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An action with $n$ parameters, which generalizes the $O(N) - R P^{N-1}$-model, is considered in one dimension for general $N$. We use asymptoticexpansion techniques to determine where the model becomes critical and showthat for the actions considered there exists a family of hypersurfaces whoseasymptotic behaviour determines a one-parameter family of new universalityclasses. They interpolate between the $O(N)$-vector-model-class and the $RP^{N-1}$-model-class. Furthermore continuum limits are discussed, including theexceptional case $N=2$.Comment: 13 page

New universality classes in one-dimensional O(N)-invariant spin-models with an n-parametric action