| ai.viXra.org > Functions and Analysis > 2608 |
|
 |
| ai.viXra.org > Functions and Analysis > 2608 |
|
|
[2] ai.viXra.org:2608.0071 [pdf] submitted on 2026-08-22 23:43:07
Authors: Ahcene Aït saadi
Comments: 17 Pages. (Note by ai.viXra.org Admin: Please cite listed scientific references!)
Many remarkable curves roses, the lemniscate, the cardioid, the limaçon have a heavyCartesian equation: implicit polynomials of degree 4, 6, up to 10, hard to plot point by pointand often multivalued. This paper presents a method that makes them drawable by hand,with no software and no polar paper: in an analytic frame (an abscissa axis together withan origin of ordinates Ou2032set o the axis by a distance λ), each of these curves is written asa single explicit ordinate y = f(x), which one tabulates and plots with a ruler. We provethe transition formulas and the pivot identity X2 + Y2 = y2 the engine guaranteeing thatthe one-line expression does describe the heavy curve then the equation of the line, of thecircle, the pseudo-derivative, and a proportional relation in the triangle. One section comparesthe two methods on six classical curves. The scope is honest and bounded: this is a methodof construction and of teaching, in the service of simpler access to objects reputed dicult.
Category: Functions and Analysis
[1] ai.viXra.org:2608.0065 [pdf] submitted on 2026-08-18 19:28:43
Authors: Ahcene Ait Saadi
Comments: 11 Pages.
Abstract. We study two elementary periodic functions — the scaled fractional part spp x (a pure ramp of period p) and its complement cpp x = p−spp x — together with the single algebraic identity spp x + cpp x = p. Because their derivatives are the constants +1 and −1, they turn a range of nonlinear differential equations into closed one-line solutions, they solve first-order transport and inviscid Burgers equations exactly, and they let one write intricate periodic curves — half-circle chains, staircases, asymmetric arcs, modulated ramps — each from a single algebraic expression. A complex generator Z(x) = sp x+i cp x opens a second, geometric part: its powers Z n are explicit algebraic curves of degree n, and the normalized family {Z α/pα} covers an eye-shaped region with an exactly empty core bounded by the unit circle. The aim is not to compete with Fourier analysis but to offer a light, exact notation for the angular, piecewise-affine patterns where Fourier series are heavy and only approximate. All identities have been verified symbolically and numerically (sympy, machine precision).
Category: Functions and Analysis