(n) checking all vertices: O(n 2 ) overall O(n 3 ) Algorithm 2: Triangulation by finding diagonals Idea: Find a diagonal, output it, recurse. A diagonal can be found in O(n) time (using the proof that a diagonal exists) O(n 2 )
9 Polygon triangulation: First steps Algorithm 3: Triangulation by identifying ears in O(n 2 ) Find an ear, output the diagonal, delete the ear tip, repeat. Avoid recomputing ear status for all vertices every time When you remove a ear tip from the polygon, which vertices might change their ear status?
10 History of Polygon Triangulation Early algorithms: O(n 4 ), O(n 3 ), O(n 2 ) Several O(n lg n) algorithms known Many papers with improved bounds 1991: Bernard Chazelle (Princeton) gave an O(n) algorithm practical not practical Ridiculously complicated, not practical O(1) people actually understand it (and I m not one of them) No algorithm is known that is practical enough to run faster than the O( n lg n) algorithms OPEN problem A practical algorithm that s better than O(n lg n).
11 An O(n lg n) Polygon Triangulation Algorithm Ingredients Consider the special case of triangulating a monotone/unimonotone polygon Convert an arbitrary polygon into monotone/unimonotone polygons
12 Monotone chains A polygonal chain is x-monotone if any line perpendicular to x-axis intersects it in one point (one connected component).
13 Monotone chains A polygonal chain is x-monotone if any line perpendicular to x-axis intersects it in one point (one connected component). one point one connected component
14 Monotone chains Not x-monotone
15 Monotone chains Claim: Let u and v be the points on the chain with min/max x-coordinate. The vertices on the boundary of an x-monotone chain, going from u to v, are in x-order. u x-monotone v
16 Monotone chains x-monotone not x-monotone As you travel along this chain, your x- coordinate is staying the same or increasing
17 Monotone chains A polygonal chain is y-monotone if any line perpendicular to y-axis intersects it in one point (one connected component). y-monotone not y-monotone
18 Monotone chains A polygonal chain is L-monotone if any line perpendicular to line L intersects it in one point (one connected component). L y-monotone not L-monotone
19 Monotone polygons A polygon is x-monotone if its boundary can be split into two x-monotone chains.
20 Monotone polygons A polygon is x-monotone if its boundary can be split into two x-monotone chains. xmin The vertices on each chain are sorted w.r.t. x-axis. xmax
21 Monotone polygons x-monotone y-monotone
22 Monotone Mountains A polygon is an x-monotone mountain if it is monotone and one of the two chains is a single segment.
23 Monotone Mountains A polygon is an x-monotone mountain if it is monotone and one of the two chains is a single segment. xmin Both endpoints have to be convex. xmax
24 Monotone mountains are easy to triangulate! Class work: come up with an algorithm and analyze it.
25 Monotone mountains are easy to triangulate!
26 Monotone mountains are easy to triangulate!
27 Monotone mountains are easy to triangulate! Yes!
28 Monotone mountains are easy to triangulate!
29 Monotone mountains are easy to triangulate!
30 Monotone mountains are easy to triangulate! NO!
31 Monotone mountains are easy to triangulate!
32 Monotone mountains are easy to triangulate!
33 Monotone mountains are easy to triangulate! Yes!
34 Monotone mountains are easy to triangulate!
35 Monotone mountains are easy to triangulate! Yes!
36 Monotone mountains are easy to triangulate!
37 Monotone mountains are easy to triangulate!
38 Monotone mountains are easy to triangulate!
39 Monotone mountains are easy to triangulate! No!
40 Monotone mountains are easy to triangulate!
41 Monotone mountains are easy to triangulate!
42 Monotone mountains are easy to triangulate!
43 Monotone mountains are easy to triangulate!
44 Monotone mountains are easy to triangulate!
45 Monotone mountains are easy to triangulate!
46 Monotone mountains are easy to triangulate!
47 Monotone mountains are easy to triangulate!
48 Monotone mountains are easy to triangulate!
49 Monotone mountains are easy to triangulate!
50 Monotone mountains are easy to triangulate!
51 Monotone mountains are easy to triangulate!
52 Monotone mountains are easy to triangulate!
53 Monotone mountains are easy to triangulate!
54 Monotone mountains are easy to triangulate!
55 Monotone mountains are easy to triangulate!
56 Monotone mountains are easy to triangulate!
57 Monotone mountains are easy to triangulate!
58 Monotone mountains are easy to triangulate!
59 Monotone mountains are easy to triangulate!
60 Monotone mountains are easy to triangulate! Analysis: O(n) time
61 Triangulating Monotone Polygons Similar idea, pick the next vertex in x-order. It can be on upper or lower chain. O(n) time
62 Towards an O(n lg n) Polygon Triangulation Algorithm? polygon P Partition into monotone polygons??? Triangulate monotone polygons triangulated P polygon P?? Partition into unimonotone polygons Triangulate unimonotone polygons triangulated P??
63 How can we partition a polygon into (uni)monotone pieces?
64 Intuition x-monotone not x-monotone What makes a polygon not monotone?
65 Intuition x-monotone not x-monotone What makes a polygon not monotone?
66 Intuition Cusp: a reflex vertex v such that the vertices before and after are both smaller or both larger than v (in terms of x-coords). x-monotone not x-monotone What makes a polygon not monotone?
67 Intuition Cusp: a reflex vertex v such that the vertices before and after are both smaller or both larger than v (in terms of x-coords). x-monotone not x-monotone Theorem: If a polygon has no cusps, then it s monotone. Proof: Intuitively clear, but proof a little tedious. Maybe later..
68 We ll get rid of cusps using a trapezoidation of P.
69 Trapezoid partitions Shoot vertical rays If polygon is above vertex, shoot vertical ray up until reaches boundary If f polygon is below vertex, shoot down If polygon is above and below vertex, shoot both up and down
70 Trapezoid partitions Shoot vertical rays If polygon is above vertex, shoot vertical ray up until reaches boundary If f polygon is below vertex, shoot down If polygon is above and below vertex, shoot both up and down
71 Trapezoid partitions Properties Each polygon in the partition is a trapezoid, because: It has one or two threads as sides. If it has two, then they must both hit the same edge above, and the same edge below. At most one thread through each vertex => O(n) threads => O(n) trapezoids
72 Trapezoid partitions Each trapezoid has precisely two vertices of the polygon, one on the left and one on the right. They can be on the top, bottom or middle of the trapezoid.
73 Trapezoid partitions Each trapezoid has precisely two vertices of the polygon, one on the left and one on the right. They can be on the top, bottom or middle of the trapezoid.
74 Trapezoid partitions Each trapezoid has precisely two vertices of the polygon, one on the left and one on the right. They can be on the top, bottom or middle of the trapezoid.
75 Trapezoid partitions Each trapezoid has precisely two vertices of the polygon, one on the left and one on the right. They can be on the top, bottom or middle of the trapezoid.
76 Diagonals In each trapezoid: if its two vertices are not on the same edge, they define a diagonal.
77 Diagonals In each trapezoid: if its two vertices are not on the same edge, they define a diagonal.
78 Diagonals In each trapezoid: if its two vertices are not on the same edge, they define a diagonal.
79 We can use the trapezoid partition of P to split the cusps
80 Removing cusps
81 Removing cusps 1. Identify cusp vertices
82 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. For each cusp vertex, add diagonal in trapezoid before/after the cusp
83 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. For each cusp vertex, add diagonal in trapezoid before/after the cusp
84 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. For each cusp vertex, add diagonal in trapezoid before/after the cusp
85 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. For each cusp vertex, add diagonal in trapezoid before/after the cusp
86 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. For each cusp vertex, add diagonal in trapezoid before/after the cusp
87 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. For each cusp vertex, add diagonal in trapezoid before/after the cusp This creates a partition of P. Claim: The resulting polygons have no cusps and thus are monotone (by theorem).
88
89
90 Removing cusps Another example
91 Removing cusps 1. Identify cusp vertices
92 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P
93 Removing cusps 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. Add obvious diagonal before/after each cusp
94 Removing cusps This partitions the polygon into monotone pieces.
95 Partition P into monotone polygons 1. Identify cusp vertices 2. Compute a trapezoid partition of P 3. Add obvious diagonal before/after each cusp x
96 Towards an O(n lg n) Polygon Triangulation Algorithm polygon P Given a trapezoid partition of P, we can triangulate it in O(n) time.?? Trapezoid partition?? O(n) add cusps diagonals O(n) Triangulate monotone polygons triangulated P
97 Towards an O(n lg n) Polygon Triangulation Algorithm polygon P Given a trapezoid partition of P, we can triangulate it in O(n) time.?? Trapezoid partition?? O(n) add cusps diagonals O(n) Triangulate monotone polygons triangulated P OR add all diagonals O(n) Triangulate monotone mountains O(n) triangulated P
98 Partitioning into monotone mountains 1. Compute a trapezoid partition of P 2. Output all diagonals.
99 Partitioning into monotone mountains 1. Compute a trapezoid partition of P 2. Output all diagonals. Claim: All diagonals partition the polygon into monotone mountains.
100 Partitioning into monotone mountains 1. Compute a trapezoid partition of P 2. Output all diagonals. Claim: The diagonals partition the polygon into monotone mountains.
101 Claim: The diagonals partition the polygon into monotone mountains. Proof: Each polygon is monotone One of the chains must be a segment (because if it had another point, that point would generate a diagonal)
102 polygon P?? Trapezoid partition?? How do we get the trapezoid partition?
103 Computing the trapezoid partition
104 Computing the trapezoid partition in O(n lg n) Plane sweep
105 Computing the trapezoid partition in O(n lg n) Plane sweep
106 Computing the trapezoid partition in O(n lg n) Plane sweep
107 Computing the trapezoid partition in O(n lg n) Plane sweep
108 Computing the trapezoid partition in O(n lg n) Plane sweep
109 Computing the trapezoid partition in O(n lg n) Plane sweep
110 Computing the trapezoid partition in O(n lg n) Plane sweep
111 Computing the trapezoid partition in O(n lg n) Plane sweep
112 Computing the trapezoid partition in O(n lg n) Plane sweep
113 Computing the trapezoid partition in O(n lg n) Plane sweep Events: polygon vertices Status structure: edges that intersect current sweep line, in y-order Events: How do we determine the trapezoids?
114 Computing the trapezoid partition in O(n lg n) Algorithm
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