Inflection by Vanishing Curvature

Summary: §§319–322. When the denominator $A^{2}E-ABD+B^{2}C$ of the radius-of-curvature formula vanishes, the osculating radius is infinite — the osculating circle degenerates into a straight line, and curvature alone cannot tell whether the curve has an inflection at $M$ or merely an arc with locally constant tangent direction. Euler resolves this by going to higher-order terms: in the rotated $(r,s)$ coordinates of §307, the curve satisfies $r\sqrt{A^2+B^2}=\alpha s^2+\beta s^3+\gamma s^4+\delta s^5+\cdots$, and the first non-vanishing coefficient among $\alpha ,\beta ,\gamma ,\delta ,\dots$ classifies the configuration. Odd exponent of $s$ in the leading term → inflection point (figure 61, serpentine). Even exponent → both branches on the same side of the tangent (figure 62), no inflection.

Sources: chapter14 (§§319–322). Figures 61, 62 (in figures61-67).

Last updated: 2026-05-11.

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Continuous curvature requires no inflection (§319)

Recall that the local approximation in $(r,s)$ from §307 gives the same equation $s^2=(\dots )r$ for both halves of the curve through $M$ — the arc $Mm$ on one side and $Mn$ on the other. Each abscissa $Mr=r$ has two corresponding ordinates, one positive and one negative. Hence both arcs have the same osculating circle and the same curvature.

Whenever the osculating radius — which equals $\frac{1}{2\alpha }$ — is finite, the curvature is uniform for at least a very small space. There will be no sudden formation of a cusp with the curve reflecting back, nor a sudden change of curvature with one portion $Mn$ convex with respect to $N$ and the other portion $Mm$ concave with respect to $N$. — §319

Inflections — points of contrary bending in Euler’s terminology — therefore can occur only where the osculating radius is not finite.

The case of infinite radius (§320)

When $A^{2}E-ABD+B^{2}C=0$, the formula $R=(A^2+B^2{)}^{3/2}/(2(A^{2}E-ABD+B^{2}C))$ yields $R=\infty$ and the osculating circle degenerates to the tangent line itself. At second order the curve appears to be straight — but the cubic and higher terms determine whether it is locally serpentine (curving one way then the other), arc-like (curving the same way on both sides), or something more exotic.

To analyze, substitute the rotation formulas

$t=\frac{-Ar+Bs}{\sqrt{A^2+B^2}},u=\frac{-As-Br}{\sqrt{A^2+B^2}}$

into the cubic terms $F{t}^3+G{t}^{2}u+Ht{u}^2+I{u}^3$ as well, and (since all terms with $r$ vanish next to $r\sqrt{A^2+B^2}$) eliminate them in favor of pure powers of $s$. The result is an expansion

$r\sqrt{A^2+B^2}=\alpha s^2+\beta s^3+\gamma s^4+\delta s^5+\cdots ,$

valid as long as the second-order data alone vanishes. The osculating radius reads off the leading coefficient: $R=\sqrt{A^2+B^2}/(2\alpha )$.

The classification by leading exponent (§§321–322)

If $\alpha =0$, the radius is infinite and the leading term shifts to $\beta s^3$. The new local equation is

$r\sqrt{A^2+B^2}=\beta s^3.$

A negative $r$ now corresponds to a negative $s$; a positive $r$ to a positive $s$. The curve has a serpentine configuration $mM\mu$ as in figure 61: the two halves of the curve swap sides of the tangent line as one passes through $M$. Hence $M$ is a point of inflection.

If $\alpha =\beta =0$, the leading term is $\gamma s^4$:

$r\sqrt{A^2+B^2}=\gamma s^4.$

Each value of $r$ corresponds to two values of $s$ (one positive, one negative) but the abscissa $r$ cannot take both signs (it is determined positively by $\gamma s^4>0$). So both halves of the curve $Mm$ and $M\mu$ lie on the same side of the tangent — figure 62. No inflection.

If $\alpha =\beta =\gamma =0$, the leading term is $\delta s^5$ and the configuration is again serpentine (figure 61): inflection.

If $\alpha =\beta =\gamma =\delta =0$, the leading term is $ϵs^6$ and both halves lie on the same side (figure 62): no inflection.

General rule (§322):

If the exponent on $s$ in the leading term is odd, then the curve has a point of inflection at $M$. If the exponent is even, then the curve has no point of inflection at $M$.

Summary table

Leading term	Configuration	Inflection?	Figure
$\alpha s^2$, $\alpha \ne 0$	regular curvature	no	(osculating circle)
$\beta s^3$	serpentine	yes	61
$\gamma s^4$	both sides same	no	62
$\delta s^5$	serpentine	yes	61
$ϵs^6$	both sides same	no	62
$\zeta s^{2k+1}$	serpentine	yes	61
$\eta s^{2k}$	both sides same	no	62

Why parity is the sole criterion

Each value of $r$ near zero corresponds, via $r\sim s^k/\text{const}$, to two values of $s$ if $k$ is even (since $s^k>0$ always so $r$ has fixed sign and $\pm s$ both work) and to one continuous swap of sign if $k$ is odd (since $s^k$ takes the sign of $s$, so positive $r\leftrightarrow$ positive $s$ and the curve crosses through the tangent). The parity of $k$ thus controls whether the local curve sits on one side of the tangent or crosses it.

Connection to the chapter-13 jet picture

The same parity-of-exponent rule appears in curvature-at-multiple-points for branches of order $j>1$, where the local equation has the form $r^j=\alpha s^n$. There, parity of $n$ relative to $j$ classifies inflection vs cusp at each branch. The simple-point case here is the special case $j=1$.

In modern language: the leading exponent $k$ in $r=(\text{const})s^k$ is the order of contact of the curve with its tangent. A point of contact with even order is regular; with odd order $\ge 3$ is an inflection.

Figures

Figures 61–67

Related pages

• chapter-14-on-the-curvature-of-a-curve — chapter summary.
• osculating-circle — the formula whose denominator vanishing triggers this analysis.
• osculating-parabola — derivation of the rotation that produces the $r$-versus-$s$ expansion.
• curvature-at-multiple-points — the parallel parity rule for branches of higher order.
• three-genera-of-local-curvature — the §332 trichotomy in which inflection is one of three possible local genera.
• singular-points-by-jet — chapter 13’s leading-jet classification of singular points.