We study a constructive method to find an algebraic curve in the real projectiveplane with a (possibly singular) topological type given in advance. Our methodworks if the topological modelto be realized has only double singularitiesand gives an algebraic curve of degree, whereandare thenumbers of double points and connected components of. This degree isoptimal in the sense that for any choice of the numbersandthere existmodels which cannot be realized algebraically with lower degree. Moreover, wecharacterize precisely which models have this property. The construction is basedon a preliminary topological manipulation of the topological model followedby some perturbation technique to obtain the polynomial which defines thealgebraic curve. This paper considers only the case in whichhas an orientableneighborhood. The non-orientable case will appear in a separate paper.
to be realized has only double singularities
and gives an algebraic curve of degree 
, where 
and 
are the
numbers of double points and connected components of 
. This degree is
optimal in the sense that for any choice of the numbers 
and 
there exist
models which cannot be realized algebraically with lower degree. Moreover, we
characterize precisely which models have this property. The construction is based
on a preliminary topological manipulation of the topological model followed
by some perturbation technique to obtain the polynomial which defines the
algebraic curve. This paper considers only the case in which 
has an orientable
neighborhood. The non-orientable case will appear in a separate paper.