Let $$\mathbb {D}$$ D be the algebra of dual numbers and $$G=M^{+}(\mathbb {D^{*}}), M(\mathbb {D^{*}})$$ G=M+(D∗),M(D∗) be linear similarity groups generated by the algebra $$\mathbb {D}$$ D in two-dimensional real vector space $$R^{2}$$ R2 . The present paper is devoted to solutions of problems of global G-equivalence of paths and curves in $$R^{2}$$ R2 for groups $$G=M^{+}(\mathbb {D^{*}}), M(\mathbb {D^{*}})$$ G=M+(D∗),M(D∗) . Complete systems of global G-invariants of a path and a curve in $$R^{2}$$ R2 are obtained. Existence and uniqueness theorems are given. Evident forms of a path and a curve with the given complete system of G-invariants are obtained.