A Study on New Roulette and Special Forms of Cycloid and Laithoidal Curves

Abstract

This article deals with a new roulette of special curves formed by a circle rolling along a line which are given the name of Laithoid curves. The new curve is a new special form of cycloid produced by rolling a circle along a horizontal line of 4 times the rolling circle's radius. It is the locus traced out by a point fixed to a circle (where the point may be on, inside, or outside the circle), as it rolls along a straight line. In this paper, a set of 6 forms of new curvatures within two groups are produced depending on a rolling circle on the Laithoid's curve, and their geometrical and algebra proportions are graphically formed. Furthermore, the article provides the coordinate equations that govern the points along these curves. With the potential to pave the way for exploring additional geometric aspects relevant to this class of curves, and to enable comparative analyses across diverse mathematical and geometric domains, particularly in three-dimensional contexts in the future.