1. On the singularity of a class of parametric curves Imre Juhász University of Miskolc, Hungary CAGD, In Press, Available online 5 July 2005 Reporter: Chen Wenyu Thursday, Oct 13, 2005

2. About the author • Introduction • Cases for parametric curves • Apply to Bezier curves • Apply to C-Bezier curves • Conclusions

3. About the author • Imre Juhász , associate professor • Department of Descriptive Geometry at the University of Miskolc in Hungary. • His research interests are constructive geometry and computer aided geometric design.

4. About the author • Introduction • Cases for parametric curves • Apply to Bezier curves • Apply to C-Bezier curves • Conclusions

5. Introduction • To detect singular points of curves • Singularities: inflection points, cusps, loops

6. Introduction • 苏步青，刘鼎元，汪嘉业， 平面三次Bezier曲线的分类及形状控制。CAGD Book

7. Introduction • Stone, M.C., DeRose, T.D., 1989. A geometric characterization of parametric cubic curves. ACM Trans. Graph. 8 (3), 147–163.

8. This paper • consider parametric curves: • Change one pointzero curvature points the ruled surface.loop points the loop surface. • Apply to Bezier curves and C-Bezier curves.

9. About the author • Introduction • Ruled surfaces and loop surfaces • Apply to Bezier curves • Apply to C-Bezier curves • Conclusions

10. Construct of the ruled surface • Considering: • Move one point, then

11. Construct of the ruled surface • The curvature • zero curvature

12. Construct of the ruled surface

13. Construct of the ruled surface • So the moving point, • It is the parametric representation of a straight line with parameter λ. • As utakes all of its permissible values lines form a ruled surface.

14. Construct of the ruled surface • Let • Then its tangent line iswhere

15. Construct of the ruled surface • So, the ruled surface is a tangent surface. • is called a discriminant curve.

16. Construct of the loop surface • Considering: • Move one point, then

17. Construct of the loop surface • Let • Then • So the loop surface

18. Construct of the loop surface • The loop surface is a triangular surface. Its boundary curves are:

19. About the author • Introduction • Ruled surfaces and loop surfaces • Apply to Bezier curves • Apply to C-Bezier curves • Conclusions

20. Bezier curves • Consider i = 0, • Then • Let t=u/(1-u), we obtain its power basis form

21. Bezier curves Result (n=2): • c0(u) is a parabolic arc starting at d1 with tangent direction d2−d1. • c3(u) is also a parabolic arc, whilec1(u) and c2(u) are hyperbolic arcs.

22. Bezier curves • Consider i = 0, • Then the loop surface:

23. Bezier curves Result (n=2): • l0(u,1-u)is a parabolic arc. • l0(0,δ)is an elliptic arc.

24. About the author • Introduction • Ruled surfaces and loop surfaces • Apply to Bezier curves • Apply to C-Bezier curves • Conclusions

25. C-Bezier curves • Zhang, J., 1996. C-curves: an extension of cubic curves. CAGD. • Definition • where

26. C-Bezier curves • Denote C-Bezier curves as • Then the discriminant curve c3(u) is

27. C-Bezier curves • Applying the parameter transformation • then

28. C-Bezier curves

29. C-Bezier curves

30. C-Bezier curves • The loop surface l3(u,δ)of a C-Bézier curve is of the form

31. C-Bezier curves Conclusions • R1 one inflection point • R2 two inflection points • R3 loop • R4 no singularity

32. C-Bezier curves • Yang, Q., Wang, G., 2004. Inflection points and singularities on C-curves. CAGD 21 (2), 207–213.

33. C-Bezier curves

34. About the author • Introduction • Ruled surfaces and loop surfaces • Apply to Bezier curves • Apply to C-Bezier curves • Conclusions

35. Conclusions • The locus of the moving control point that yields vanishing curvature on the curve is the tangent surface of that curve which yields cusps on the curve. • Specified the locus of the moving control point that guarantees a loop on the curve.

36. Future work • It would be interesting to find how discriminant curves are related in this more general case.

37. Thank you !