Given a polyhedron, the set of spanning line segments (SLS) satisfies two properties: (1) completeness property — the set SLS must cover all the extreme vertices on the convex hull of the polyhedron; (2) inseparability property — there cannot be a plane that separates the set SLS into two nonempty subsets without intersecting one of them. Owing to the above two properties, the set SLS is proposed as a representation for testing the intersection between a plane and a three-dimensional (3D) polyhedron. Given a 3D polyhedron with N vertices, this paper presents an incremental O( N)-time algorithm for constructing the set SLS. The proposed algorithm has the same time complexity as the previous best result [Wang ME, Woo TC, Chen LL, Chou SY. Computing spanning line segments in three dimensions, The Visual Computer 12 (1996) 173–180], but it reduces the working memory required in the previous work from O( N) to O(1).