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Business Geographics


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Business Geographics

  1. 1. Geometric Algorithms in Business Geographics Presented by T. N. Badri
  2. 2. <ul><li>Geometric algorithms came to the frontline in the early 1970s in the field of algorithm design. </li></ul><ul><li>They are useful in graphics, motion planning of robots, CAD/CAM, Geographic Information Systems and also molecular biology. </li></ul><ul><li>These applications can be used in some business situations, to make more informed decisions. </li></ul>
  3. 3. <ul><li>This presentation describes three geometric ideas and then addresses their usefulness in the corporate world </li></ul><ul><li>Computational Geometry Algorithms </li></ul><ul><li>Their uses in GIS/Molecular Biology/CAD/Robotics </li></ul><ul><li>Three Examples </li></ul><ul><ul><li>Voronoi Diagram and Delaunay Triangulation </li></ul></ul><ul><ul><ul><li>Use of the Voronoi Diagram in Product mix for stores </li></ul></ul></ul><ul><ul><ul><li>Use of Delaunay Triangulation in finding Steiner Tree </li></ul></ul></ul><ul><ul><li>Convex Hull algorithms </li></ul></ul><ul><ul><li>Use in range and gap detection </li></ul></ul><ul><ul><ul><li>Segment Intersection </li></ul></ul></ul><ul><ul><ul><li>Use of segment intersection in overlaid maps </li></ul></ul></ul>
  4. 4. Voronoi Diagram for a set of vertices in the plane <ul><li>For a given set of vertices the Voronoi Diagram gives us cells identified with each vertex </li></ul><ul><li>All points in the Voronoi cell are closer to the vertex than to any other vertex </li></ul>
  5. 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. 6. Stacked barchart for sub-areas Source of map: Altavision Geographics
  7. 7. <ul><li>The Delaunay Triangulation is the dual of the Voronoi Diagram </li></ul><ul><li>It can be obtained from the Voronoi Diagram </li></ul><ul><li>The Delaunay is special among meshes because it maximizes the </li></ul><ul><li>the minimum angle among the triangles in the mesh. </li></ul>What is a Delaunay Triangulation?
  8. 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. 9. <ul><li>The Delaunay Triangulation can be used as the scaffolding for the Steiner Tree </li></ul>Steiner Tree on the Delaunay Triangulation
  10. 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. 11. Convex hull can be used to determine gaps in Network range <ul><li>What is a Convex Hull? </li></ul><ul><li>Suppose there is a set P of n vertices in the plane: P = (x 1 , x 2 , …., x n ). </li></ul><ul><li>The smallest convex set enclosing all the vertices is called the convex hull. </li></ul>
  12. 12. The Convex hull gives an empire map
  13. 13. <ul><li>We can use discs of various diameters instead of vertices </li></ul>x x x x x x x If the discs represent broadcast range, this picture shows the gaps in the overall range.
  14. 14. Segment Intersection and Subregion Intersection <ul><li>Given a map of roads and a map of rivers one may be interested in the intersection points, as possible locations for bridges </li></ul><ul><li>Overlay of two or more geographical maps can be as helpful as querying a relational database </li></ul>
  15. 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. 16. THANK YOU