On the singularity of a class of parametric curves

On the singularity of a class of parametric curves

Imre JuhászUniversity of Miskolc, Hungary

CAGD, In Press, Available online 5 July 2005

Reporter: Chen WenyuThursday, 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 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.

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., 1989. A geometric

characterization of parametric cubic curves. ACM Trans. Graph. 8 (3), 147–163.

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.

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

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): 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.

Bezier curves

Consider i = 0,

Then the loop surface:

Bezier curves

Result (n=2): l0(u,1-u) is a parabolic arc. l0(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., 1996. C-curves: an extension of cubic curves. CAGD.

Definition

where

C-Bezier curves Denote C-Bezier curves as

Then the discriminant curve c3(u) is

C-Bezier curves Applying the parameter

transformation

then

C-Bezier curves

C-Bezier curves

C-Bezier curves

The loop surface l3(u,δ) of a C-Bézier 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., 2004. Inflection points and si

ngularities 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

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 !