Low-cost experimental application of real-time kinematic positioning for increasing the benefits in cereal crops

In this paper, we introduce the notion of infinity branches and approaching surfaces. We obtain an algorithm that compares the behavior at the infinity of two given algebraic surfaces that are defined by an irreducible polynomial. Furthermore, we show that if two surfaces have the same asymptotic behavior, the Hausdorff distance between them is finite. All these concepts are new and represent a great advance for the study of surfaces and their applications.

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, $\mathcal{C}$, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.

In this paper we present two algorithms for computing the g-asymptotes or generalized asymptotes, of a plane algebraic curve, $$\mathscr {C}$$C , implicitly or parametrically defined. The asymptotes of a curve $$\mathscr {C}$$C reflect the status of $$\mathscr {C}$$C at points with sufficiently large coordinates. It is well known that an asymptote of a curve $$\mathscr {C}$$C is a line such that the distance between $$\mathscr {C}$$C and the line approaches zero as they tend to infinity. However, a curve $$\mathscr {C}$$C may have more general curves than lines describing the status of $$\mathscr {C}$$C at infinity. These curves are known as g-asymptotes or generalized asymptotes. The pseudocodes of these algorithms are presented, as well as the corresponding implementations. For this purpose, we use the algebra software . A comparative analysis of the algorithms is carried out, based on some properties of the input curves and their results to analyze the efficiency of the algorithms and to establish comparative criteria. The results presented in this paper are a starting point to generalize this study to surfaces or to curves defined by a non-rational parametrization, as well as to improve the efficiency of the algorithms. Additionally, the methods developed can provide a new and different approach in prediction (regression) or classification algorithms in the machine learning field.

This paper presents an interpretation for the extraneous factors appearing in the computation of iterated resultants with respect to a variable for three multivariate polynomials defined from the rational parametrization of a surface. We introduce a method for deriving the implicit equation of a surface using only two univariate resultants, whose polynomials match the degree of the parametrization. Our results provide a new approach to extracting the implicit representation of rational surfaces.

In this paper, we first summarize the existing algorithms for computing all the generalized asymptotes of a plane algebraic curve implicitly or parametrically defined. From these previous results, we derive a method that allows to easily compute the whole branch and all the generalized asymptotes of a “special” curve defined in n-dimensional space by a parametrization that is not necessarily rational. So, some new concepts and methods are established for this type of curves. The approach is based on the notion of perfect curves introduced from the concepts and results presented in previous papers.