Efficiently computing geodesic offsets on triangle meshes by the extended Xin–Wang algorithm

Geodesic offset curves are important for many industrial applications, such as solid modeling, robot-path planning, the generation of tool paths for NC machining, etc. Although the offset problem is well studied in classical differential geometry and computer-aided design, where the underlying surface is sufficiently smooth, very few algorithms are available for computing geodesic offsets on discrete representation, in which the input is typically a polyline curve restricted on a piecewise linear mesh. In this paper, we propose an efficient and exact algorithm to compute the geodesic offsets on triangle meshes by extending the Xin–Wang algorithm of discrete geodesics. We define a new data structure called parallel-source windows, and extend both the “one angle one split” and the filtering theorem to maintain the window tree. Similar to the original Xin–Wang algorithm, our extended algorithm has an O(n) space complexity and an O(n²logn) asymptotic time complexity, where n is the number of vertices on the input mesh. We tested our algorithm on numerous real-world models and showed that our algorithm is exact, efficient and robust, and can be applied to large scale models with complicated geometry and topology.