We present a technique for memory-efficient and time-efficient volume rendering of curvilinear adaptive mesh refinement data defined within extrudable computational spaces. One of the main challenges in the ray casting of curvilinear volumes is that a linear viewing ray in physical space will typically correspond to a curved ray in computational, space. The proposed method utilizes a specialized representation of curvilinear space that provides for the compact representation of parameters for transformations between computational space and physical space, without requiring extensive preprocessing. By simplifying the representation of computational space positions using an extrusion of a profile surface, the requisite transformations can be greatly simplified. Our implementation achieves interactive rates with minimal load time and memory overhead using commodity graphics hardware with real-world data.