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Exploring Computational Mathematics: Unfolding Polyhedra

Exploring Computational Mathematics: Unfolding Polyhedra . Brittany Terese Fasy, Duke University David Millman, UNC-Chapel Hill. Standard Curriculum. Undergraduate: Courses Quizzes and Exams Projects Senior Project. Graduate: Research Presentations Papers Proposals Courses.

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Exploring Computational Mathematics: Unfolding Polyhedra

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  1. Exploring Computational Mathematics: Unfolding Polyhedra Brittany Terese Fasy, Duke University David Millman, UNC-Chapel Hill MathFest 2008

  2. Standard Curriculum Undergraduate: • Courses • Quizzes and Exams • Projects • Senior Project Graduate: • Research • Presentations • Papers • Proposals • Courses MathFest 2008

  3. An Undergraduate Research Course: BRIDGING THE GAP MathFest 2008

  4. Student Objectives • Learn about Mathematical Research • Understand an Open Problem • Identify and Discuss Areas of Computational Mathematics • Read an Academic Paper • Present to Peers • Write a Proposal • Decide Upon an Area for Further Investigation MathFest 2008

  5. Vertex Unfolding Vertex Unfolding of Simplicial Manifolds, E. Demaine, D. Eppstein, J. Erickson, G. Hart, and J. O’Rourke MathFest 2008

  6. Course Components • Lectures on Topic Areas • Bi-Weekly Assignments • Final Project • Paper Presentation • Future Investigation Proposal MathFest 2008

  7. Topic 1: Computer Science • Basic Programming Skills • Data Structures • Parsing and File I/O • Debugging Tools and Technique • Documentation • Optional • Inheritance and Templates • Pointers MathFest 2008

  8. Topic 2: Graph Theory • Planar Graph • Searching • Graph Data Structures • Euler Tour and Hamiltonian Cycle • Optional: • NP Complete • Shortest Path • Min Flow / Max Cut MathFest 2008

  9. Topic 3: Topology • Basis for a Topology • Covering Set • Homotopy Equivalence • Fundamental Group • Optional: • Persistence Homology • Mesh Generation MathFest 2008

  10. Topic 4: Numerical Analysis • Floating Point Numbers and Arithmetic • Error Propagation • Root Finding • Convergence • Optional: • Matlab • Solving ODEs • Interpolation MathFest 2008

  11. Topic 5: Computational Geometry • Predicates (e.g. inCircle) • Voronoi Diagrams • Delaunay Triangulations • Convex Hulls • Triangle and Object Intersections • Optional: • Arrangements • Range Queries MathFest 2008

  12. Topic 6: Visualization • Programming with OpenGL • Matrix Transformations • User Interactions and Callbacks • Optional: • Rendering Versus Ray Tracing • Applications of Previous Topics MathFest 2008

  13. Project Assignments • Computer Science • Debugging Exercise • Create Shape Object • Graph Theory • Design & Create Graph Data Structure • Create Planar Graph from Shape Object • Topology • Euler Tour of Graph MathFest 2008

  14. Project Assignments (continued) • Numerical Analysis • Numerical Stability • Computational Geometry • Compute Planar Coordinates • Visualization • Visualize the Output • Add User Interaction MathFest 2008

  15. Presentations & Proposals • Academic Papers for Each Topic Covered • Group Presentations of Papers • Proposal Writing for Project Extension MathFest 2008

  16. Acknowledgements • Olga Brezhneva • Dave Coulliette • Robert W. Vallin • Jack Snoeyink MathFest 2008

  17. QUESTIONS? MathFest 2008

  18. How to Contact Us: • brittany@cs.duke.edu • dave@cs.unc.edu MathFest 2008

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