28 - The Construction of Curves

Abstract

This chapter explains the way to construct curves. Many interesting curves can be constructed in the Cartesian coordinates. Particularly simple is the construction of curves, whose equations are given in the parametric form. The construction of the curve is preceded by the calculation of the values of x and y corresponding to the values of the parameter t of sufficient degree of approximation. In constructing Lissajous curves, it is useful to predetermine the points of contact of the curve with the sides of the corresponding rectangle, and also to draw beforehand auxiliary straight lines inside the rectangle. A Lissajous curve is open if, for some value of the parameter t, the curve becomes wedged at a vertex of the rectangle. Cycloid, epicycloid and hypocycloid are the names of the curves described by the point M situated on the circumference of a circle, radius r, rolling along a straight line (cycloid), along the circumference of a stationary circle of radius R, touching it on the outside (epicycloid) and along the circumference of a stationary circle of radius R, touching it on the inside (hypocycloid).