Osculating Curves

Summary: An osculating curve is a simpler curve that coincides with a given curve at a point so closely that, for at least a very small distance, the two are the same. Just as the tangent line is the simplest curve that touches at a point, the osculating curve is the simplest curve that kisses — agrees in direction and in higher-order behavior. Euler develops the notion as a refinement of the tangent algorithm of chapter 13: given a point $M$ on a curve, the osculating curve at $M$ comes from keeping more terms in the local expansion than just the linear truncation.

Sources: chapter14 (§304).

Last updated: 2026-05-11.

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The motivating analogy (§304)

In chapter 13, Euler studied straight lines that indicate the direction of a curve at a chosen point — tangents. He now investigates simpler curves that coincide with a given curve at some place so closely that for a very small distance the two are the same. From a knowledge of the simpler curve, we then learn something about the proposed curve.

First we consider the tangent line, then we consider a simpler curve which is much closer to the proposed curve, in that it is not only tangent, but rather kisses. Curves of this kind with such a tight fit are usually called OSCULATING. — §304

The method parallels the chapter-7 method for branches at infinity: there, after the tangent line at infinity (the rectilinear asymptote), Euler refined to a parabolic asymptote, then a curvilinear asymptote of order $k$. Here, after the tangent line at a finite point, Euler refines first to an osculating parabola and then to the osculating circle of curvature.

What “kissing” means concretely

If the curve’s local equation, after [[tangent-by-translation|translating the origin to $M$]], is

$0=At+Bu+C{t}^2+Dtu+E{u}^2+F{t}^3+\cdots ,$

then:

• Tangent line = keep linear terms only: $At+Bu=0$. Agrees with the curve to first order at $M$.
• Osculating parabola = keep linear and quadratic terms: $At+Bu+C{t}^2+Dtu+E{u}^2=0$. Agrees to second order. Reduces to a parabola in suitably rotated normal coordinates (osculating-parabola).
• Osculating circle = the circle that has the same curvature as the osculating parabola at its vertex. Its radius is the radius of curvature.
• Higher osculating curves = retain even more terms; needed only when the curvature itself vanishes (inflection-by-vanishing-curvature) or the point is singular (curvature-at-multiple-points).

Why a parabola, then a circle

A general second-order curve through $M$ tangent to a given line is a parabola, and Euler defines the curvature of the given curve at $M$ to be the curvature of this osculating parabola at its vertex. He then introduces the osculating circle because the curvature of a circle is the simplest possible — uniform along the curve and inversely proportional to the radius — and so it is convenient to define the curvature of any curve by the radius of the circle that has the same curvature. This is the radius of curvature, the most consequential single quantity introduced in the chapter.

Related pages

• chapter-14-on-the-curvature-of-a-curve — chapter summary.
• tangent-by-translation — the chapter-13 substitution $x=p+t$, $y=q+u$ that this chapter extends to second order and beyond.
• osculating-parabola — §§305–307: derivation of the second-order osculating curve.
• osculating-circle — §§308–310, §318: replacing the parabola with a circle of the same vertex-curvature.
• branches-at-infinity — the analogous refinement program for branches going to infinity.