Probabilistic Nodes Combination: Modeling and Interpolation of 2D Curve

1. Introduction

Curve interpolation and extrapolation (Collins II, 2003) represents one of the most important problems in mathematics: how to model the curve (Chapra, 2012) via discrete set of two-dimensional points (Ralston & Rabinowitz, 2001)? Also the matter of curve representation and parameterization is still opened in mathematics and computer sciences (Zhang & Lu, 2004). The author wants to approach a problem of curve modeling by characteristic points. Proposed method relies on functional modeling of curve points situated between the basic set of the nodes or outside the nodes. The functions that are used in calculations represent whole family of elementary functions with inverse functions: polynomials, trigonometric, cyclometric, logarithmic, exponential and power function. These functions are treated as probability distribution functions in the range [0;1]. Nowadays methods apply mainly polynomial functions, for example Bernstein polynomials in Bezier curves, splines and NURBS (Schumaker, 2007). Numerical methods for data interpolation or extrapolation are based on polynomial or trigonometric functions, for example Lagrange, Newton, Aitken and Hermite methods. These methods have some weak sides (Dahlquist & Bjoerck, 1974) and are not sufficient for curve interpolation and extrapolation in the situations when the curve cannot be build by polynomials or trigonometric functions. Proposed 2D curve interpolation and extrapolation is the functional modeling via any elementary functions and it helps us to fit the curve during the computations.

The author presents novel Probabilistic Nodes Combination (PNC) method of curve interpolation and extrapolation. This paper takes up new PNC method of two-dimensional curve modeling via the examples using the family of Hurwitz-Radon matrices (MHR method). The method of PNC requires minimal assumptions: the only information about a curve is the set of at least two nodes. Proposed PNC method is applied in curve modeling via different coefficients: polynomial, sinusoidal, cosinusoidal, tangent, cotangent, logarithmic, exponential, arcsin, arccos, arctan, arcctg or power. Function for PNC calculations is chosen individually at each interpolation and it represents probability distribution function of parameter α ∈ [0;1] for every point situated between two interpolation knots. PNC method uses two-dimensional vectors (x,y) for curve modeling - knots (x_i,y_i) ∈ R² in PNC method:

• 1.

  PNC version with no matrices (N = 1) needs 2 knots or more;

• 2.

  At least five knots (x₁,y₁), (x₂,y₂), (x₃,y₃), (x₄,y₄) and (x₅,y₅) if PNC in MHR version is implemented with matrices of dimension N = 2;

• 3.

  For more precise modeling knots ought to be settled at key points of the curve, for example local minimum or maximum and at least one node between two successive local extrema.

Condition 2 is connected with important features of MHR method: MHR version with matrices of dimension N = 2 (MHR-2) requires at least five nodes, MHR version with matrices of dimension N = 4 (MHR-4) needs at least nine nodes and MHR version with matrices of dimension N = 8 (MHR-8) requires at least 17 nodes. Condition 3 means for example the highest point of the curve in a particular orientation, convexity changing or curvature extrema (Figure 1). So this paper wants to answer the question: how to interpolate end extrapolate the curve by a set of knots?

Figure 1.

Knots of the curve before modeling

Coefficients for curve modeling are computed using probability distribution functions: polynomials, power functions, sinus, cosine, tangent, cotangent, logarithm, exponent or arcsin, arccos, arctan, arcctg.