Cohen's kappa curves, new geometrical forms of dual curves

Abstract

In this article, we introduce the concepts of taxicab and uniform products in the context of dual curves associated with Cohen's kappa, primarily defined by a set of inflection curvatures of an ellipse and a circle using parallel asymptotes. The novel curve under scrutiny, denominated as the "Like-Bulb Filament" (LBF) curve, is delineated as the locus of dual vertices originating from a couple of conic curvatures. The emergence of LBF transpires through the orchestrated arrangement of line segments emanating from a predetermined central focal point upon an elliptical form concomitant with a circular entity possessing a radius equivalent to the ellipse's minor axis. The LBF’s curve is intricately choreographed through the dynamic interplay of a constant unit circle and three asymptotic lines. Notably, two of these asymptotes achieve tangential intersections with the LBF curve, while the third gracefully traverses its central core. Additionally, we embark on a comprehensive algebraic examination complemented by a geometrically informed construction methodology. In these instances, a consistent conic curvature of the uint circle and an elliptical structure assume pivotal roles in the genesis of the LBF’s curve. Also, a geometric connection is speculated between these curve configurations and their relevance to engineering processes across fields. However, the document acknowledges the need for more intensive study on the presented traits. Hence, it emphasizes addressing the existing research gap in subsequent investigations.