Footpoint distance as a measure of distance computation between curves and surfaces

1. Footpoint distance as a measure of distance computation between curves and surfaces Bharath Ram Sundar*, AbhijitChunduru*, RajatTiwari*, AshishGupta^ and RamanathanMuthuganapathy*. *Department of Engineering Design Indian Institute of Technology Madras Chennai, India ^Renishaw, Pune India (Formerly worked in India Science lab, General Motors, India)

2. Overview • Introduction • Statement • Curve-curve case • Distance query- Surface-Surface • Distance query- Curve-Surface

3. Introduction • Most of CAD design requirements can be modeled as geometric queries, such as distance to edge, planarity, gap, interference and parallelism. • Typically done in discrete domain, thus there is need to solve in continuous domain. • Should be scalable efficiently for a larger domain.

4. Motivation • Commercial CAD packages offer elementary computations, difficult to scale and generally discretely computed.

5. Disadvantages with discrete computation • Approximation- Queries made are approximate as faceted models are approximate representation of geometry. • Computational complexity-Computational expense increases with densely faceted model. • Result remapping- Mapping back to original geometry further adds to approximation.

6. Problem Statement • Surface-Surface • Given two freeform surfaces, compute regions on each surface, such that, for any point (P) in a region on one surface there lies a corresponding point (P’) on the other surface at a distance less than a threshold value.

7. Curve-Surface • Given a freeform curve and a set of freeform surfaces, compute segments of the curve where the minimum distance between the curve and any of the surfaces is more than a threshold value.

8. Existing Distance functions • Typical minimum distance computation is performed. • Hausdorff distance.

9. Contributions • To the best of our knowledge, no work seems to exist that compute corresponding patches of curves/surfaces satisfying above or below a certain distance value, which is the focus of this work. Our major contributions are: • Footpoint distance measure has been proposed as a measure for distance computation. • Established points of correspondence through footpoints was explored in the case of curve-curve case and found to be an useful tool. • Corresponding surface patches for the surface surface case are identified using footpoint distance. Alpha shape has been used to detect boundaries including island regions. • A lower-envelope based approach has been proposed and demonstrated for the distance query between a curve and a set of surfaces.

10. Curve-curve • It is desired to find the exact distance bounds for curves C1(t) and C2(r) that correspond. • Let d1 be Minimum of the antipodal distances.Let d2be the subsequent minima. • Distance for the shown segments of is bound by the distances d1 and d2.

11. Surface-Surface Let S1(u1, v1) and S2(u2, v2) be the surfaces and D1(u1, v1, u2, v2) be the distance function. The basic partial differential equations for extremum are • Symbolic representation of bisector surface is possible for curve-curve case. • Such a representation for the bisector of a pair of surfaces and subsequently for D1(u1, v1, u2, v2) has not been shown to be possible yet. • Using antipodal points as the start looks infeasible and this motivated us to directly work on the footpoint distance, given a query distance.

13. Surface-Surface Distance query • Footpoint distance and α-shape • Solving distance query • Boundary detection using α-shape • Boundary identification for islands

14. Footpoint distance and α-shape

15. Solving distance query

16. Boundary detection using α-shape Patches in the form of point sets on both surfaces for Dq =0.8

17. Boundary detection using α-shape Points Sets in parametric space Dq =0.8

18. α-shape α-shape for S1 α-shape for S2 α-shapes for points in parametric space for Dq =0.8

19. Boundary identification for islands Island boundaries identified in parametric space. Regions on the surfaces for the identified boundaries.

20. Surface patches for various Distance queries

21. Surface surface results

22. Curve and set of free-form Surfaces • We initially find all the bisector points B between a curve C(t) and a surface S = S(u,v), which can be identified by solving the following equations

23. Input Curve-Surface

24. Footpoints Lines joining minimum distance footpoints.

25. Min footpoint distance function • For a point on the curve, there are several footpoints on the surface. • We take the minimum distance footpoint (MinF)..

26. Curve-Surface Lower envelope • Given a distance query valuDq, Lower envelope technique is then computed about Dq .

27. Curve segments for Dq= 0.55

28. Various Distance queries

29. Conclusions • Algorithms for computing distance between curves and surfaces that satisfy a distance input value has been proposed and implemented. • Footpoint distance has been shown to be an appropriate distance measure for the intended problems. • Implementation Results have been provided.

30. Thank you