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In this paper, we give characterization of quadratic $\varepsilon-$canonical number system ($\varepsilon-$CNS) polynomials for all values $\varepsilon \in\lbrack0,1)$. Our characterization provides a unified view of the well-known characterizations of the classical quadratic CNS polynomials ($\varepsilon=0$) and quadratic SCNS polynomials ($\varepsilon=1/2$). This result is a consequence of our new characterization results of $\varepsilon -$shift radix systems ($\varepsilon-$SRS) in the two-dimensional case and their relation to quadratic $\varepsilon-$CNS polynomials., Comment: 46 pages, 2 figures