Blending Functions Generating a Cubic Curve that Closely Approximates a Polygon from Four Control Points

We introduce a set of cubic blending functions that generate a curve that allows closer approximation of a control polygon defined by four control points. The curve interpolates both endpoints and the direction of the tangent line at each endpoint follows the direction of the respective end of the control polygon. Our curve also interpolates the midpoint of the two inner control points and the direction of the tangent line at this point is in the same direction as the central segment of the control polygon. Importantly, our blending functions satisfy the Partition of Unity, Positivity, Local Support and Variation Diminishing properties. All these make our scheme suitable for the constrained piecewise interpolation of 2-dimensional data points.