=ADD=
=reftype=
14
=number=
04-05
=url=
ftp://ftp.risc.uni-linz.ac.at/pub/techreports/2004/04-05.ps.gz
=year=
2004
=month=
04
=author=
Shalaby; Mohamed
=title=
Apline Implicitization of Planar Shapes and Applications
=abstract=
 A new method for constructing algebraic spline curves by approximate
implicitization
is proposed. Our research aims at generating a low degree $C^m$ implicit
spline representation
of a given parametric planar curve for $m=0, 1$.  To ensure the low
degree condition, quadratic B--splines 
are used to approximate the given curve via orthogonal projection in
Sobolev spaces. 
Adaptive knot removal, which is based on spline wavelets, is used to
reduce the number
of segments. The resulting spline segments are implicitized.\\

In order to generate a continuous bivariate function, the implicitized
segments are joined  along suitable transversal lines. This yields a
globally continuous bivariate function.  As shown by analyzing the
asymptotic behavior of these transversal lines for step size $h
\rightarrow 0$, the given curve can be implicitized with any desired
accuracy.

In order to generate a $C^1$ bivariate function, we multiply the
implicitized segments  with
suitable polynomial factors. The resulting bivariate functions are
joined along suitable transversal lines. This yields a globally $C^1$
bivariate function.

We conclude the thesis by discussing some applications for the proposed
method.
Our method yields a simple and powerful tool for the spline
implicitization of planar curves.            
=note=
PhD Thesis
=sponsor=
FWF project SFB013 15
=keywords=
implicitization, B-spline, approximation, foot point, hierarchy of shapes