A preliminary study of planar rational systems can be found in the paper [7] by Camouzis et al. In the work, they give some results and provide some open questions for systems of equations of the following type: x n + 1 = α 1 + β 1 x n + γ 1 y n A 1 + B 1 x n + C 1 y n, y n + 1 = α 2 + β 2 x n + γ 2 y n A 2 + B 2 x n + C 2 y n, n = 0, 1, 2, …, where the parameters are taken to be nonnegative.
In this paper we study situations when non-rational parameterizations of planar or space curves as results of certain geometric operations or constructions are obtained, in general.
We consider the problem of finding parametric rational Bézier cubic spirals (planar curves of monotonic curvature) that interpolate end conditions consisting of positions, tangents and curvatures.
The goal of this paper is to apply these two new techniques to analyze global behavior of solutions to the more general family of first-order planar systems of rational difference equations (1) with nonnegative parameters.
Consider the class of planar systems of first-order rational difference equations.
This gives rise to 100 explicit planar systems of first-order rational difference equations with positive parameters.
Mainly, we focus on one particular application of recently introduced rational envelope curves, newly constructed over an arbitrary planar rational curve in space.
Moreover, the rational concept of artificial atoms can readily be transferred to planar semiconductor electronics, allowing for the exploration of electrons in a well-defined fractal geometry, including interactions and external fields.
The paper brings together the well-known kinematics of planar two- and three-DOF kinematic chains with revolute and/or prismatic joints and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions.
Boundaries between small compounds, peptides, and peptide mimetics are blurred; small compound PPI disruptors (PPIDs) are heavier, more hydrophobic, more rigid, and more planar than conventional small compound drugs, and peptide motifs are often included in their structure as part of their rational design [ 59, 60].
