CG Lecture 3 CG Lecture 3 Motivation: Art gallery problem Motivation: Art gallery problem Definition : two points q and r in a Polygon decomposition simple polygon P can see each R other if the open segment qr lies entirely within P . p 1. Polygon triangulation q • Triangulation theory A point p guards a region • Monotone polygon triangulation R ⊆ P if p sees all q ∈ R r 2. Polygon decomposition into monotone pieces Problem: Given a polygon P , what 3. Trapezoidal decomposition is the minimum number of guards required to guard P , and what are 4. Convex decomposition their locations? 5. Other results 1 2 Simple observations Simple observations Art gallery problem: upper bound Art gallery problem: upper bound • Convex polygon : all points • Theorem : Every simple planar are visible from all other polygon with n vertices has a points � only one guard in triangulation of size n -2 ( proof any location is necessary! later ). convex • Star-shaped polygon: all • n -2 guards suffice for an n -gon: points are visible from any • Subdivide the polygon into n –2 point in the kernel � only triangles (triangulation). one guard located in its • Place one guard in each triangle. star-shaped kernel is necessary. 3 4 Art gallery problem: lower bound Art gallery problem: lower bound Simple polygon triangulation Simple polygon triangulation • There exists a Input : a polygon P polygon with n described by an ordered vertices, for which sequence of vertices ⎣ n /3 ⎦ guards are < v 0 , …v n–1 >. necessary. Output : a partition of P into n –2 non-overlapping Can we improve the upper Can we improve the upper • Therefore, ⎣ n /3 ⎦ triangles and the bound? bound? guards are needed in adjacencies between Yes! In fact, at most ⎣ n /3 ⎦ Yes! In fact, at most the worst case. them. guards are necessary. guards are necessary. 5 6 1
Ideas? Ideas? Simple polygon triangulation: observations Simple polygon triangulation: observations • The triangulation is not unique. One of them suffices. • The triangulation is always possible. • No new vertices are required. • The triangulation adds new edges, called diagonals , between existing vertices. Not all diagonals are valid! Not all diagonals are valid! 7 8 Triangulation theory Diagonals in polygons Diagonals in polygons Triangulation theory • A vertex is convex if its interior angle < π , Proof: let v be a convex vertex and let Case 1 Case 1 otherwise it is concave. a a a and b its adjacent vertices. Since P • A diagonal is a new edge between two polygon is a simple polygon and n >3, there is b b vertices that is entirely inside the polygon. no edge between a and b . v v Consider the following two cases: • Lemma 1 : every polygon has a convex vertex. 1. the new edge ab is a diagonal Proof: the highest vertex (the one with the Case 2 Case 2 a a L largest y coordinate) is convex. 2. Otherwise, there exists a vertex x L x x which is the closest to v with • Lemma 2 : Every polygon with n >3 vertices has b b respect to a line L parallel to ab a diagonal. v v which is a diagonal. 9 10 Triangulation theory Triangulation theory Triangulation dual Triangulation dual Theorem: Every simple polygon with Definition: The triangulation dual T of a triangulation of a simple n vertices has a triangulation with P P 1 polygon P is a graph whose nodes 1 n –3 diagonals and n –2 triangles. are triangles and whose edges are Proof: By induction on n : adjacencies between triangles sharing an edge. • Basis: A triangle ( n =3) has a Property: the triangulation dual is a triangulation (itself) with no v v tree whose node degree is ≤ 3. P 2 P diagonals and one triangle. 2 • Induction on n : Proof: Degree ≤ 3 by construction: no triangle has more than For an n -vertex polygon, construct a diagonal dividing the polygon • three neighbors. into two polygons P 1 and P 2 with n 1 and n 2 vertices such that n 1 + n 2 –2 = n . • No cycles: by contradiction. If it has a cycle, it is a Diagonals: ( n 1 –3)+( n 2 –3)+1 = ( n 1 + n 2 –2)–3 = n –3 polygon with a hole (not a simple polygon). Triangles: ( n 1 –2)+( n 2 –2) = ( n 1 + n 2 –2)–2 = n –2 • • If fact, T If fact, T is a binary tree with root degree one or two! is a binary tree with root degree one or two! 11 12 2
Simple triangulation algorithm Simple triangulation algorithm Simple triangulation algorithm (1) Simple triangulation algorithm (1) v 1 v proc triangulate(P) Idea : Reduce the polygon by clipping a 1 v 2 v 2 if | P | ≤ 3 output(P) and return; triangle at each iteration. i 0; The clipped triangle will be formed by v 0 v while diagonal( v i ,v i+2 ) is not legal i ++; 0 three consecutive vertices output( v i ,v i+1 ,v i+2 ); ( v i ,v i+1 ,v i+2 ). The diagonal is ( v i ,v i+2 ). remove v i+1 from P; triangulate( P ); Test for validity : 1. The diagonal does not intersect other Complexity: n times n diagonal tests that take polygon edges. each O( n ) � O ( n 3 ). Sources of inefficiencies: 2. The diagonal must be inside the • repeated diagonal tests polygon ( test that diagonal is inside normal cone ). • diagonals are not sorted or ordered � Precompute diagonals in O ( n 2 ). 13 14 Triangulation algorithms O( n O( n log log n n ) )- -time triangulation algorithm time triangulation algorithm Triangulation algorithms • What is the lower bound? 1) Partition the polygon • O ( n 2 ) by precomputing diagonals into y -monotone pieces • Improve to O ( n log n ) by ordering them (see later). (“ תוינו טונומ תוכיתח ”). • Is less than O ( n log n ) possible? 2) Triangulate each y -monotone piece separately. 15 17 Monotone polygons: definition Monotone polygons: definition Property of monotone polygons Property of monotone polygons L • A simple polygon is called L Definition : a vertex v is an interior cusp iff it is a concave L L monotone with respect to a line L if vertex whose adjacent vertices are both at or above (at for any line L’ perpendicular to L or below) line L . the intersection of the polygon with v v L’ is connected. L L v v • A polygon is called monotone if Theorem : If a polygon P has no interior cusps with there exists any such line ℓ . respect to a line L , then it is monotone with respect to L . • A polygon that is monotone with Proof sketch : partition P into two chains connecting the respect to the x/y -axis is called x/y- top and bottom vertices. Assume one of them is not monotone. monotone with respect to L . Then P must contain an y -monotone but not Question: How can we check in O( n ) interior cusp above or below. x -monotone polygon time whether a polygon is y -monotone? 18 19 3
Triangulating a Y- Triangulating a Y -monotone polygon monotone polygon Triangulating a y- -monotone polygon monotone polygon Triangulating a y • Continue sweeping while one • Sweep the polygon from top to bottom. left chain right chain chain contains only one edge, left chain right chain • Greedily triangulate anything possible above the while the other edge is concave . top top sweep line, and then forget about this region. • When a “convex edge” appears • When procesing a vertex v , the unhandled region in the concave chain (or a above it always has a simple structure: second edge appears in the other Two y -monotone (left and right) chains, each one), triangulate as much as containing at least one edge. If a chain consists of two possible using a “fan”. or more edges, it is reflex , and the other chain consists of a single edge whose bottom endpoint has not been • Time complexity: O( k ), where k handled yet. is the complexity of the polygon. bottom bottom • Each diagonal is added in O (1) time. 20 21 Monotone polygon triangulation Monotone polygon triangulation Stack operations on chains Stack operations on chains 22 23 Y- -monotone polygons monotone polygons Y Vertex classification Vertex classification Polygon vertices classification: • A start (resp., end ) vertex is a start vertex whose interior angle is less than π and its two neighboring vertices both lie merge below (resp., above) it. regular • A split (resp., merge ) vertex is a vertex whose interior angle is split greater than π and its two neighboring vertices both lie end below (resp., above) it. • All other vertices are regular . 24 25 4
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