4.1 PARAMETRIC REPRESENTATION

Consider a coordinate system S( x, y). A position vector that is drawn from the origin of coordinate system S to a current point of the curve is represented by vector function

where i and j are the unit vectors of coordinate axes. The symbol C ⁰ means that x( ?) and y( ?) are continuous functions; G designates the open interval a < ? < b for the variable parameter ?. Functions x( ?) and y( ?) associate the point of the curve with the variable parameter ?.

A simple curve means that there is one-to-one correspondence between the point of the curve and parameter ?. A simple curve does not have points of self-intersection. Examples of self-intersecting curves are an extended involute curve (Fig. 1.6.2) and an extended epicycloid (Fig. 1.6.1). In some cases, to avoid the appearance of a point of self-intersection it is sufficient to just limit the interval ( a, b) for the variable parameter ?.

A parametric curve is a regular curve if

Here,

where

The inequality r _? ? 0 means that

Symbol C ¹ means that functions x( ?) and y( ?) have continuous derivatives to the first order at least.

4.2 REPRESENTATION BY IMPLICIT FUNCTION

An equation

does not necessarily represent a planar curve. Rather it merely represents...