We propose new approaches to ray tracing that greatly reduce the required number of operations while strictly preserving the geometrical correctness of the solution. A hierarchical "beam" structure serves as a proxy for a collection of rays. It is tested against a kd-tree representing the overall scene in order to discard from consideration the sub-set of the kd-tree (and hence the scene) that is guaranteed not to intersect with any possible ray inside the beam. This allows for all the rays inside the beam to start traversing the tree from some node deep inside thus eliminating unnecessary operations. The original beam can be further sub-divided, and we can either continue looking for new optimal entry points for the sub-beams, or we can decompose the beam into individual rays. This is a hierarchical process that can be adapted to the geometrical complexity of a particular view direction allowing for efficient geometric anti-aliasing. By amortizing the cost of partially traversing the tree for all the rays in a beam, up to an order of magnitude performance improvement can be achieved enabling interactivity for complex scenes on ordinary desktop machines.