Voronoi Diagram

A Captivating Geometrical Construct

Presented by: Lamour RobertsPreceptor: Dr. Bruno Guerrieri (Associate Professor of Mathematics)Department of Mathematics, Florida A&M UniversityTallahassee, FL, 32307

FGLSAMP SUMMER RESEARCH EXPERIENCE FOR UNDERGRADUATES

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

Summer 2005 Research

• Area of focus: The Voronoi Diagram• Research Mentor: Dr. Bruno Guerrieri (Associate Professor

of Mathematics)• Expertise: Computational Geometry

Computational Geometry

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

• 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

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;

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

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

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

Demonstration

• Demo of Fortune’s Algorithm

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

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

Questions and Suggestions