1. Voronoi Diagram A Captivating Geometrical Construct Presented by: Lamour Roberts Preceptor: Dr. Bruno Guerrieri (Associate Professor of Mathematics) Department of Mathematics, Florida A&M University Tallahassee, FL, 32307 FGLSAMP SUMMER RESEARCH EXPERIENCE FOR UNDERGRADUATES

2. FGLSAMP Summer Research Philosophy FGLSAMP is an alliance of 13 institutions committed to: –Increase the number of undergraduate degrees award in STEM disciplines –Provide activities that compliment classroom learning such as: Undergraduate Research Experience –Provide performance based Financial assistance –Provide development Faculty and graduate mentoring

3. Summer 2005 Research Area of focus: The Voronoi Diagram Research Mentor: Dr. Bruno Guerrieri (Associate Professor of Mathematics) Expertise: Computational Geometry

4. Computational Geometry Computational geometry is concerned with the solving of geometrical problems through the efficient design and analysis of algorithms.

5. A well mentioned construct of computational geometry Given n points called sites in a plane, their Voronoi diagram is a tessellation of the plane according to the nearest neighbor rule (Aurenhammer). Each site is associated with the Voronoi polygon closest to it. The Voronoi Construct

6. Why Focus on the Voronoi Construct Some of the reasons: –Several natural processes results in the formation of Voronoi Diagrams –Can be used to develop robust tools for solving unrelated problems in computational science;

7. Applications of the Voronoi Diagram Anthropology and Archeology – Neolithic clans, chiefdoms, ceremonial centers, or hill forts. Astronomy – stars, and galaxies. Biology, Ecology, and Forestry – Plant competition, protein folding. Crystallography and Chemistry – Metallic Sodium, sphere packings Geography – Settlements Marketing– US metropolitan areas; individual retail stores. Mathematics – Quadratic forms Robotics – path planning Statistics and Data Analysis – “Natural Neighbors” Gravitational Influence of Stars. Descartes. 1644

8. Characterization of Voronoi Diagram What is P? –collinear set of point sites: Voronoi = n-1 parallel lines –otherwise: Voronoi is a connected planar graph, in which all edges are line segments or half-infinite lines

9. Computing the Voronoi Diagram for each site: The sweep algorithm (Fortune’s Algorithm) Sweep a horizontal line from top to bottom across the sites on the plane

10. Demonstration Demo of Fortune’s Algorithm

11. Challenges Faced Resources: –Lack of well documented information concerning the computation of the Voronoi Diagram The Source Code: –I faced a few challenges when writing the JAVA source code from Fortune’s algorithm –The Data Structure used to construct the Voronoi diagram was very challenging

12. Overall Experience Research –Gained invaluable research experience –Improved my JAVA programming language skills Seminars –More insight into graduate school culture –Improved my oral presentation skills Project –Learned how to plan and regulate a project that is due to be completed within 10 weeks –Happy with the experience I have gained

13. Questions and Suggestions