Many systems near criticality can be described by Hamiltonians involving se
veral relevant couplings and possessing many nontrivial fixed points. A sim
ple and physically appealing characterization of the crossover lines and su
rfaces connecting different nontrivial fixed points is presented. Generaliz
ed crossover is related to the vanishing of the renormalization group funct
ion Z(t)(-1). An explicit example is discussed in detail based on the tetra
gonal. Landau-Ginzburg-Wilson Hamiltonian.