In BRep geometric modeling system, the positions of end-points of the resultant intersection curves (called key points as well) are very important for many other operations, such as boolean operations. This paper provides a method to determine the positions of the end-points of the intersection curves between a cylinder and a quadric surface. Our algorithm ¯rst turns the original intersection problem into another intersec- tion problem between a set of line segments and a quadric surface, and obtains the parametric form of the intersection curves. Then, it computes all the key positions. Finally, it calculates the intersec- tion intervals and their corresponding intersection curves by mid-point judging method. The algo- rithm has been applied in the commercial software Gems6.0, and practical experiences show both ro- bustness and e±ciency of the new algorithm.