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

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

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

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

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

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

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

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

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

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

Construct of the ruled surface The curvature zero curvature

Construct of the ruled surface

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

Construct of the ruled surface Let Then its tangent line is where

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

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

Construct of the loop surface Let Then So the loop surface

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

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

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

Bezier curves Result ( n=2 ): c 0 (u) is a parabolic arc starting at d 1 with tangent direction d 2 −d 1. c 3 (u) is also a parabolic arc, while c 1 (u) and c 2 (u) are hyperbolic arcs.

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

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

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

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

C-Bezier curves Denote C-Bezier curves as Then the discriminant curve c 3 (u) is

C-Bezier curves Applying the parameter transformation then

C-Bezier curves

The loop surface l 3 (u,δ) of a C-B é zier curve is of the form

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

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

C-Bezier curves

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

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

Thank you !