A Second-Order Algorithm for Curve Parallel Projection on Parametric Surfaces

A second-order algorithm is presented to calculate the parallel projection of a parametric curve onto a parametric surface in this chapter. The essence of our approach is to transform the problem of computing parallel projection curve on the parametric surface into that of computing parametric projection curve in the two-dimensional parametric domain of the surface. First- and second-order differential geometric characteristics of the parametric projection curve in the parametric domain of the surface are firstly analyzed. A marching method based on second-order Taylor Approximation is formulated to calculate the parametric projection curve. A first-order correction technique is developed to depress the error caused by the truncated higher order terms in the marching method. Several examples are finally implemented to demonstrate the effectiveness of the proposed scheme. Experimental results indicate that both the computational efficiency and accuracy of the presented method have dominant performance as compared with the first-order differential equation method.