Computing nodes for plane data points by constructing cubic polynomial with constraints

Abstract

To construct a parametric polynomial curve for interpolating a set of data points, the interpolation accuracy and shape of the constructed curve are influenced by two principal factors: parameterization of the data points (computing a node for each data point) and interpolation method. A new method of computing nodes for a set of data points was proposed. In this paper, the functional relationship between data points and corresponding nodes in cubic polynomials was established. Using this functional relationship, a functional cubic polynomial with one degree of freedom can pass through four adjacent data points. The degree of the freedom can be represented by two adjacent node intervals can be obtained by minimizing the cubic terms of the cubic polynomial. Since each node is computed in different node spaces, a method for constructing a quadratic curve is presented, which transforms all the quadratic curves into a unified form to compute nodes. Nodes computed using the new method exhibit quadratic polynomial precision, i.e., if the set of data point is taken from a quadratic polynomial $F\left(t\right)$, the nodes by the new method are used to construct a interpolation curve, an interpolation method reproducing quadratic polynomial gives quadratic polynomial $F\left(t\right)$. The primary advantage of the proposed method is that the constructed curve has a shape described by data points. Another advantage of the new method is that the nodes computed by it have affine invariance. The experimental results indicate that the curve constructed by the nodes using the new method has a better interpolation accuracy and shape compared to that constructed using other methods.