By Hans Hagen
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Further Work There remain several problems that must be addressed if free-form blending is to be a truly general technique for geometric modeling. A method for performing Boolean set operations on volumes created by free-form blending is essential. In this respect, the mesh on which the interpolant is defined is both a boon and a hindrance. If two volumes share the same mesh, then the subdivision property of the interpolant allows for a simple recursive algorithm for set operations. However, if the volumes do not share a common mesh (say one of the volumes has been rotated), methods for set operations are much trickier.
The input is an octahedron represented by vertices v t -,i = 0 - - - 5 , edges e j , j = 0 - - - 1 1 and faces f k , k = 0 • • • 7, together with vertex normals n/,/ = 0 • • -5. 0). e0 = (t> 4 ,t>o), e C2 = (VQ,V5), e3 = (v5,Vi), 64 es es ew = ^3,^5), = (v3,v4), = (v3,v2), = (vi,v 2 ), i = Os,^), e5 e7 e9 en = (vi,v3), = (v2,v4), = (v0,vi), = (v 2 ,t>o). /o = (e 0 ,6 4 ,e 5 ), /i = (ei,e 5 ,e 3 ), /2 = (65,64,63), /3 = (64,62,63), /4 = (61,60,65), /5 = (61,63,62), /6 = (62,64,60), /7 = (62,60,61).
The emphasis being algebraic space curves, the "normals" and higher order derivatives along curves are restricted to polynomials of some degree. 2 Surface Fitting 25 with its 1 • • • fcth-order derivatives, respectively, in the same direction as the specified "normal" vectors and its derivatives along the entire span of the CjS. [This is one natural generalization into space of the usual two-dimensional Hermite interpolation, applied to fitting curves through point data and matching derivatives at those points] 2.