Rational Parametrization of the Circle

Summary: The identity

$z=\frac{a(1-x^2)}{1+x^2},y=\frac{2ax}{1+x^2}$

parametrizes the circle $y^2+z^2=a^2$ by rational functions of $x$. Euler derives it in §46 (with $a=1$) and again as a special case of the §50 substitution.

Sources: chapter3

Last updated: 2026-04-23

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Statement

For all real $x$,

${\left(\frac{a(1-x^2)}{1+x^2}\right)}^2+{\left(\frac{2ax}{1+x^2}\right)}^2=a^2,$

so the map $x\mapsto (z,y)$ traces points on the circle of radius $a$. Every point except $(-a,0)$ is hit exactly once as $x$ ranges over $\mathbb{R}$.

Derivation from §46

Euler opens Chapter 3 with the example $y=\frac{1-z^2}{1+z^2}$. Substituting $z=\frac{1-x}{1+x}$ gives $y=\frac{2x}{1+x^2}$ (source: chapter3, §46). This is the parametrization with $a=1$, in the “$z$-as-function-of-$x$” form.

Derivation from §50

In §50 Euler handles $y=\sqrt{(a+bz)(c+dz)}$ by letting $\sqrt{(a+bz)(c+dz)}=(a+bz)x$. For $y=\sqrt{a^2-z^2}=\sqrt{(a+z)(a-z)}$ — so $b=1$, $c=a$, $d=-1$ — the general formula

$z=\frac{c-a{x}^2}{b{x}^2-d},y=\frac{(bc-ad)x}{b{x}^2-d}$

specializes to

$z=\frac{a-a{x}^2}{x^2+1}=\frac{a(1-x^2)}{1+x^2},y=\frac{2ax}{1+x^2}$

(source: chapter3, §50). This is exactly the classical half-angle parametrization: with $x=\tan (\theta /2)$, one recovers $z=a\cos \theta$, $y=a\sin \theta$.

Geometric meaning

The substitution $\sqrt{(a+z)(a-z)}=(a+z)x$ is the projection from the point $(-a,0)$: it parametrizes the circle by the slope $x$ of the line through that point. The excluded value corresponds to the point at infinity in the direction of the line — equivalently, to $(-a,0)$ itself.

Significance

• First rational parametrization explicitly written down in the Introductio and used repeatedly in later chapters.
• A template for Euler’s general strategy: remove a radical by exploiting one “obvious” point (or factor) and projecting from it.
• Modern reading: every smooth conic is rationally parametrizable because it has genus zero; the circle is the canonical example.

Related pages

• substitution
• rationalizing-substitutions
• chapter-3-on-the-transformation-of-functions-by-substitution
• folium-of-descartes — another famous rational parametrization, from §52.