The hyperbolic multiquadric method is defined by 
n 
GMG, y) = > ej v d, y)+r (2) 
1= 
where d; 2x, y) is given above, r must be specified, 
and the coefficients c; are found by solving the n by 
pi»Lb-,.n GMis 
localized as follows: for each data point (xi, yi^ 
GM is determined by the nineteen points closest to 
(xj, y) plus (x;, y) itself. 
n system GM(x, y) - F 
The weighted quadratic least squares method GQ 
is determined by the ten points closest to G5, Y) 
plus (x;, y;) itself, for each data point (xj, yj). The 
weights are 
wi%, ¥) = (1; - da, y? 3) 
where r; equals the distance between Gu, y;) and 
the most distant of the ten closest points. In 
addition, the weighted planar least squares 
method GP is determined by the eight points 
closest to (xi, y;) plus Gu, yj) itself, for each data 
point (x;, y;). The weights as above are used with 
r; equal to the distance between (x, yj) and the 
most distant of the eight closest points. 
The triangular Shepard method is defined by the n 
data (x, y; Filz and a triangulation (v1;, 
v2, v3); 4, where m is the number of triangles 
in the triangulation. It is given by 
m 
GT, y) 2 Y wj, y) LF;(x, y) (4) 
i=l 
where 
  
1 1 
w;(x, y) = 3 LN 2 
2 izk 9 
[la en  [[ae» 
j=1 j=1 
LF;(x, y) is the linear interpolant over the i-th 
triangle, and di is the Euclidean distance from (x, 
y) to vertex j of the triangle i. Note that no 
parameters need be specified for LF, although it is 
dependent on the triangulation. 
930 
4. SURFACE INTERPOLANTS 
Several interpolation schemes that assume 
prescribed values and their gradients on the 
boundary of a planar triangle have been 
developed. Barnhill, Birkhoff, and Gordon (1973) 
first developed such a method using lines parallel 
to the triangle sides, and Nielson (1979) has 
developed method for line segments joining 
vertices to their opposite sides. This method 
consists of using the planar interpolant on the 
underlying planar triangle with boundary data 
obtained by projecting values and gradients onto 
the plane. 
For such an interpolation, let p be a point of the 
triangle with vertices, in counterclockwise order 
v1, v2, and v3, and let p' be the central projection of 
p onto the underlying planar triangle having the 
same vertices T - (v1,v2,v3), i.e., boundary points 
of the triangle project to boundary points of T. 
Denote by bl, b2, and b3 the barycentric 
coordinates of p' with respect to T. These are 
defined by 
3 
EE (5) 
1=1 
and 
3 
Y bi Vi = p (6) 
i=l 
This is a basic scheme of piecewise linear surface 
interpolation. This scheme is given a 
triangulation of a set of points in the plane and 
computes the value at (x, y) of a piecewise linear 
surface. Equivalently, b; = A;/A, where A is the 
area of t and A; is the area of the subtriangle (p', 
vi VE) for (i, j, ke s = {(1,2,3), (2,3,1), (3,1,2)). 
Consider the line defined by v1 and p'. This line 
intersects the edge opposite v1 at the point 
_b2v2+b3v3 
qu=-""2+h3 (7) 
The points q2' and q3' lying on the sides opposite 
v2 and v3, respectively, are defined similarly. The 
following side-vertex-based interpolant is derived 
by Lawson (1977). For a bivariate function f on T, 
the interpolant defines 
3 
F(p) » Y, w; Hp) (8) 
i=1