Algorithms For Orthogonal And Directional Projections Of Parametric Curves Onto Free-form Surfaces

Projection of curves onto surfaces plays an important role in curve and surfacemodeling, solid modeling, artistic design and industrial manufacture. However, un-til now, there are few papers dealing with this problem. Meanwhile, there are somedrawbacks in the existing projection algorithms. For example, the approximate curvethey got is not on the base surface, and the approximation precision of the approximatecurve cannot be controlled, etc. Moreover, the projections have been implemented inmany CAD (computer aided design) systems as an important part. But according toour experiments, they may fail in the orthogonal projection, when the projection curvelies out of the surface and the test curve passes the curvature center of the surface. Thispaper mainly discusses two kinds of projections–orthogonal projection and directionalprojection.This paper presents an approximate algorithm for the orthogonal projection ofparametric curves onto B-spline surfaces. The algorithm first compute the point projec-tion of the starting point of the test curve onto the base surface, and generates a polylinein the parametric domain of the base surface, using the second order tracing methodof orthogonal projection, with which the pre-image curve of the orthogonal projectioncurve is approximated. Then the final3D approximate curve can be obtained by map-ping the polyline onto the base surface. This paper compares this algorithm with theexisting first order algorithms in the experiments. And the experimental results showthat the accuracies of the two kinds of algorithms are comparable, but this algorithm ismuch faster than the first order algorithms. Meanwhile the approximate precision andthe continuity of the approximate curve can be controlled in this algorithm, which over-comes the drawbacks of the existing first order algorithms. According the experimentresults, this algorithm can also correctly deal with the two kinds of special projections–the projection curve lies out of the surface and the test curve passes the curvature centerof the surface. For the directional projection of curves onto surfaces, there are no related papersso far. So this paper generalizes the approximate algorithm for the orthogonal projec-tion above to the problem of directional projection. The algorithm first compute thedirectional projection of the starting point of the test curve onto the base surface usingthe line-surface intersection method. And the approximate polyline is generated withthe second order tracing method of directional projection. Then the approximationprecision and the continuity of the approximate curve are controlled in the same wayas the orthogonal projection. The experimental results show that the efciency of thealgorithm has met the real-time requirement. And it can also control the approximateprecision and the continuity of the approximate curve.