The orbit of an n -variate polynomial f ( x ) over a field \({\mathbb {F}} \) is the set \(\text {orb}(f) := \lbrace f(A{\mathbf {x}}+{\mathbf {b}}) : A \in {\mathrm{GL}}(n,{\mathbb {F}}) \text{ and } {\mathbf {b}} \in {\mathbb {F}}^n\rbrace \) . The orbit of a polynomial f is a geometrically interesting subset of the set of affine projections of f . Affine projections of polynomials computable by seemingly weak circuit classes can be quite powerful. For example, the polynomial \(\mathsf {IMM}_{3,d} \) – the (1, 1)-th entry of a product of d generic 3 × 3 matrices – is computable by a constant-width read-once oblivious algebraic branching program (ROABP), yet every polynomial computable by a size- s general arithmetic formula is an affine projection of \(\mathsf {IMM}_{3,\text {poly}(s)} \) as shown by Ben-or and Cleve [12]. To our knowledge, no efficient hitting set construction was known for \(\text {orb}(\mathsf {IMM}_{3, d}) \) before this work. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of: (1) Low-individual-degree polynomials computable by commutative ROABPs . This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials and the orbits of multilinear sparse polynomials . (2) Multilinear polynomials computable by constant-width ROABPs . This implies a quasi-polynomial time hitting set for the orbits of the family \(\lbrace \mathsf {IMM}_{3,d}\rbrace _{d \in {\mathbb {N}}} \) . (3) Polynomials computable by constant-depth, constant-occur formulas . This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in , and also polynomial-time hitting sets for the orbits of the power symmetric polynomials and the sum-product polynomials . (4) Polynomials computable by occur-once formulas . We say a polynomial has low individual degree if the degree of every variable in the polynomial is at most \(\text {poly}(\log n) \) , where n is the number of variables. The first two results are obtained by building upon and strengthening the rank concentration by translation technique of Agrawal, Saha, and Saxena [6]; the second result additionally uses the merge-and-reduce idea from Forbes and Shpilka [30], Forbes, Shpilka, and Saptharishi [27]. The proof of the third result adapts the algebraic independence based technique of Agrawal, Saha, Saptharishi, and Saxena [5], Beecken, Mittmann, and Saxena [11] to reduce to the case of constructing hitting sets for the orbits of sparse polynomials. A similar reduction using the Shpilka-Volkovich (SV) generator based argument in Shpilka and Volkovich [90] yields the fourth result. The SV generator plays an important role in all the four results.