With the rapid evolution of Internet technology, fog computing has taken a major role in managing large amounts of data. The major concerns in this domain are security and privacy. Therefore, attaining a reliable level of confidentiality in the fog computing environment is a pivotal task. Among different types of data stored in the fog, the 3D point and mesh fog data are increasingly popular in recent days, due to the growth of 3D modelling and 3D printing technologies. Hence, in this research, we propose a novel scheme for preserving the privacy of 3D point and mesh fog data. Chaotic Cat mapbased data encryption is a recently trending research area due to its unique properties like pseudo-randomness, deterministic nature, sensitivity to initial conditions, ergodicity, etc. To boost encryption efficiency significantly, in this work, we propose a novel Chaotic Cat map. The sequence generated by this map is used to transform the coordinates of the fog data. The improved range of the proposed map is depicted using bifurcation analysis. The quality of the proposed Chaotic Cat map is also analyzed using metrics like Lyapunov exponent and approximate entropy. We also demonstrate the performance of the proposed encryption framework using attacks like brute-force attack and statistical attack. The experimental results clearly depict that the proposed framework produces the best results compared to the previous works in the literature.