  We present quantum algorithms to realize geometric transformations (two-point
swappings, symmetric flips, local flips, orthogonal rotations, and
translations) based on an $n$-qubit normal arbitrary superposition state
(NASS). These transformations are implemented using quantum circuits consisting
of basic quantum gates, which are constructed with polynomial numbers of
single-qubit and two-qubit gates. Complexity analysis shows that the global
operators (symmetric flips, local flips, orthogonal rotations) can be
implemented with $O(n)$ gates. The proposed geometric transformations are used
to facilitate applications of quantum images with low complexity.
