In this paper, we first consider a family of constraints given by straight lines. For a uniform probability distribution, we determine the constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors for all positive integers $n$. In addition, we calculate the constrained quantization dimension and the constrained quantization coefficient with respect to this family of constraints. Next, we turn to another family of constraints, consisting of concentric circles. For the same probability distribution, we present a methodology to compute the constrained optimal sets of $n$-points and the corresponding $n$th constrained quantization errors for all positive integers $n$. Finally, we conclude the paper with a summary of the results and a discussion of future research directions.