Enhanced techniques applicable to a ray tracing hardware accelerator for traversing a hierarchical acceleration structure are disclosed. The traversal efficiency of such hardware accelerators are improved, for example, by transforming a ray, in hardware, from the ray's coordinate space to two or more coordinate spaces at respective points in traversing the hierarchical acceleration structure. In one example, the hardware accelerator is configured to transform a ray, received from a processor, from the world space to at least one alternate world space and then to an object space in hardware before a corresponding ray-primitive intersection results are returned to the processor. The techniques disclosed herein facilitate the use of additional coordinate spaces to orient acceleration structures in a manner that more efficiently approximate the space occupied by the underlying primitives being ray-traced.