Affine Curves

Summary: Euler’s weakening of similarity. If abscissa and ordinate are scaled by different ratios — $x\mapsto X/m,y\mapsto Y/n$ with $m\ne n$ — the resulting curves are no longer similar, but they share order, genus, and many qualitative properties. Headline example: every ellipse is affine to a circle, and every two ellipses are affine to each other. Unlike similarity, affinity is not intrinsic to the curve but depends on the chosen axis.

Sources: chapter18 §§442–446.

Last updated: 2026-05-12.

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Definition (§442)

Similar curves (similar-curves) preserve a single scaling ratio $AP/ap=PM/pm=1/n$. If we relax this and allow abscissas to follow ratio $1/m$ and ordinates to follow a different ratio $1/n$, the curves so obtained from a given $AMB$ are called affine to it.

Affinity contains similarity as the special case $m=n$. Or put differently: similarity is the species of affinity where the two scaling ratios coincide.

For each independent choice of $1/m$ and $1/n$, a single affine companion is constructed: pick $ap$ so that $AP/ap=1/m$, then erect the perpendicular $pm$ with $PM/pm=1/n$. Varying $1/m,1/n$ generates a two-parameter family of affines.

Algebraic recipe (§443)

Given $y=P(x)$ for the original curve $AMB$, substitute $x=X/m,y=Y/n$ to get an equation in $(X,Y)$ with free parameters $m,n$. Equivalently: $Y/n$ is a zero-degree function of $X$ and $m$.

The thread-degree picture: where similarity required $(a,x,y)$ to appear with equal degree per term (see thread-constants), affinity allows two independent scaling “thread directions”, one for the $x$-axis and one for the $y$-axis.

The asymmetry that matters (§444)

Similarity is intrinsic to a pair of curves: if $AMB$ and $amb$ are similar with respect to one pair of axes, they are similar with respect to any other pair of axes or homologous points.

Affinity is frame-dependent: $AMB$ and $amb$ are affine only with respect to the specific axis $AB$ used to define their abscissas. Choose a different axis and the affinity relation is generally lost.

Nonetheless: all curves affine to each other share order and genus. The affine recipe preserves the algebraic structure of the equation up to independent rescaling, and rescaling cannot change degree (cf. degree-invariance) — only relative metric proportions of the axes.

The headline example: circles, ellipses, hyperbolas (§445)

Start with the circle on a diameter, $y^2=2cx-x^2$.

Similar curves. Substitute $x=X/n,y=Y/n$: $\frac{Y^2}{n^2}=\frac{2cX}{n}-\frac{X^2}{n^2},Y^2=2ncX-X^2.$ This is still a circle, with diameter $2nc$. All curves similar to the circle are circles.

Affine curves. Substitute $x=X/m,y=Y/n$: $\frac{Y^2}{n^2}=\frac{2cX}{m}-\frac{X^2}{m^2},m^2{Y}^2=2m{n}^{2}cX-n^2{X}^2.$ This is the general equation of an ellipse referred to one of its principal axes. So:

• every ellipse is affine to the circle;
• every ellipse is affine to every other ellipse;
• two ellipses are similar iff their principal-axis ratios coincide.

The same reasoning shows all hyperbolas are affine to each other; equal-axis-ratio hyperbolas are similar.

Degenerate case: parabola and friends (§446)

For the parabola $y^2=cx$, substituting $x=X/m,y=Y/n$: $\frac{Y^2}{n^2}=\frac{cX}{m},Y^2=\frac{n^{2}c}{m}X.$ This is still a parabola, with leading coefficient $n^{2}c/m$. Affinity collapses to similarity — all affine images of a parabola are similar parabolas.

The same collapse happens for any two-term equation of the form $y^p=c^{p-q}{x}^q$: with two terms, the system of degrees in $(X,m),(Y,n)$ leaves only a single effective scaling ratio. So $y^3=c^{2}x$, $y^2=c{x}^2$, $y^{3}x=c^3$, etc., all share the property that affinity is the same as similarity.

The contrast with the ellipse case is sharp: a three-term equation like $y^2=2cx-x^2$ has enough algebraic room for two independent scaling ratios; a two-term equation does not.

Why the chapter cares

Two threads (no pun intended) run through later geometry:

• In chapter 19 and following — and in subsequent analytic geometry — affinity is the natural notion of “same conic up to choice of axes”. Chapter 18 is where Euler isolates it.
• The collapse of affinity to similarity in two-term cases (§446) is the first instance of a recurring theme: degeneracies that erase the distinction between coarser and finer equivalence relations.

Related pages

• similar-curves — the stricter relation, recovered when $m=n$.
• thread-constants — the dimensional bookkeeping.
• classification-of-conics — chapter-6 framework where ellipse, parabola, hyperbola first appeared.
• ellipse — the principal-axis form $b^2{x}^2+a^2{y}^2=a^2{b}^2$ that §445 recovers.
• degree-invariance — the reason affine curves share order.
• chapter-18-on-the-similarities-and-affinities-of-curves