Polygon Triangulation Computational Geometry The Art Gallery - PowerPoint PPT Presentation

Yazd Univ. Polygon Triangulation Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation 1392-1 Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 1 / 37

Motivation: The Art Gallery Problem Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 2 / 37

Motivation: The Art Gallery Problem Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 2 / 37

Motivation: The Art Gallery Problem Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 2 / 37

Triangulating Polygons Definitions Simple polygon: Regions enclosed by a single Yazd Univ. closed polygonal chain that does not intersect itself. Question: How many cameras do we need to guard Computational Geometry a simple polygon? Answer: Depends on the polygon. The Art Gallery One solution: Decompose the polygon to parts which Problem Guarding and are simple to guard. Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon Simple Polygon Non-Simple Polygons 3 / 37

Triangulating Polygons Definitions Simple polygon: Regions enclosed by a single Yazd Univ. closed polygonal chain that does not intersect itself. Question: How many cameras do we need to guard Computational Geometry a simple polygon? Answer: Depends on the polygon. The Art Gallery One solution: Decompose the polygon to parts which Problem Guarding and are simple to guard. Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon Star-shaped Convex 3 / 37

Triangulating Polygons Definitions Simple polygon: Regions enclosed by a single Yazd Univ. closed polygonal chain that does not intersect itself. Computational Question: How many cameras do we need to guard Geometry a simple polygon? Answer: Depends on the polygon. The Art Gallery Problem One solution: Decompose the polygon to parts which Guarding and Triangulations are simple to guard. Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 3 / 37

Triangulating Polygons Definitions diagonals: Yazd Univ. Triangulation: A decomposition of a polygon into triangles by a maximal set of non-intersecting Computational Geometry diagonals. diagonals The Art Gallery Problem Guarding and Triangulations not diagonal Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 4 / 37

Triangulating Polygons Definitions diagonals: Yazd Univ. Triangulation: A decomposition of a polygon into triangles by a maximal set of non-intersecting Computational Geometry diagonals. The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 4 / 37

Triangulating Polygons Definitions diagonals: Yazd Univ. Triangulation: A decomposition of a polygon into triangles by a maximal set of non-intersecting Computational Geometry diagonals. The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 4 / 37

Triangulating Polygons Definitions diagonals: Yazd Univ. Triangulation: A decomposition of a polygon into triangles by a maximal set of non-intersecting Computational Geometry diagonals. The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 4 / 37

Triangulating Polygons Definitions Guarding after triangulation: Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 5 / 37

Triangulating Polygons Definitions Guarding after triangulation: Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 5 / 37

Triangulating Polygons Definitions Guarding after triangulation: Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 5 / 37

Triangulating Polygons Definitions Guarding after triangulation: Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 5 / 37

Triangulating Polygons Definitions Guarding after triangulation: Yazd Univ. Computational Geometry The Art Gallery Problem Guarding and Triangulations Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 5 / 37

Questions: Does a triangulation always exist? How many triangles can there be in a triangulation? Yazd Univ. Theorem 3.1 Computational Geometry Every simple polygon admits a triangulation, and any triangulation of a simple polygon with n vertices consists The Art Gallery Problem of exactly n − 2 triangles. Guarding and Triangulations Proof. By induction. Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 6 / 37

Questions: Does a triangulation always exist? How many triangles can there be in a triangulation? Yazd Univ. Theorem 3.1 Computational Geometry Every simple polygon admits a triangulation, and any triangulation of a simple polygon with n vertices consists The Art Gallery Problem of exactly n − 2 triangles. Guarding and Triangulations Proof. By induction. Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone Polygon 6 / 37

Questions: Does a triangulation always exist? How many triangles can there be in a triangulation? Yazd Univ. Theorem 3.1 Computational Geometry Every simple polygon admits a triangulation, and any triangulation of a simple polygon with n vertices consists The Art Gallery Problem of exactly n − 2 triangles. Guarding and Triangulations Proof. By induction. Computing triangulation Partitioning a Polygon into Monotone Pieces Triangulating a Monotone w w Polygon v v v ′ u u 6 / 37

Guarding a triangulated polygon T P : A triangulation of a simple polygon P . Select S ⊆ the vertices of P , such that any triangle in Yazd Univ. T P has at least one vertex in S , and place the Computational cameras at vertices in S . Geometry To find such a subset: find a 3 -coloring of a triangulated polygon. The Art Gallery Problem In a 3-coloring of T P , every triangle has a blue, a red, Guarding and Triangulations and a black vertex. Hence, if we place cameras at all Computing triangulation red vertices, we have guarded the whole polygon. Partitioning a Polygon into Monotone Pieces Triangulating a Monotone By choosing the smallest color class to place the Polygon cameras, we can guard P using at most ⌊ n/ 3 ⌋ cameras. 7 / 37

Guarding a triangulated polygon T P : A triangulation of a simple polygon P . Select S ⊆ the vertices of P , such that any triangle in Yazd Univ. T P has at least one vertex in S , and place the Computational cameras at vertices in S . Geometry To find such a subset: find a 3 -coloring of a triangulated polygon. The Art Gallery Problem In a 3-coloring of T P , every triangle has a blue, a red, Guarding and Triangulations and a black vertex. Hence, if we place cameras at all Computing triangulation red vertices, we have guarded the whole polygon. Partitioning a Polygon into Monotone Pieces Triangulating a Monotone By choosing the smallest color class to place the Polygon cameras, we can guard P using at most ⌊ n/ 3 ⌋ cameras. 7 / 37

Guarding a triangulated polygon T P : A triangulation of a simple polygon P . Select S ⊆ the vertices of P , such that any triangle in Yazd Univ. T P has at least one vertex in S , and place the cameras at vertices in S . Computational Geometry To find such a subset: find a 3 -coloring of a triangulated polygon. The Art Gallery Problem In a 3-coloring of T P , every triangle has a blue, a red, Guarding and Triangulations and a black vertex. Hence, if we place cameras at all Computing red vertices, we have guarded the whole polygon. triangulation Partitioning a Polygon into By choosing the smallest color class to place the Monotone Pieces Triangulating a Monotone cameras, we can guard P using at most ⌊ n/ 3 ⌋ Polygon cameras. 7 / 37