A plane algebraic curve can be represented as the zero-set of a polynomial in two or if one takes homogenous coordinates: three variables. The coefficients of the polynomial determine the curve uniquely. Thus features of the curve, like for instance rotation symmetry, must find their correspondence in the algebraic structure of the coefficients of the polynomial. In this article we will investi...

A curve that can be parametrized by polynomials is called a polynomial curve. It is well-known that a polynomial curve has only one place at infinity. Sathaye indicated the Abhyankar’s question for curves with one place at infinity. Let C be a curve with one place at infinity. Is there a polynomial curve associated with the semigroup generated by pole orders of C at infinity? We found a counter...

A discrete polynomial curve is defined as a set of points lying between two polynomial curves. This paper deals with the problem of fitting a discrete polynomial curve to given integer points in the presence of outliers. We formulate the problem as a discrete optimization problem in which the number of points included in the discrete polynomial curve, i.e., the number of inliers, is maximized. ...

We can generalize some of the constructions in the previous section to higher order curves. One problem with using parabolas as the basic curve element for modelling is that a parabola is a convex curve, convexity being defined as the curve lying on one one side of its tangent lines. This means that in order to model a curve that is not convex like S-curve one must use piecewise parabolic curve...

This paper deals with the problem of fitting a discrete polynomial curve to 2D noisy data. We use a discrete polynomial curve model achieving connectivity in the discrete space. We formulate the fitting as the problem to find parameters of this model maximizing the number of inliers i.e., data points contained in the discrete polynomial curve. We propose a method guaranteeing inclusion-wise max...