Abstract: For the purpose of the curves reuse and redesign, an array replication method of curves on manifold triangulation surfaces is proposed. In our method, the curves on mesh surfaces are denoted by geodesic B-splines, which can transform the operation for curves to the operation for control points of curves. By introducing the discrete exponential mapping theory, the control points of the source curves can be mapped into its tangent space, and their normal coordinates can be obtained. Taking advantage of the invariable property of the normal co- ordinates, the corresponding relation between the pre- and post-array control points are established, with which the regular multiple replication of curves can be realized. The normal coordinates can retain the geodesic distance and relative position of the control points, thus ensuring the shape retention of the curves in reuse and array. The position from the shape of the curve array processing is seperated, making sure that the curve generated only related to the local area and ignore the overall size of the surface. What’s more, it is not only easy to ensure the shape retention of the curve, but also reduce the calculation. Experimental results show that, our method is robust, effective, and can meet the requirements of interactive design of curves on mesh surfaces.

Key words: Geodesic B-spline, Logarithmic mapping, Mesh surface, Pattern, Reuse, grinding surface quality, grinding temperature, intermittent grinding, superalloy, pressurized internal-cooling