| Summary: In this study, first of all, the similarity group B(n) and its important subgroups SB(n), LB(n), SLB(n), H(n), LH(n) are introduced in the dimension n. The G- orbits of each mentioned subgroups of the similarity group B(n) are studied and sketched in the dimension 1 and dimension 2. In addition, invariant rational functions and their generator sets for the most mentioned subgroups of the group B(n) in dimension 2 are obtained. The LB(1)- and B(1)-equivalence conditions are investigated for given two-point systems in 1-dimensional linear space R. Later the LB(2) and B(2)- invariant rational functions and their generators are obtained. Then the LB(2) and B(2)- equivalence conditions are investigated for given two points systems in 2-dimensional linear space RxR. So any two systems can be tested easily using LB(2) and B(2)- invariant rational functions.
Key Words: Invariant,Similarity Transformation, Similarity Group, Homotethy, Similarity Geometry.
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