This paper can be considered as a research on Algebraic Differential Geometry. It is about differential rational invariants of subgroups of the Affine group over the constant fields of partial differential fields (characteristic zero). The obtained results can be formulated in terms of Differential Geometry as follows: 1. For any motion group represented by a subgroup H of the Affine group it is shown that systems of generators of a field of H-invariant (not differential) rational functions can be used to construct systems of generators for the differential field of H-invariant differential rational functions of parameterized surface (patch). 2. For some classic motion groups H the generating systems of the field of H-invariant differential functions are presented. 3. For motion groups, including all classical subgroups of the Affine group, separating systems of invariants, uniqueness and existence theorems are offered.

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