By Hans Hagen
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A degree three algebraic surface does not suffice since the rank of the resulting linear system is greater than 19, the number of independent unknowns. Next as a possible interpolant one considers a degree four algebraic surface with 34 independent unknown coefficients. Repeating the interpolation for the original curves results in 52 equations. The rank of this linear system is 33, and thus there is a one afrine independent parameter family of quartic Hermite interpolating surface, which is f ( x , y , z ) - r\z4 -\- r2+112°ri yz3 + r2+112°ri xz^ ~ Topics in Surface Modeling 48 An instance of this family (TI = 1, r?
Characterizations of C° surface fits of points and curves using implicitly defined algebraic surfaces is also given by [55]. Furthermore, implicit algebraic surfaces have been used for Cl blend fits [19], [41], [45], [51], [58]. Other approaches to parametric surface fitting and transfinite interpolation are also mentioned in that paper, as well as in [25], [47], [44], [56], [60]. Papers [11 [12] generalize the results of [50], [55]. It provides conditions for exact Ck fits of implicitly defined algebraic surfaces through given points and space curves together with derivative information ("normals") along the curves.
With the constraint that AW = W7 — W is a solution of the computed system of invariant equations. Next, suppose that a surface designer wants to see how the surface shape changes with a value change of w\ alone. However, a change in value of iui automatically changes the value of additional W{S related to it by the invariants JjS. Usually, the linear system of invariant equations are underdetermined, yielding an infinite number of choices of Atuz-s (i = 2,3, • • • ,c). Then how does the designer mak a choice of values for the WiS that reflects the influence of a change of only w\, as clearly as possible?