1. CAD/Graphics 2013, Hong Kong Computation of Voronoi diagram of planar freeform closed convex curves using touching discs Bharath Ram Sundar and Ramanathan Muthuganapathy Department of Engineering Design Indian Institute of Technology Madras, Chennai, India

2. 2 CAD/Graphics 2013 Hong Kong Voronoi diagram of curves Definition :Given a number of disjoint planar regions bounded by free-form curve segments C0(t), C1(r1),..., Cn(rn), their Voronoi diagram [Aurenhammer 1991] is defined as a set of points that are equidistant but minimal from two different regions.

3. 3 CAD/Graphics 2013 Hong Kong The bisector between pair of curves contributes to the Voronoi diagram. Typically constructed from the lower envelope of all pairs of bisectors. Some portions of the curves my not be required. Why a branch point is difficult to compute? Current Approach: Use bisector or portions of them to compute branch points. Here, it is shown that branch points can be identified without explicitly solving the bisector.

4. 4 CAD/Graphics 2013 Hong Kong Voronoi neighborhood between two curves is created/changed at minimum distance point/branch point. Hence these special points are solved for directly. Minimum distance as antipodal or two touch disc. Branch disc (BD) as three touch disc (TTD) Our methodology

5. 5 CAD/Graphics 2013 Hong Kong Initially all pairs of minimum antipodal discs (MADs) are solved and store in a list. MADs are processed in increasing order of radius in the list. Whenever discs are added connectivity information is maintained. Three touch discs (TTDs) is solved for only when relevant neighborhood is formed and inserted into the list. All consistent antipodal lines Minimum radius antipodal

6. 6 Illustration of the basic idea CAD/Graphics 2013 Hong Kong Initial Radius list After processing R ab, R bc TTD of (Ca, Cb, Cc) added TTD is processed to decide if it is a branch disc

7. 7 Emptiness check of ADs CAD/Graphics 2013 Hong Kong

8. 8 Algorithm steps CAD/Graphics 2013 Hong Kong

9. 9 Algorithm continued CAD/Graphics 2013 Hong Kong 1 4 2 3 5

10. 10 Results CAD/Graphics 2013 Hong Kong

11. 11 Salient features Given a curve of degree m, the degree of the bisector is 4m − 2. Computing TTD or AD has a degree of m+(m−1). Instead of step sizes or intersection of bisectors, a simple directed edge existence is used. CAD/Graphics 2013 Hong Kong

12. 12 Conclusion It has been shown that bisector computation is not required to compute the branch points, effectively reducing computational complexity to a large extent. Algorithm avoids explicit intersection checks with each curve for checking the emptiness of antipodal discs and branch discs. Relevant portions that contribute to the Voronoi are identified and local bisectors are computed only for corresponding pairs. Curves on the convex hull are obtained as a by product of the algorithm. Extension of the algorithm to non-convex curves is currently underway. CAD/Graphics 2013 Hong Kong

13. 13 Thank you CAD/Graphics 2013 Hong Kong

14. 14 Backup slides Definition 1: A disc of radius R is essentially a circle of radius R. Definition 2: When a disc is tangential to a curve, the point on the curve that just touches the disc is called footpoint. Definition 3: Radius vector (RV) is the vector having the tail as the center of the disc and head as its footpoint. Definition 4: A disc is said to be consistent if outward normal of the curve is opposite to the RV at every footpoint. Definition 5: An antipodal disc (AD) is a disc where the normals at the two footpoints are opposite to each other. Minimum antipodal disc (MAD) is the minimum in radius of all ADs and its corresponding footpoints are termed minimum antipodal points (MAPs). Line joining the two footpoints of the MAD is called minimum antipodal line (MAL). CAD/Graphics 2013 Hong Kong

15. 15 Definition 6: A three touch disc (TTD) is a disc having three footpoints, each one touching a different curve. Definition 7: A Voronoi disc is a disc at any point on the VD. It is always empty. Definition 8: A branch disc (BD) is an empty disc whose center is a branch point. It has three footpoints. Definition 9: A Voronoi segment is a contiguous set of points having two footpoints bound on either side by a branch point and a center of MAD or from a branch point and ∞. CAD/Graphics 2013 Hong Kong

16. 16 Lemma 1: Only a consistent AD or TTD plays a role in the Voronoi diagram. Lemma 2: MAD is always a consistent disc. Lemma 3: Bisector radius function attains local minima only at MAPs between the two curves. Lemma 4: Voronoi radius function attains local minima only at MAPs. Lemma 5: Voronoi disc will touch a convex curve at most at one point. i.e. the disc will have footpoints from different curves. Lemma 6: Radius of cTTD of any given three curves is not less than the radius of MAD of any pair of curves among the three. CAD/Graphics 2013 Hong Kong