Surface Fitting Using Implicit Algebraic Surface Patches

1 Introduction 2 Interpolation and least-squares approximation provide! efficient ways of generating Ck.continuous meshes of surface patches, necessary for the consLrucLion of accurate computer geometric models of solid physical objects [see for e.g. [8.7]. Two surfaces J(:I;, Y,z) = 0 and g(x, y, z) = 0 meet with Ck-continuity along a curve C if and only if there exists functions a:(x, Y, z) and J3(x, v, z) such that all derivatives upta order ,;; of Ct.!-tJg equals zero [~ce for e.g.[G2]]. Ck-conl.inuity of two surface patches follows if the above conditioll is true along the common boundary curves between the two patches. This paper surveys the use of low dCF;L'Ce. implicitly defined, algebraic surfaces and surface patches in three dimensional real space m. 3 for various scat.tered data Ck-fitting problems. The use of low degree algebraic surface patches to constrHct models of physical ol>jects stems from the advantage of faster computations in sul>sequent geometric model manipulation operations such as computer graphics display, animation. and physica.l ol>ject simulations, see for e.g. (10]. Why algebraic surfaces? A real algebraic surface S in m. 3 is implicitly defined by a single polynomial equation F: !(x,y, z) = 0, where coeIfici<lllts of! arc over the real numlJers m... 1fanipulating polynomials, as opposed to arlJiLrary anaJytic fUllctions. is computationally more efficient. Furthermore algebraic surfaces provide enough gencl'illi\.y lo accurately model almost all complicated rigid objects. IThc oUlpul of olle opera.tion acts as [he inpul. 10 another operaLioll