osculating-radius-of-ellipse

Osculating Radius of the Ellipse

Summary: §§316–317. Worked example: the radius of curvature of the ellipse at an arbitrary point. With center $A$, transverse semiaxis $a=AD$, conjugate semiaxis $b=AC$, and equation $a^2{y}^2+b^2{x}^2=a^2{b}^2$, Euler computes $A=2b^{2}p$, $B=2a^{2}q$, $C=b^2$, $D=0$, $E=a^2$ at a generic point $M=(p,q)$ and obtains the radius $R=\frac{(a^4{q}^2+b^4{p}^2{)}^{3/2}}{a^4{b}^4}=\frac{a^2\cdot M{N}^3}{b^4}=\frac{a^2{b}^2}{M{O}^3},$ where $MN$ is the normal length and $MO$ is the perpendicular from the center to the tangent. The third form makes the formula symmetric between the two principal axes.

Sources: chapter14 (§§316–317). Figure 60 (in figures55-60).

Last updated: 2026-05-11.

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Setup (§316)

Let the curve be the ellipse with equation $a^2{y}^2+b^2{x}^2=a^2{b}^2$, taken on the axis $AD$ from the center $A$ (figure 60). Take an abscissa $AP=p$ with corresponding ordinate $PM=q$, so $a^2{q}^2+b^2{p}^2=a^2{b}^2$. Substitute $x=p+t$, $y=q+u$:

$a^2{q}^2+2a^{2}qu+a^2{u}^2+b^2{p}^2+2b^{2}pt+b^2{t}^2=a^2{b}^2,$

so

$2b^{2}pt+2a^{2}qu+b^2{t}^2+a^2{u}^2=0.$

Read off the local coefficients:

$A=2b^{2}p,B=2a^{2}q,C=b^2,D=0,E=a^2.$

Tangent geometry from $A,B$

The subtangent ratio gives $PM/PN=-B/A=-a^{2}q/(b^{2}p)$ (negative because the normal $MN$ meets the axis on the other side of $P$ from the abscissa direction — a feature of the ellipse). Equivalently, $PN=b^{2}p/a^2$, the classical Apollonian subnormal of the ellipse.

Because of the coefficients of $t$ and $u$, the normal $MN$ intersects the axis before $P$. — §316

The discriminant denominator

Compute

$A^{2}E-ABD+B^{2}C=4b^4{p}^2\cdot a^2-0+4a^4{q}^2\cdot b^2=4a^2{b}^2(a^2{q}^2+b^2{p}^2).$

Apply the ellipse equation $a^2{q}^2+b^2{p}^2=a^2{b}^2$ to simplify:

$A^{2}E-ABD+B^{2}C=4a^4{b}^4.$

This positive quantity confirms (via convexity-from-osculating-circle) that the curve is concave with respect to $N$ — as expected for the convex side of an ellipse arc.

The radius (§317)

Compute also

$A^2+B^2=4(a^4{q}^2+b^4{p}^2),$

then

$R=\frac{(A^2+B^2)\sqrt{A^2+B^2}}{2(A^{2}E-ABD+B^{2}C)}=\frac{4(a^4{q}^2+b^4{p}^2)\cdot 2\sqrt{a^4{q}^2+b^4{p}^2}}{2\cdot 4a^4{b}^4}=\frac{(a^4{q}^2+b^4{p}^2{)}^{3/2}}{a^4{b}^4}.$

The two geometric reformulations

In terms of the normal length $MN$: from $PN=b^{2}p/a^2$ and $PM=q$,

$M{N}^2=P{M}^2+P{N}^2=q^2+\frac{b^4{p}^2}{a^4}=\frac{a^4{q}^2+b^4{p}^2}{a^4}.$

So $\sqrt{a^4{q}^2+b^4{p}^2}=a^2\cdot MN$, and

$R=\frac{a^2\cdot M{N}^3}{b^4}.$

In terms of the central perpendicular $MO$: extend $MN$ and drop the perpendicular $AO$ from the center $A$. By similar triangles $MNP$ and $ANO$ (right-angled at $P$ and $O$ respectively, sharing angle $N$), with $AN=p-b^{2}p/a^2=p(a^2-b^2)/a^2$,

$NO=\frac{a^2{b}^2{p}^2-b^4{p}^2}{a^4\cdot MN},MO=NO+MN=\frac{a^2{q}^2+b^2{p}^2}{a^2\cdot MN}=\frac{b^2}{MN}.$

(Using $a^2{q}^2+b^2{p}^2=a^2{b}^2$.) Hence $MN=b^2/MO$, and substituting back,

$R=\frac{a^2{b}^2}{M{O}^3}.$

Why the $MO$ form is preferred

This expression is accommodated to both of the axes, $AD$ and $AC$. — §317

The form $R=a^4{b}^4/(a^4{b}^4\cdot M{O}^3/(a^2{b}^2))=a^2{b}^2/M{O}^3$ is symmetric in the two semiaxes: had Euler taken his abscissa along $AC$ instead of $AD$, the same formula in $MO$ would have resulted. The intermediate forms $(a^4{q}^2+b^4{p}^2{)}^{3/2}/(a^4{b}^4)$ and $a^{2}M{N}^3/b^4$ each privilege one of the axes.

Endpoint values

• At a vertex of the major axis ($p=a$, $q=0$): $MN=b^2/a$, $R=a^2(b^2/a{)}^3/b^4=b^2/a$. The radius equals half the latus rectum of the conic — recovering the classical fact that the osculating circle at the vertex matches the conic to fourth order.
• At a vertex of the minor axis ($p=0$, $q=b$): $MN=a^2/b$ (with $b$ swapped for $a$ in the perpendicular calculation), $R=a^2/b$.
• At any point: $R$ scales as $M{O}^{-3}$, i.e., grows as the central perpendicular shrinks — captures the obvious fact that the ellipse is “flattest” near the major-axis vertex when $b<a$.

Figures

Figures 55–60

Related pages

• chapter-14-on-the-curvature-of-a-curve — chapter summary.
• osculating-circle — the general formula being applied.
• osculating-parabola — predecessor construction.
• ellipse — the underlying curve and its tangent properties.
• principal-axes-and-foci — the symmetric role of $a$ and $b$ that the $MO$-form exposes.