1. Algorithm for computing positive α-hull for a set of planar closed curves Vishwanath A. Venkataraman, Ramanathan Muthuganapathy Advanced Geometric Computing Lab, Department of Engineering Design, Indian Institute of Technology Madras, India- 600036 SMI 2015, University of Lille, France

2. Hulls Bounding hulls Convex hull Minimum enclosing disc (MED) α-hull Wide applications such as shape matching, collision detection etc. Quite a few algorithms exist for point-set. SMI 2015, University of Lille, France

3. α- Hull Generalization of convex hull α-disc from α-neighbors α-disc SMI 2015, University of Lille, France

4. Positive and negative α-hulls Let α be a positive real so that there exists a disc of radius that contains all input curves. The positive α-hull of S is the intersection of all such discs. In the actual computation, only the α-discs are identified and not their intersections. SMI 2015, University of Lille, France

5. α- hull (negative) for a point set Typically uses Delaunay triangulation (DT) (dual of closest Voronoi diagram (VD)) for a point-set. α = 0, point set. α ∞, convex hull is obtained SMI 2015, University of Lille, France

6. Computation of hulls for curves Mainly computed for point-sets. Recently convex hull and MED have been computed for curves. SMI 2015, University of Lille, France

7. α- hull for curves VD is computationally intensive (polynomial of a bisector is of high degree). Approximation of curves by points or polylines leads to inaccurate computation. As the VD need not contain straight lines, the dual of VD i.e. DT does not exist. This work addresses the computation of positive α- hull using curves as such as well as without using VD/DT. SMI 2015, University of Lille, France

8. Key question(s) for curves Can α–hull be computed without using farthest VD? Also, can this be done without sampling the curves into points/polylines? Since there is no DT for curves, what kind of data structure is required for computation? What constraint equations are required? SMI 2015, University of Lille, France

9. Positive α-hull Only points on the convex hull will be involved in positive α–hull. Similarly, in the curves as input, the curves in the convex hull are only required for computation. FromTo SMI 2015, University of Lille, France

10. Key Idea SMI 2015, University of Lille, France

11. Algorithm details SMI 2015, University of Lille, France

12. Starting Triplet Proposition 1: The largest radius disc tangential to any three curves in the convex hull is definitely a part of the positive alpha-hull. Proposition 2: Largest disc exists for only for curves that are sequential in the convex hull. SMI 2015, University of Lille, France

13. Update Triplet data Proposition 3: Once a ‘central’ curve has been used for a triplet disc computation, the curve plays no further part in the computation of triplet discs. The positive alpha-hull extends from the convex hull to the minimum enclosing disc. SMI 2015, University of Lille, France

19. DAG SMI 2015, University of Lille, France

22. α-neighbours Triplets in increasing order Omit the central curve and the pair is added, From [C 4 C 2 C 1 ], [C 4 C 1 ] are added Omit descendant nodes [C 4 C 3 C 2 ] present in R-list From [C 1 C 7 C 6 ], [C 1 C 6 ] is added. Triplets along with its siblings. [C 5 C 4 C 1 ] is split into [C 5 C 4 ] and [C 4 C 1 ] and its sibling [C 6 C 5 C 4 ] is split into [C 6 C 5 ] and [C 5 C 4 ] Greater than given α (say 6) Lesser than given α SMI 2015, University of Lille, France

23. Implementation using IRIT SMI 2015, University of Lille, France dot product

24. α- disc computation SMI 2015, University of Lille, France

25. Results SMI 2015, University of Lille, France

26. Result for Split curves SMI 2015, University of Lille, France

27. For non-convex curves SMI 2015, University of Lille, France

28. Time Taken SMI 2015, University of Lille, France Intel i5 2.5GHz 6 GB ram 64 bit Windows operating system using IRIT solid modeling kernel

29. Computational Effort Number of triplets = One less than the number of curves in the convex hull At each iteration, only two triplets are updated and one deleted. DAG can be easily maintained. Constraint equations are `dot product’. Overall complexity is O(n 2 ). SMI 2015, University of Lille, France

30. Conclusion Computation can be done without using (farthest) Voronoi diagram (and its corresponding DT) using the concept of enclosing discs. Algorithm can handle convex, non-convex and intersecting curves. It is also shown that curves need not be discretized for computation. SMI 2015, University of Lille, France

31. Questions ?