Implicit Blending Based On High-dimension Morgan-Scott Partition

In CAGD, CAD/CAM, blending, that is, constructing transitional surfaces between several given surfaces has been widely used in the geometric design of machinery, aircrafts, ships, automobiles and so forth, to enhance flow fluid, diminish stress concentration and sometimes even to improve aesthetics. It involves in partial geometry, partial differential equations, multivariate splines, computer graphics, and so forth.Further, as there are basically two ways of defining surfaces in geometric modeling (parametric and implicit), so, correspondingly, there are two kinds of blending:parametric blending and implicit blending.As parametric surfaces are easy to drawing, tessellation, subdivision and texture mapping and so forth, so parametric blending has always been paid many attentions and has a lot of methods. Here we only give a simply introduction of some classic methods for parametric blending.Besides, as implicit surfaces have the properties: easier to decide the position of a point with the surface, easier to show the shape of the object and so forth, which make implicit blending become more and more attractive. Several methods have been given, which can be generalized into four categories: the functional splines method, the blending operators method, the potential method and the piecewise algebraic surfaces method.In this paper, we first present and analyze the existing blending method, then introduce some preliminaries of the B-net method, the multivariate splines and the high-dimension Morgan-Scott partition. Finally, we develop further to give a new space partition of R_+~n and apply it to simultaneous blending of convex corners and polyhedra. For getting theexpression of the spline on each cell of the partition, we first use the B-net method to get the one on the corresponding n-1 dimension Morgan-Scott partition and then implicitize and homogenize it. Several examples are presented in the end to demonstrate the validity of our method.