32. Curves in a plane

We have hitherto used the notation $$\begin{array}{}\text{(1)}& y=f(x)\end{array}$$ to express functional dependence of $y$ upon $x$. It is evident that this notation is most appropriate in the case in which $y$ is expressed explicitly in terms of $x$ by means of a formula, as when for example $$y=x^2,\sin x,a{\cos }^{2}x+b{\sin }^{2}x.$$

We have however very often to deal with functional relations which it is impossible or inconvenient to express in this form. If, for example, $y^5–y–x=0$ or $x^5+y^5–ay=0$, it is known to be impossible to express $y$ explicitly as an algebraical function of $x$. If $$x^2+y^2+2Gx+2Fy+C=0,$$ $y$ can indeed be so expressed, viz. by the formula $$y=-F+\sqrt{F^2–x^2–2Gx–C};$$ but the functional dependence of $y$ upon $x$ is better and more simply expressed by the original equation.

It will be observed that in these two cases the functional relation is fully expressed by equating a function of the two variables $x$ and $y$ to zero,  by means of an equation $$\begin{array}{}\text{(2)}& f(x,y)=0.\end{array}$$

We shall adopt this equation as the standard method of expressing the functional relation. It includes the equation  as a special case, since $y–f(x)$ is a special form of a function of $x$ and $y$. We can then speak of the locus of the point $(x,y)$ subject to $f(x,y)=0$, the graph of the function $y$ defined by $f(x,y)=0$, the curve or locus $f(x,y)=0$, and the equation of this curve or locus.

There is another method of representing curves which is often useful. Suppose that $x$ and $y$ are both functions of a third variable $t$, which is to be regarded as essentially auxiliary and devoid of any particular geometrical significance. We may write $$\begin{array}{}\text{(3)}& x=f(t),y=F(t).\end{array}$$ If a particular value is assigned to $t$, the corresponding values of $x$ and of $y$ are known. Each pair of such values defines a point $(x,y)$. If we construct all the points which correspond in this way to different values of $t$, we obtain the graph of the locus defined by the equations . Suppose for example $$x=a\cos t,y=a\sin t.$$ Let $t$ vary from $0$ to $2\pi$. Then it is easy to see that the point $(x,y)$ describes the circle whose centre is the origin and whose radius is $a$. If $t$ varies beyond these limits, $(x,y)$ describes the circle over and over again. We can in this case at once obtain a direct relation between $x$ and $y$ by squaring and adding: we find that $x^2+y^2=a^2$, $t$ being now eliminated.