Zernike moments (ZM) have potentials in pattern recognition, image analysis/processing, and in computer vision. Their invariance and orthogonality are attractive in many applications. Unfortunately, direct computation of factorials in them is time consuming and discouraging. Also, people have shown poor invariance for Cartesian ZM in the discrete case. Need of a faster computing algorithm with better rotational invariance is already felt in various research communities. To find a solution, we have examined a completely new approach which comprises a unique mapping of a gray-level image on a discrete circular disk to compute Cartesian as well as polar ZM through a new recursive relation. Such a map does not require any special sampling. The proposed new recursion is based on the computational structure of the radial polynomial embedded in the ZM. Fast computation of ZM, in our case, uses the developed new recursion and the properties of discrete circles, e.g., symmetry, uniqueness, and hole-free generation of discrete disk. This discrete circular map helps not only the fast computation of ZM but also lends its support to provide better rotational invariance for polar ZM over that of Cartesian ZM. The invariance is useful in recognition.