In this paper we examine three new surfaces associated to a given torus in three-dimensional Euclidean space. These surfaces are obtained either by the coefficients of the second fundamental form or by the Gaussian and mean curvatures of the original torus. Anyone of the associated surfaces and the torus form a pair which is invariant under arbitrary orientation-preserving similarity transformation of the Euclidean space. Differences among the three obtained surfaces are also discussed.
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