Regular algebraic curve segments (III)—applications in interactive design and data fitting

In this paper (part three of the trilogy) we use low degree G 1 and G 2 continuous regular algebraic spline curves defined within parallelograms, to interpolate an ordered set of data points in the plane. We explicitly characterize curve families whose members have the required interpolating properties and possess a minimal number of inflection points. The regular algebraic spline curves considered here have many attractive features: They are easy to construct. There exist convenient geometric control handles to locally modify the shape of the curve. The error of the approximation is controllable. Since the spline curve is always inside the parallelogram, the error of the fit is bounded by the size of the parallelogram. The spline curve can be rapidly displayed, even though the algebraic curve segments are implicitly defined.