In this paper, we present a novel geometric method for e?ciently and robustly computing intersections between a ray and a triangular Bézier patch defi ned over a triangular domain, called the hybrid clipping (HC) algorithm. If the ray pierces the patch only once, we locate the parametric value of the intersection to a smaller triangular domain, which is determined by pairs of lines and quadratic curves, by using a multi-degree reduction method. The triangular domain is iteratively clipped into a smaller one by combining a subdivision method, until the domain size reaches a prespecifi ed threshold. When the ray intersects the patch more than once, Descartes’ rule of signs and a split step are required to isolate the intersection points. The algorithm can be proven to clip the triangular domain with a cubic convergence rate after an appropriate preprocessing procedure. The proposed algorithm has many attractive properties, such as the absence of an initial guess and insensitivity to small changes in coe?cients of the original problem. Experiments have been conducted to illustrate the e?cacy of our method in solving ray-triangular Bézier patch intersection problems.