We present an efficient and robust algorithm for computing the intersection curve of two freeform surfaces using a Bounding Volume Hierarchy (BVH), where the leaf nodes contain osculating toroidal patches. The covering of each surface by a union of tightly fitting toroidal patches greatly simplifies the geometric operations involved in the surface–surface-intersection computation, i.e., the bounding of surface normals, the detection of surface binormals, the point projection from one surface to the other surface, and the intersection of local surface patches. Moreover, the hierarchy of simple bounding volumes (such as rectangle-swept spheres) accelerates the geometric search for the potential pairs of surface patches that may generate some curve segments in the surface–surface-intersection. We demonstrate the effectiveness of our approach by using test examples of intersecting two freeform surfaces, including some highly non-trivial examples with tangential intersections. In particular, we test the intersection of two almost identical surfaces, where one surface is obtained from the same surface, using a rotation around a normal line by a smaller and smaller angle θ = 10 − k degree, k = 0 , . . . , 5. The intersection results are often given as surface subpatches in some highly tangential areas, and even as the whole surface itself, when θ = 0 . 00001 ◦ .