Curvature at Multiple Points

Summary: §§323–331. The §305 derivation of the osculating parabola assumes a simple point: $A$ and $B$ not both zero. When $A=B=0$ — the case of a multiple point, figure 56 — Euler treats each branch through $M$ separately. For a double point, each branch reduces to a leading equation $r=\alpha s^2+\beta s^3+\cdots$ and the simple-point analysis applies branch-by-branch. When two or more branches share the same tangent, the leading equation is $r^j=\alpha s^n+\cdots$ with $j\ge 2$. The branch is then of order $j$, and the relation between $n$ and $j$ classifies the local picture into a small zoo: cusps (figure 63), pairs of internally or externally tangent circular arcs (figures 64, 65), and higher-order analogues.

Sources: chapter14 (§§323–331). Figures 56, 63, 64, 65 (in figures55-60, figures61-67).

Last updated: 2026-05-11.

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Branch-by-branch reduction (§§323–324)

If both $A=0$ and $B=0$, the curve has two or more branches intersecting at $M$ (figure 56), as treated in chapter 13’s jet analysis. Suppose $mt+nu=0$ is the equation of the tangent of one of the branches. Look for the equation of that branch in the rotated coordinates $(r,s)$ where $r$ is taken on the normal to that tangent. As before $r$ is infinitely small compared to $s$. With

$t=\frac{-mr+ns}{\sqrt{m^2+n^2}},u=\frac{-ms-nr}{\sqrt{m^2+n^2}},$

substitute into the local equation and eliminate the infinitesimal terms. For a double point $M$ this gives

$rs=\alpha s^3+\beta s^4+\gamma s^5+\delta s^6+\cdots ,$

dividing by $s$:

$r=\alpha s^2+\beta s^3+\gamma s^4+\delta s^5+\cdots .$

For a triple point the analogous reduction gives $r{s}^2=\alpha s^4+\cdots$, so $r=\alpha s^2+\cdots$ — same leading shape. In every case where the branch has its own distinct tangent, the reduced single-branch equation is

$r=\alpha s^2+\beta s^3+\gamma s^4+\delta s^5+\cdots ,$

identical in form to the simple-point case. The osculating radius of this branch at $M$ is $1/(2\alpha )$, and the leading-exponent parity rule decides inflection vs no inflection.

In this way we judge individually each branch passing through the point $M$, provided each such branch has a different tangent. — §324

When two branches share a tangent (§325)

If two or more branches have the same tangent at $M$, both $A$ and $B$ vanish and the second-degree form $C{t}^2+Dtu+E{u}^2$ has two equal factors. Suppose $C{t}^2+Dtu+E{u}^2=(mt+nu{)}^2$. Change to coordinates $(r,s)$ as above. We obtain an equation of the form

$r^2=\alpha rs+\beta s^3+\gamma r{s}^3+\delta s^4+ϵr{s}^4+\zeta s^5+\cdots .$

Terms in which $r$ has degree $\ge 2$ vanish next to $r^2$, but terms with one $r$ and many $s$ also vanish next to $r^2$ when $s\to 0$ slowly enough — Euler keeps only the dominant ones case by case.

Case I: $\beta \ne 0$ — cusp (§326, figure 63)

If the term $\beta s^3$ is present, it dominates all the others, so the leading balance is

$r^2=\beta s^3.$

Then $r=s\sqrt{\beta s}=s^{3/2}\sqrt{\beta }$. The osculating radius of each branch at $M$ is $\frac{1}{2}\sqrt{s/\beta }$. Since $s$ vanishes at $M$, the radius is zero there, i.e., curvature is infinitely large — the branch becomes an arc of an infinitely small circle. Since the ordinate $s$ has the same sign for positive and negative $r$, both halves are on the same side; the two branches $Mm$ and $M\mu$ are tangent to a common line $Mt$ at $M$ and lie on the same side of it, separated by their mutual tangent. Cusp at $M$ (figure 63).

Case II: $\beta =0$, $\delta \ne 0$ — tangent arcs (§327, figures 64, 65)

If $\beta =0$ and the term $\delta s^4$ is present, the leading balance becomes

$r^2=\alpha r{s}^2+\delta s^4⟺(r-f{s}^2)(r-g{s}^2)=0,$

provided the discriminant ${\alpha }^2+4\delta \ge 0$. (If ${\alpha }^2+4\delta <0$, no real branches and $M$ is a conjugate point — an isolated double point.) Otherwise the equation factors into two:

$r=f{s}^2\text{and}r=g{s}^2,$

each describing a parabolic-vertex branch. The osculating radii of the two branches at $M$ are $1/(2f)$ and $1/(2g)$ respectively — both finite and generally unequal.

• If $f$ and $g$ have the same sign (both branches concave the same way): two internally tangent circular arcs (figure 64).
• If $f$ and $g$ have opposite signs: two externally tangent circular arcs (figure 65).

Higher-order branches (§§328–331)

If even $\delta$ vanishes — and the equation does not factor over the reals into two equations $r=\alpha s^m$ — then the branch is no longer “first-order.” The leading non-trivial form is one of

$r^2=\alpha s^5,r^2=\alpha s^7,r^2=\alpha s^9,\dots ,$

i.e., $r^2=\alpha s^n$ with $n$ odd, $n\ge 5$. Each such branch has a cusp at $M$ (figure 63 again), and the osculating radius is infinitely large (in contrast to the $r^2=\alpha s^3$ case, where it is infinitely small).

Euler’s classification (§§328–331):

• Branches of the first order: $r=\alpha s^m$. These cover all cases where the branch reduces single-valuedly.
• Branches of the second order: $r^2=\alpha s^n$, $n$ odd, $n\ge 3$. All have a cusp at $M$ (figure 63). Osculating radius is infinitely small if $n<4$ and infinitely large if $n>4$ — except the borderline $n=3$ case.
• Branches of the third order: $r^3=\alpha s^n$, $n$ an integer $>3$ not divisible by $3$. These arise when three tangents of branches at $M$ coincide. If $n$ is odd, $M$ is an inflection point on this branch; if $n$ is even, it is not. Osculating radius infinitely small if $n<6$, infinitely large if $n>6$.
• Branches of the fourth order: $r^4=\alpha s^n$, $n$ odd, $n>4$. Cusp at $M$, like the second-order branches. Osculating radius infinitely small if $n<8$, infinitely large if $n>8$.
• General order $j$: $r^j=\alpha s^n$ with $\gcd (j,n)=1$ and $n>j$. Odd $j$ branches behave like first-order branches (inflection or no inflection by parity of $n$); even $j$ branches behave like second-order (cusp at $M$). Osculating radius is infinitely small if $n<2j$, infinitely large if $n>2j$.

Why three coincident tangents do not always give a third-order branch

If three tangents of branches passing through $M$ coincide, there are several possibilities (§329):

• three branches of the first order, all mutually tangent at $M$;
• one branch of the second order (cusp) and one branch of the first order, mutually tangent;
• one branch of the third order ($r^3=\alpha s^n$).

Similarly for four coincident tangents (§330): four first-order branches, or two first + one second-order (cusp), or two second-order, or one first + one third, or finally one branch of the fourth order. The branch order is always at most the multiplicity of the coincident tangent direction.

The recipe in capsule

Given a multiple point $M$ on a curve:

1. Find each branch’s tangent direction (equivalently, factor the leading homogeneous form of singular-points-by-jet).
2. For each distinct tangent direction, rotate so the new abscissa lies along the normal to that tangent and reduce the local equation to $r^j=\alpha s^n+\cdots$ where $j$ is the multiplicity of that tangent factor.
3. Read off the branch type from $(j,n)$:
   • $j=1$: simple branch on this tangent — apply the simple-point parity rule to $n$.
   • $j=2$, $n=3$: cusp with infinitely small osculating radius (figure 63).
   • $j=2$, $n\ge 5$ odd: cusp with infinitely large osculating radius.
   • higher $j$: cusp-like (even $j$) or first-order-like (odd $j$), with $n<2j$ giving small radius and $n>2j$ giving large.
4. Sum the local picture across all distinct tangent factors.

Figures

Figures 55–60

Figures 61–67

Related pages

• chapter-14-on-the-curvature-of-a-curve — chapter summary.
• singular-points-by-jet — chapter-13 classification of multiple points by lowest-jet form, prerequisite to this analysis.
• osculating-parabola — the simple-point algorithm extended here to each branch.
• osculating-circle — the radius formula whose case-by-case behavior is catalogued.
• inflection-by-vanishing-curvature — the parity rule for $j=1$ branches reused here for higher $j$.
• three-genera-of-local-curvature — the §332 master classification subsuming all cases here.
• multiple-points-on-curves — chapter 12’s classification of multiple points.