The indices of the basis functions of a term are all less
then 3, then the associated b vector denotes a point When one and only one of the indices of a term is 3 or 4
then the associated b vector denotes a tangent vector When two and only two of the indices are 3 or 4 then the
associated b vector denotes a twist vector
When in the case of tricubic solid indices of a term are 3
or 4 then the associated b vector denotes the vector defined by the third order mixed partial derivatives of the function p(u,v,w) Our current method of constricting the arrays of
geometric coefficients suggests the following empirical interpretation For each odd index on b assign a zero and for each
even index assign a one
The transformation form geometric to algebraic
coefficients is given by 4
4
4
aijk M il M jm M knblmn l 1 m 1 n 1
More compactly as
aijk M il M jm M knblmn here M is the Hermite basis transformation matrix and the various subscripted indices denote specific element of the matrix
Stanton and Crain developed a slightly different approch
in 1974 Treating the 64 hyper patch parameters as 4 sets of 16
parameters This allows us to understand more easily the basic
function properties and the relationship between patches and hyper patches
Algebraic form: Geometric form: Point form:
p (u , v, w) Fi a (u ) F ja (v) Fka ( w)aijk p (u , v, w) Fi b (u ) F jb (v) Fkb ( w)bijk p (u , v, w) Fi p (u ) F jp (v) Fkp ( w) pijk
Fi a (u), Fja (v), Fka (w) are basic functions applied to the algebraic coefficients defined by
F1a (u ) u 3 , F2a (u ) u 2 , F (u ) u , a 3
Similarly for v and w
F4a (u ) 1.
Fi b (u), Fjb (v), Fkb (w) are basic functions applied to the
geometric coefficients defined by
F (u ) 2u 3u 1, b 1
3
2
F2b (u ) 2u 3 u 2 , F (u ) u 2u u, b 3
3
2
F (u ) u u . b 4
3
2
p p p Fi (u), Fj (v), Fk (w) are basic functions applied to the
point coefficients defined by
9 3 2 11 F1 (u ) u 9u u 1, 2 2 27 3 45 2 p F2 (u ) u u 9u, 2 2 27 3 p 2 9 F3 (u ) u 18u u, 2 2 9 3 9 2 p F4 (u ) u u u. 2 2 p
