  We describe geometric algorithms that generalize the classical continued
fraction algorithm for the torus to all translation surfaces in hyperelliptic
components of translation surfaces. We show that these algorithms produce all
saddle connections which are best approximations in a geometric sense, which
generalizes the notion of best approximation for the classical continued
fraction. In addition, they allow to list all systoles along a Teichmueller
geodesic and all bispecial words which appear in the symbolic coding of linear
flows. The elementary moves of the described algorithms provide a geometric
invertible extension of the renormalization moves introduced by S. Ferenczi and
L. Zamboni for the corresponding interval exchange transformations.
