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\begin{document}
\title{Parametric Curves and Applications in CAGD: Introduction and some 
Special Issues}
\author{ $\begin{array}{ccc} \mbox{T. Recio}\thanks{
Partially supported by CICyT PB 92/0498/C02/01 (Geometr\'{\i}a 
Real y Algoritmos), Esprit/Bra 6846 (Posso), and TIC-1026-CE} & & 
\mbox{J.R. Sendra \& J. Sendra}\thanks{Partially supported by 
Univ. Alcal\'a Proy. 030/95} \\ 
\mbox{Dpto. de Matem\'aticas} & & \mbox{Dpto. de Matem\'aticas} \\ 
\mbox{Universidad de Cantabria} & & \mbox{Universidad de Alcal\'a} \\ 
\mbox{Santander 39071 (Spain)} & & \mbox{Madrid 28871 (Spain)}   \\ 
\mbox{recio@matsun1.unican.es} & & \mbox{mtsendra@alcala.es} 
\end{array}$}
\date{}
\maketitle
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\centerline{\bf Abstract}

\medskip
In this talk we briefly introduce the notion and main 
problems related to parametric curves, and we focus on two issues specially 
interesting in applications: the problem of computing real reparametrization 
of real parametric curves, and the rationality analysis and parametrization 
of offset curves of parametric curves.

Parametric curves play an important role in the field of Computer Aided 
Geometric Design. They arise in practical applications as computer 
graphics, geometric modeling or manipulation of offset varieties. The 
natural field of applications of rational curves is Computer Aided 
Design.  Indeed, real varietes are the most common ones in Geometric 
Design. However, since Algebraic Geometry is mainly developed over 
algebraically closed fields, the majority of the previous 
algorithms are checked only over  the complex numbers. We 
present an algorithm that decides whether a complex rational 
parametrization of a plane curve can be reparametrized over the reals, and 
if so,  it computes a real parametrization of the curve.

An interesting problem to mention is the algebraic manipulation of offset  
curves. Offset curves of a given curve, or more formerly parallel curves, 
are defined as the envelopes of systems of circles with centres on the 
original curve and fixed radious. They  arise in practical applications 
as tolerance a\-na\-ly\-sis, geometric control, robot path-planning and 
numerical-control machining problems. One of the main 
problems in offset curves manipulation is that the rationality of the 
original curve is not preserved (in general) when the offset curve is 
considered. In this paper,  a rationality analysis of the offset 
curves is presented, and symbolic algorithms to parametrized the rational 
components of the offset curves to parametric curves are derived.
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{\bf keywords:} parametric curves, cagd, introduction, real parametrization,
real curves, offset curves, rational offset curves. 

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