# Computational Geometry

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1 Motivation 1

2 Motivation Polygons and visibility Visibility in polygons Triangulation Proof of the Art gallery theorem Two points in a simple polygon can see each other if their connecting line segment is in the polygon 2

3 Art gallery problem Motivation Visibility in polygons Triangulation Proof of the Art gallery theorem Art Gallery Problem: How many cameras are needed to guard a given art gallery so that every point is seen? 3

4 Art gallery problem Motivation Visibility in polygons Triangulation Proof of the Art gallery theorem In geometry terminology: How many points are needed in a simple polygon with n vertices so that every point in the polygon is seen? The optimization problem is computationally difficult Art Gallery Theorem: n/3 cameras are occasionally necessary but always sufficient 4

5 Art gallery problem Motivation Visibility in polygons Triangulation Proof of the Art gallery theorem Art Gallery Theorem: n/3 cameras are occasionally necessary but always sufficient 5

6 Motivation Triangulation, diagonal Visibility in polygons Triangulation Proof of the Art gallery theorem Why are n/3 cameras always enough? Assume polygon P is triangulated: a decomposition of P into disjoint triangles by a maximal set of non-intersecting diagonals Diagonal of P : open line segment that connects two vertices of P and fully lies in the interior of P 6

7 Motivation A triangulation always exists Visibility in polygons Triangulation Proof of the Art gallery theorem Lemma: A simple polygon with n vertices can always be triangulated, and always with n 2 triangles u Proof: Induction on n. If n = 3, it is trivial Assume n > 3. Consider the leftmost vertex v and its two neighbors u and w. Either uw is a diagonal (case 1), or part of the boundary of P is in uvw (case 2) Case 2: choose the vertex t in uvw farthest from the line through u and w, then vt must be a diagonal v v u w w 7

8 Motivation A triangulation always exists Visibility in polygons Triangulation Proof of the Art gallery theorem In case 1, uw cuts the polygon into a triangle and a simple polygon with n 1 vertices, and we apply induction In case 2, vt cuts the polygon into two simple polygons with m and n m + 2 vertices, 3 m n 1, and we also apply induction By induction, the two polygons can be triangulated using m 2 and n m = n m triangles. So the original polygon is triangulated using m 2 + n m = n 2 triangles 8

9 Motivation A 3-coloring always exists Visibility in polygons Triangulation Proof of the Art gallery theorem Observe: the dual graph of a triangulated simple polygon is a tree Dual graph: each face gives a node; two nodes are connected if the faces are adjacent 9

10 Motivation A 3-coloring always exists Visibility in polygons Triangulation Proof of the Art gallery theorem Lemma: The vertices of a triangulated simple polygon can always be 3-colored Proof: Induction on the number of triangles in the triangulation. Base case: True for a triangle Every tree has a leaf, in particular the one that is the dual graph. Remove the corresponding triangle from the triangulated polygon, color its vertices, add the triangle back, and let the extra vertex have the color different from its neighbors 10

11 Motivation A 3-coloring always exists Visibility in polygons Triangulation Proof of the Art gallery theorem 11

12 Motivation n/3 cameras are enough Visibility in polygons Triangulation Proof of the Art gallery theorem For a 3-colored, triangulated simple polygon, one of the color classes is used by at most n/3 colors. Place the cameras at these vertices This argument is called the pigeon-hole principle 12

13 Motivation n/3 cameras are enough Visibility in polygons Triangulation Proof of the Art gallery theorem Question: Why does the proof fail when the polygon has holes? 13

14 Motivation Two-ears for triangulation Using the two-ears theorem: (an ear consists of three consecutive vertices u, v, w where uw is a diagonal) Find an ear, cut it off with a diagonal, triangulate the rest iteratively Question: Why does every simple polygon have an ear? Question: How efficient is this algorithm? 14

15 Overview Motivation A simple polygon is y-monotone iff any horizontal line intersects it in a connected set (or not at all) Use plane sweep to partition the polygon into y-monotone polygons Then triangulate each y-monotone polygon 15

16 Monotone polygons Motivation A y-monotone polygon has a top vertex, a bottom vertex, and two y-monotone chains between top and bottom as its boundary Any simple polygon with one top vertex and one bottom vertex is y-monotone 16

17 Vertex types Motivation What types of vertices does a simple polygon have? start stop split merge regular... imagining a sweep line going top to bottom start merge regular split end 17

18 Sweep ideas Motivation Find diagonals from each merge vertex down, and from each split vertex up A simple polygon with no split or merge vertices can have at most one start and one end vertex, so it is y-monotone 18

19 Sweep ideas Motivation explored explored unexplored unexplored 19

20 Sweep ideas Motivation Where can a diagonal from a split vertex go? Perhaps the upper endpoint of the edge immediately left of the merge vertex? 20

21 Sweep ideas Motivation Where can a diagonal from a split vertex go? Perhaps the upper endpoint of the edge immediately left of the merge vertex? 21

22 Sweep ideas Motivation Where can a diagonal from a split vertex go? Perhaps the upper endpoint of the edge immediately left of the merge vertex? 22

23 Sweep ideas Motivation Where can a diagonal from a split vertex go? Perhaps the last vertex passed in the same component? 23

24 Sweep ideas Motivation Where can a diagonal from a split vertex go? Perhaps the last vertex passed in the same component? 24

25 Sweep ideas Motivation Where can a diagonal from a split vertex go? Perhaps the last vertex passed in the same component? 25

26 Helpers of edges Motivation The helper for an edge e that has the polygon right of it, and a position of the sweep line, is the lowest vertex v above the sweep line such that the horizontal line segment connecting e and v is inside the polygon 26

27 Status of sweep Motivation The status is the set of edges intersecting the sweep line that have the polygon to their right, sorted from left to right, and each with their helper: the last vertex passed in that component 27

28 Motivation Status structure, event list The status structure stores all edges that have the polygon to the right, with their helper, sorted from left to right in the leaves of a balanced binary search tree The events happen only at the vertices: sort them by y-coordinate and put them in a list (or array, or tree) 28

29 Main algorithm Motivation Initialize the event list (all vertices sorted by decreasing y-coordinate) and the status structure (empty) While there are still events in the event list, remove the first (topmost) one and handle it 29

30 Event handling Motivation Start vertex v: Insert the counterclockwise incident edge in T with v as the helper 30

31 Event handling Motivation End vertex v: Delete the clockwise incident edge and its helper from T 31

32 Event handling Motivation Regular vertex v: If the polygon is right of the two incident edges, then replace the upper edge by the lower edge in T, and make v the helper If the polygon is left of the two incident edges, then find the edge e directly left of v, and replace its helper by v 32

33 Event handling Motivation Merge vertex v: Remove the edge clockwise from v from T Find the edge e directly left of v, and replace its helper by v e 33

34 Event handling Motivation Split vertex v: Find the edge e directly left of v, and choose as a diagonal the edge between its helper and v Replace the helper of e by v Insert the edge counterclockwise from v in T, with v as its helper e 34

35 Efficiency Motivation Sorting all events by y-coordinate takes O(n log n) time Every event takes O(log n) time, because it only involves querying, inserting and deleting in T 35

36 Degenerate cases Motivation Question: Which degenerate cases arise in this algorithm? 36

37 Representation Motivation A simple polygon with some diagonals is a subdivision use a DCEL Question: How many diagonals may be chosen to the same vertex? 37

38 More sweeping Motivation With an upward sweep in each subpolygon, we can find a diagonal down from every merge vertex (which is a split vertex for an upward sweep!) This makes all subpolygons y-monotone 38

39 Result Motivation Theorem: A simple polygon with n vertices can be partitioned into y-monotone pieces in O(n log n) time 39

40 Motivation How to triangulate a y-monotone polygon? 40

41 Motivation How to triangulate a y-monotone polygon? 41

42 Motivation How to triangulate a y-monotone polygon? 42

43 Motivation How to triangulate a y-monotone polygon? 43

44 Motivation How to triangulate a y-monotone polygon? 44

45 Motivation How to triangulate a y-monotone polygon? 45

46 Motivation How to triangulate a y-monotone polygon? 46

47 Motivation How to triangulate a y-monotone polygon? 47

48 Motivation How to triangulate a y-monotone polygon? 48

49 Motivation How to triangulate a y-monotone polygon? 49

50 Motivation How to triangulate a y-monotone polygon? 50

51 Motivation How to triangulate a y-monotone polygon? 51

52 Motivation How to triangulate a y-monotone polygon? 52

53 Motivation How to triangulate a y-monotone polygon? 53

54 Motivation How to triangulate a y-monotone polygon? 54

55 Motivation How to triangulate a y-monotone polygon? 55

56 Motivation How to triangulate a y-monotone polygon? 56

57 Motivation How to triangulate a y-monotone polygon? 57

58 The algorithm Motivation Sort the vertices top-to-bottom by a merge of the two chains Initialize a stack. Push the first two vertices Take the next vertex v, and triangulate as much as possible, top-down, while popping the stack Push v onto the stack 58

59 Result Motivation Theorem: A simple polygon with n vertices can be partitioned into y-monotone pieces in O(n log n) time Theorem: A monotone polygon with n vertices can be triangulated O(n) time Can we immediately conclude: A simple polygon with n vertices can be triangulated O(n log n) time??? 59

60 Result Motivation We need to argue that all y-monotone polygons together that we will triangulate have O(n) vertices Initially we had n edges. We add at most n 3 diagonals in the sweeps. These diagonals are used on both sides as edges. So all monotone polygons together have at most 3n 6 edges, and therefore at most 3n 6 vertices Hence we can conclude that triangulating all monotone polygons together takes only O(n) time Theorem: A simple polygon with n vertices can be triangulated O(n log n) time 60

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