1. On the singularity of a class of parametric curves Speaker: Xu Hui 2007.11.8

2. About the author---Imre Juh á sz Associate professor, head of dept. Department of Descriptive Geometry, University of Miskolc. Miskolc- Egyetemv á ros H3515. Hungary Research interests:computer aided geometric design, constructive geometry, computer graphics

3. About the author---Imre Juh á sz Publication Juh á sz, I., Cubic parametric curves of given tangent and curvature, Computer-Aided Design, 30 (1), pp. 1-9., 1998. Juh á sz, I., Weight-based shape modification of NURBS curves, Computer Aided Geometric Design, 16 (5), pp. 377-383., 1999. Juh á sz I., Hoffmann M. Modifying a knot of B-spline curves, Computer Aided Geometric Design, 20 (5), pp. 243-245., 2003. Juh á sz I., Hoffmann M.: Constrained shape modification of cubic B-spline curves by means of knots, Computer-Aided Design, 36 (5), pp. 437-445., 2004. Juh á sz I., On the singularity of a class of parametric curves, Computer Aided Geometric Design, 2005 ， 23 (2), pp 146-156.

4. Introduction Manocha and Canny studied for polynomial and rational parametric curves of arbitrary degree. 1992. Detecting cusps and inflection points in curves. Computer Aided Geometric Design Li and Cripps studied for rational curves. 1997. Identification of inflection points and cusps on rational curves. Computer Aided Geometric Design Monterde is devoted to the description of singularities of rational B é zier curves. 2001. Singularities of rational B é zier curves. Computer Aided Geometric Design Yang, Q. and Wang, G., 2004. Inflection points and singularities on C-curves. Computer Aided Geometric Design

5. The ruled surface of vanishing curvature positions

6. vanishes at

7. The ruled surface of vanishing curvature positions Dicriminants: Tangent line of : is tangent line of

8. The ruled surface of vanishing curvature positions The result: has a cusp at control point is on the curve A vanishing curvature at the point the incidence of and the tangent line of

9. The loop surface a self intersection point applying a loop surface the boundary curves

10. Applying Discriminant of curves B é zier curves Rational B é zier curves C- B é zier curves

11. Discriminant of B é zier curves discriminants of B é zier curves n=3 is parabolic arc starting at with tangent direction. is a parabolic arc, and are hyperbolic arcs.

12. Discriminant of B é zier curves the loop surface of B é zier curves is parabolic arc starting at. is an elliptic arc with endpoints and.

13. Discriminant of rational B é zier curves The cubic rational B é zier curves Discriminant is a rational quartic curve which is a combination of control points and. is planar. The loop surface is degenerated to a plane region but this plane is not the one spanned by control points and.

14. Singularities of C- B é zier curves

16. Singularities of C- B é zier curves

18. Singularities of C- B é zier curves

19. Comparison

20. Conclusions Advantages Get information on the existence of the singularity Obtian singularity exact place Online visual aid for desigeners

21. References

22. Thank you