The rapid growth in the complexity of geometry models has necessisated revision of several conventional techniques in computer graphics. At the heart of this trend is the representation of geometry with locally constant approximations using independent sample primitives. This generally leads to a higher sampling rate and thus a high cost of representation, transmission, and rendering. We advocate an alternate approach involving context-aware samples that capture the local variation of the geometry. We detail two approaches; one, based on differential geometry and the other based on statistics. Our differential-geometry-based approach captures the context of the local geometry using an estimation of the local Taylor's series expansion. We render such samples using programmable Graphics Processing Unit (GPU) by fast approximation of the geometry in the screen space. The benefits of this representation can also be seen in other applications such as simulation of light transport. In our statistics-based approach we capture the context of the local geometry using Principal Component Analysis (PCA). This allows us to achieve hierarchical detail by modeling the geometry in a non-deterministic fashion as a hierarchical probability distribution. We approximate the geometry and its attributes using quasi-random sampling. Our results show a significant rendering speedup and savings in the geometric bandwidth when compared to current approaches.