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On the singularity of a class parametric curves Imre Juhasz
2006 CAGD
Reporter: Wei Wang 2006-10-19
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Outline
Introduction Previous works The i-discriminant curve and loop surface Applications Conclusions and future work
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Introduction
About the author:
an associate professor at Department of Descriptive Geometry at the University of Miskolc in Hungary. his research interests are constructive geometry and computer aided geometric design.
Imre Juhász
About the paper:
the paper discusses the singularity of a class parametric curves that are linear combinations of control points and basis functions and lets a control point vary while the rest is held fixed, and show the locus of the moving control point that yields a singularity (inflection point, cusp, and loop ) point. the paper discusses singularities of Bezier curves, rational Bezier curves and C-Bezier curves.
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Previous work
Previous work:
Wang,J.Y., 1981. Shape classification of the parametric cubic curve and parametric B-spline cubic curve . Computer-Aided Design, 13(4),199-206
Su,B.,Liu,D.,1983. An affine invariant and its application in computational geometry. Scientia Sinica (ser.A)24(3),259-267. Stone, M.C., DeRose, T.D., 1989. A geometric characterization of parametric cubic curves.ACM Trans. Graph. 8 (3), 147–163. Monterde,J.,2001.Sigularities of rational Bezier curves .Computer Aided Geometric Design18(8),805-816.
Yang,Q.,Wang,G.,2004.Inflection points and singularities on C-curves. Computer Aided Geometric Design21(2),207-213.
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Singularities
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The i-discriminant and loop surface
Vanishing curvature positions
Consider parametric curves
Let control points d
i
vary , i {0 ,1, , n}, and fix the rest.
The curvature
of g(ū) vanishes at
if :
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CAGD&CG The i-discriminant
The i-discriminant and loop surface
（３）
（４）
（５）
Whenλ = 0，determine the curve And
,we call it i-discriminant .
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The i-discriminant and loop surface
tangent line of i-discriminant
( u ) Fi ( u ) r ci (u ) ( ). F ( u ) F ( u ) F ( u )
(5)
Eq.(5) is the parametric representation of a straight line with parameterλ.
λ vary Tangent line ū vary
Tangent surface
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The i-discriminant and loop surface
Characterization of singularities of g(u)
The necessary and sufficient condition for a vanishing ˆ ˆ curvature at the point g ( u ), u [ a , b ], is the incidence of d i and tangent line of c i (u ) at u . ˆ
Control point iff there is a cusp at
is on the curve c i (u ) .
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The i-discriminant and loop surface
Loop surface
is a self intersection point
if
such that
Therefore, the locus of the control point
is the loop surface
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The i-discriminant and loop surface
Loop surface
So,it is a triangular surface,
the boundary curves are
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Application to Bezier
Bezier curve
r0 ( u )
Where
is the jth Bernstein polymomial of degree n. :
So we can obtain 0-discriminant
Where
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Application to Bezier
Bezier curve
Because,
So,
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Application to Bezier
Bezier curve
Loop surface
0
Boundary curves
Where ,t=u/(1-u) .
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Application to Bezier
Results when n=3
Discriminant curves
c0 (u ) t ( d 3 d 2 ) 2t ( d 2 d1 ) d1
2
c1 ( u )
( 1 u ) d 0 2 u (1 u ) d 2 u ( d 2 d 3 ) ( 1 u ) 2 u (1 u )
2
2
2
Where ,t=u/(1-u) .
c0(u) is a parabolic arc starting at d1 with tangent direction d2−d1. c3(u) is also a parabolic arc, while c1(u) and c2(u) are hyperbolic arcs.
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Application to Bezier
Boundary of loop surface
boundary δ = 1-u u=0 δ ->0 representation
l 0 ( u ,1 u ) 3 t ( d 3 d 2 ) 3 t ( d 3 d 1 ) d 3
2
Property
t=u/(1-u) a parabolic arc w0=3,w1=3/2,w2=1 an elliptic arc
c0 (u ) t ( d 3 d 2 ) 2t ( d 2 d1 ) d1
2
a parabolic arc starting at d1
So, all boundary curves are in the plane spanned by d1,d2,d3.
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Application to Bezier
Singularities when n=3
0-discriminant is planar Tangent surface is planar Loop surface is planar
In the same plane
Plane spanned by d1,d2,d3
Only plane cubic Bezier curve can have a cusp ,a loop or a zero curvature point
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Application to rational Bezier
Discriminant curves
Monterde,2001,
Discriminant curves of rational Bezier curves of degree n are rational curves of degree 2(n-1).
C0(u) is a quartic curve which is combination of control point
d1,d2,d3.
c0(u)
the torsion vanishes everywhere
c0(u) is planar
But, d1,d2,d3 is not contained by the plane !
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Application to rational Bezier
Results of cubic rational Bezier
Loop surface l0(u,δ) is degenetated to a plane region but this plane is not the one spanned by d1,d2,d3.
Therefore, twisted rational cubic Bezier curves can have a singularity (a cusp ,a loop or a zero curvature point).
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Application to C-Bezier
C-Bezier curves:
Basis functions
where
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Application to C-Bezier
The 3-discriminant
Parameter transformation
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Application to C-Bezier
Results when n=3
c0(u) is a parabolic arc starting at d1 with tangent direction d2−d1. c3(u) is also a parabolic arc ending at d1 with tangent direction d2−d1. while c1(u) and c2(u) are hyperbolic arcs.
Therefore C-Bezier curve can have a cusp or a zero curvature point if their control points are coplanar, i.e., the curve is planar.
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Application to C-Bezier
Loop surface
So,
all in the plane spanned by d0,d1,d2.
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Application to C-Bezier
Results of loop surface
Loop surface is in the plane spanned by {d0,d1,d2,d3}\{di}, the same as the tangent surface of .
Therefore, only planar C-Bezier curves can have a loop.
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Application to C-Bezier
The four discriminant curves with their tangent surface of the same C-Bezier curve(a=1.5)
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Application to C-Bezier
The four discriminant curves with their tangent surface of the same C-Bezier curve(a=1.5)
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Application to C-Bezier
Characterization
Region C3(u)
R1 R2 R3 R4
Property of C-Bezier Has a cusp
Has an inflection point Has two inflection points Has a loop No singularity
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Comparison
Bezier C-Bezier Rational Bezier
Cubic Discriminant
Tangent surface Loop surface li(u,δ)
C0(u),C3(u) parabolic arcs Plane quartic rational curves C1(u),C2(u) hyperbolic arcs
All degenerated to plane region degenerated to plane region spaned by
{d0,d1,d2,d3}\{di}.
degenerated to plane region but not spaned by
{d0,d1,d2,d3}\{di}
Singularity
Only when the curve is planar
Even when the curve is twisted
From the aspect of singularity ,cubic Bezier and C-Bezier exhibit higher similarity than integral and rational Bezier curves do.
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Conclusions and Future work
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.
The positions of the moving control point that yields a loop on the curve is a surface.
WangWei 2009-5-1
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Conclusions and Future work
Future work
For each control point we get a discriminant curve. Stone and DeRose(1989)point out the relation between such discriminants of cubic parametric curves.
It would be interesting to find how discriminant curves are related in this more general case.
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Thanks!
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