A Uniform B-spline Collocation Method for Solving the Boundary Value Problem of Geodesic Curves

Geodesics on surfaces are widely used in image processing, robotics, NC machining, etc. In this work, the second and third order uniform B-spline collocation methods are proposed to solve the BVP (boundary value problem) of geodesics on parametric surfaces. Firstly, the discrete governing equation of geodesics is established, which can be solved by the approximate Newton iteration method after an initial approximation is provided. Then the accuracy of the B-spline collocation method is deduced. Finally, the proposed method is verified on the spherical surface, torus surface and spline surface in Matlab 2016b software environment. The computational accuracy, time cost and efficiency of different methods are investigated. The experimental results show that the third order B-spline collocation method generally requires less time to obtain the same accuracy, which indicates the method has advantages in computational efficiency. When the length error of the geodesic is 0.01%, the third order B-spline collocation method saves 5.5%~26.9% of the time cost compared with Chen’s geodesic-like method. For the second order B-spline collocation method, the computational efficiency is comparable to Kasap's central difference method. The proposed method ensures continuous geodesic curves, and no additional interpolation function is needed. A potential application of this method is trajectory planning for automated tape placement.