1. Visibility Graphs and Motion Planning Kittiphan Techakittiroj for the Degree of Master of Science Department of Computer Science, Ball State University, IN, USA
Committee:
Prof. Jay S. Bagga
Prof. Clinton P. Fuelling
Prof. J. Michael McGrew
Contributions:
Theory and Implementation
Presentation Outline:
Graph Theory
Visibility Graph
Motion Planning & Application
Committee:
Chairperson: Jay S. Bagga, Professor
Member: Clinton P. Fuelling, Professor and Chairman
Member: J. Michael McGrew, Assoc. Professor
Contributions are combinations of theory and implementation.
Theory: This part contains the details of topics from graph theory, topology and computational geometry in motion planning. It also includes new algorithms which are developed as a part of this thesis.
Implementation: This part is a software system to implement the theory as an example of real applications. This software also includes many tools to help in studying visibility graphs.
Presentation Outline:
Graph Theory will give an introduction to graph theory.
Visibility Graph contains the topics in the visibility graph area. A brief introduction to terms in computational geometry will be given.
Motion Planning & Applications: Combine graph theory and visibility graph to plan motions. Applications are given for idea of the real world.

2. Graph Theory Graph: vertices + edges Path Tree
Graphs: A graph composes of a set, V, on vertices, and a set, E, of edges each of which connect two vertices in V. It is denoted by G=(V; E).
Path: A path is a walk from one vertex to the other vertex without passing through the same vertex twice. No repeated vertices in a path.
Tree: Tree is a connected sub graph (or graph) that contains no cycle.

3. Creation & Search Creation: Search: Complete Graph Convex Hull
Simple Polygon from Points
Search:
Breadth First Search
Depth First Search
Complete Graph: The complete graph has edges connecting all pairs of distinct vertices in the graph.
Convex Hull: The convex hull of a set of points is defined as the smallest convex polygon enclosing all the points. A convex polygon is a polygon such that any line segment connecting two points inside the polygon itself lies entirely inside the polygon.
Simple Polygon from Points: A closed path is a path whose last point is the same as its first point. A closed path is also called a polygon. A simple polygon is one whose corresponding path does not intersect itself; that is, no edges of the polygon intersect except for neighboring edges at their common vertex.
Breadth First Search: Given a graph and a vertex in the graph, search the graph from the given vertex to the rest of the graph by traversing to all of its children before go deeper to their descendants.
Depth First Search: Given a graph and a vertex in the graph, search the graph from the given vertex to the rest of the graph by traversing to its child first before go back to its parent.

4. Minimum Distance Minimum Distance Spanning Tree
Single Source Shortest Path
Eccentricity of a Graph
Radius and Diameter of a Graph
Minimum Distance Spanning Tree: Given an undirected connected weighted graph G find a spanning tree T of G of minimum weight (also referred as the minimum cost spanning tree).
Single Source Shortest Path: Given an undirected graph and a vertex in the graph, find the shortest paths from the given vertex to all other vertices in the graph.
Eccentricity of a Graph: The eccentricity e(v) of a vertex v of a connected graph G is the maximum d(u; v), where d(u; v) is the distance between two vertices u and v (length of the shortest u-v path).
Radius and Diameter of a Graph: The radius, rad G, is defined as the minimum e(v) while the diameter, diam G, is the maximum e(v), where e(v) is the eccentricity of each vertex v, for all vertices in G.

5. Geometry Polygon Obstacle Path
Polygons: A closed path is a path (connections) whose last point is the same as its first point. A closed path is also called a polygon.
Obstacles: The set of obstacles is a set of points, lines, and polygons.
Paths: Paths are directions for the point to move without penetrating into any obstacle. Path in this definition is a little bit different than the one defined for graph theory in which there is no set of obstacles.

6. Visibility Graph Visibility Edge Visibility Graph
Segment Endpoint Visibility Graph
Polygon Visibility Graph
Visibility Edge: For a set of obstacles in the plane, a point u has visibility to a point v, if the line segment s connecting u and v do not touch or cross any obstacles. The segment s is called a visibility edge.
Visibility Graph: For a set of points, lines, and polygons in the plane, its visibility graph is a graph, G(V; E) such that V , the set of vertices, consists of vertices (corners) of the obstacles, and E is a set of all visibility edges.
Segment Endpoint Visibility Graph: Given a finite set S of n (>1) planar line segments in general position. Segment endpoint visibility graph Gs is the undirected graph whose vertices are the endpoints of the line segments of S and whose edges connect those pairs of vertices which are either endpoints of the same segment from S or the line segment connecting them does not intersect any element of S.
Polygon Visibility Graph: The visibility graph of a polygon is the spanning sub-graph of a visibility graph. It contains edges only in one specific region, (i.e. the interior or exterior of a polygon).

7. Advance Visibility Graph
Condensed Graph
Contracted Graph
Invisibility Edge
Invisibility Graph
Degree of Invisibility (Edge)
Degree of Invisibility (Vertex)
Condensed Graph: Given a segment endpoint visibility graph G, the condensed graph of G is a graph whose vertices represent the obstacles (segments), with two connected if all four visibility edges exits between the corresponding obstacles.
Contracted Graph: Given a segment endpoint visibility graph G for a set of obstacles, the contracted graph of G is a graph whose vertices represent the obstacles, with two connected by an edge if there is at least one visibility edge between the two corresponding obstacles in G.
Invisibility Edge: For a set of obstacles in the plane, a point u has invisibility to a point v, if the line segment s connecting u and v touches or crosses at least one of the obstacles. The segment s is called an invisibility edge.
Invisibility Graph: For a set of points, lines, and polygons in the plane, its invisibility graph is a graph, G(V; E) such that V , the set of vertices, consists of vertices (corners) of the obstacles, and E is a set of all invisibility edges.
Degree of Invisibility (Edge): If s is an invisibility edge, the degree of invisibility of s is the number of crossings on s.
Degree of Invisibility (Vertex): If v is a vertex, the degree of invisibility of v is the maximum degree of invisibility of all edges connected to v.

8. Motion Planning
Motion Planning is to find the best path for moving a point among a set of obstacle.
Process of Planning
Finding visibility graphs as a set of paths.
Applying geometric or graph algorithms.
Motion planning has a variety of meanings. This project considers motion planning in the sense of finding the best path for moving a point among a set of obstacle.
Process of Planning
Finding visibility graphs as a set of paths.
Applying geometric or graph algorithms.

9. Applications
Computer networks: A good place (by distance) for a server
Weakest points in the network
Shortest path of connecting two machines
Spanning length of a network
Consultant: Cost estimation of wiring system
Building design: One factor of a good place to put a light
One factor of a weak place to get in
Medical treatment: Amplitude of X-ray
Mobile robot: Shortest path for movement
Traveling: Shortest path for traveling
Computer networks: A good place (by distance) for a server: using the center of a graph.
Weakest points in the network: using the cutting vertices of a graph.
The shortest path of connecting two machines: using the shortest path.
The spanning length of a network: using the diameter of a graph.
Consultant: Cost estimation of wiring system: using the radius, diameter or eccentricity.
Building design: One factor of a good place to put a light: a good place to put a light is supposed to have the smallest degree of invisibility, so the light can spread through out the whole area.
One factor of a good place to put a speaker: the same reason as the previous one.
One factor of a weak place to get in: for safety reason, the weak position is the easy place to break in.
Medical treatment: The amplitude of X-ray: By using the degree of invisibility of the X-ray source and using the human body as a set of obstacles, we can know how many layers the X-ray beam has to go through.
Mobile robot: The shortest path for movement.
Traveling: The shortest path for traveling.