A plane cubic spline segment can be specified in man)/ways. In this paper a specification method that can be divided into two phases is presented. The first phase relates the segment to the geometry of a deining triangle, but leaves one degree of freedom. This degree of freedom is in the polynomial coefficients as a linear parafneter and the variation of the geometric relationship of the segment to the defining triangle as this parameter varies is explored. The second phase of the specification requires a choice of value for this parameter in accordance with a criterion of fairness that is specified. To support the method of this phase the variation of the curvature of the segment with changes in the parameter value is demonstrated.
To show how the specification method can be used, the paper shows that the interpolation problem for general points in the plane can give rise to a set of defining triangles. A n algorithm for generating such triangles for an open or closed curve problem for a given set of points is described. Results of using an implementation of this algorithm together with the specification method are presented. and these can be used to derive x (tJ= Xo + ~ (X-xoJ t + ((3 ~) (xi -xo) -o~ (X-Xo) J t 2 4-(x 1 -Xo) ยขo~-2) t 3 )/(tJ=)/o +a(Y-yo)t+((3-a)O/, Yo) --~ (Y-)/o)) t 2 + ()/, -Yo) ยขa 2) t 3