3 3 



U(x,y) = R..(x,y) =2 2 a^ (x - x f {y - y / (19) 



"I m=0 n=0 ' 



where x. . < x < x. and y. ■, < y < y. # 



i-l - - i |-1 -' -' \ 



may be determined for each R.. rectangle within R. In analogy with the cubic 



M 

 spline, the unknowns a for this two-dimensional bicubic surface are computed 

 ^ mn 



by requiring that the surface fit each data point exactly, i.e., R(x.,y.) = U.., 

 and that the partial derivatives — ft anc ' — \ be continuous. In 



order to accomplish this, the first step is to compute estimates of the partial 

 derivatives at each data point by utilizing the cubic spline formulation. Let 



3U.. 3U.. 9 2 U.. 3P.. 



n II II i c - LI - LI 



P.. = -r— l , q.. = -r— l , and S.. = . . ' 5— l . 



1 1 3x 1 1 3y 1 1 3x3y 3y 



In order to compute P.. at each grid point, equations (9,11,12) are utilized 



to obtain values of the second derivative z. from the U. data values along the 



1 in 



nth row. Equation (8) is then used with these z. to produce values for P. , 



i=0 ... 1-1 with P. being computed from equation (7). This process is 



repeated for each row of input data to generate the full P.. matrix. In 



identical fashion, q.. and S.. are computed by operating on columns of U.. 

 ■ I 1 1 ij 



and P.. respectively. Utilizing the given data values and these computed 



derivatives at each grid point, the unknown, a , may be determined for 

 r nm 



each R.. rectangle covering R by solving the matrix equation 



A(x.- Vl )K..A T ( y .- yM )= (a|i n ) (20) 



The proof for this equation is given in de Boor (1962). 



Normally, the U.. are given on a square grid, in which case we let 

 h=(x. - x. ,)=(y. - y. .). The elements of the matrices in equation (20) 

 are then 



12 



