Connections between antisymmetry, (3)-symmetry, (32)-symmetry groups and curvilinear symmetry groups, defined by P. L. Dubov, are established. Curvilinear symmetry groups of categories $\tilde{G}_{20}$, $\tilde{G}_{21}$, $\tilde{G}_2$, are derived. By applying, respectively, curvilinear symmetry groups of categories $\tilde{G}_{20}$, $\tilde{G}_{21}$ on finite segment patterns LPF and strip segment patterns LPS, corresponding curvilinear finite segment patterns LPFC and curvilinear strip segment patterns LPSC, are enumerated and visually interpreted.