Business Geographics

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  • 1. Geometric Algorithms in Business Geographics Presented by T. N. Badri
  • 2.
    • Geometric algorithms came to the frontline in the early 1970s in the field of algorithm design.
    • They are useful in graphics, motion planning of robots, CAD/CAM, Geographic Information Systems and also molecular biology.
    • These applications can be used in some business situations, to make more informed decisions.
  • 3.
    • This presentation describes three geometric ideas and then addresses their usefulness in the corporate world
    • Computational Geometry Algorithms
    • Their uses in GIS/Molecular Biology/CAD/Robotics
    • Three Examples
      • Voronoi Diagram and Delaunay Triangulation
        • Use of the Voronoi Diagram in Product mix for stores
        • Use of Delaunay Triangulation in finding Steiner Tree
      • Convex Hull algorithms
      • Use in range and gap detection
        • Segment Intersection
        • Use of segment intersection in overlaid maps
  • 4. Voronoi Diagram for a set of vertices in the plane
    • For a given set of vertices the Voronoi Diagram gives us cells identified with each vertex
    • All points in the Voronoi cell are closer to the vertex than to any other vertex
  • 5. Voronoi Diagram for Hardware Stores in a neighbourhood What sort of items should a Home Store have in stock? That depends on the modernity of houses in a sub-area. The colour gives some idea of the age of houses in an area. Darker greens indicate more recent houses. Source of map: Altavision Geographics
  • 6. Stacked barchart for sub-areas Source of map: Altavision Geographics
  • 7.
    • The Delaunay Triangulation is the dual of the Voronoi Diagram
    • It can be obtained from the Voronoi Diagram
    • The Delaunay is special among meshes because it maximizes the
    • the minimum angle among the triangles in the mesh.
    What is a Delaunay Triangulation?
  • 8. Length = 4 kms 2/3 (  3) Length = 3(2/3)  3 = 3.4641 kms a saving of about 13.4%  3 V 1 V 2 V 3 V 1 V 2 V 3 S 1 What is a Steiner Minimal Tree?
  • 9.
    • The Delaunay Triangulation can be used as the scaffolding for the Steiner Tree
    Steiner Tree on the Delaunay Triangulation
  • 10. Steiner Tree and Minimum Spanning Tree for 100 random points in unit cube We can use a Delaunay Triangulation with Steiner Tree methods
  • 11. Convex hull can be used to determine gaps in Network range
    • What is a Convex Hull?
    • Suppose there is a set P of n vertices in the plane: P = (x 1 , x 2 , …., x n ).
    • The smallest convex set enclosing all the vertices is called the convex hull.
  • 12. The Convex hull gives an empire map
  • 13.
    • We can use discs of various diameters instead of vertices
    x x x x x x x If the discs represent broadcast range, this picture shows the gaps in the overall range.
  • 14. Segment Intersection and Subregion Intersection
    • Given a map of roads and a map of rivers one may be interested in the intersection points, as possible locations for bridges
    • Overlay of two or more geographical maps can be as helpful as querying a relational database
  • 15. References 1. Voronoi Diagrams and Delaunay Triangulations, Steven Fortune, Computing in Euclidean Geometry, pp. 193-233 2. Computational Geometry, Mark de Berg, Cheong, Marc Van Kreveld and Mark Overmars, 1997, Third edition Springer Verlag.
  • 16. THANK YOU