Osculating circle

[2] The osculating circle provides a way to understand the local behavior of a curve and is commonly used in differential geometry and calculus.

More formally, in differential geometry of curves, the osculating circle of a sufficiently smooth plane curve at a given point p on the curve has been traditionally defined as the circle passing through p and a pair of additional points on the curve infinitesimally close to p. Its center lies on the inner normal line, and its curvature defines the curvature of the given curve at that point.

The center and radius of the osculating circle at a given point are called center of curvature and radius of curvature of the curve at that point.

A geometric construction was described by Isaac Newton in his Principia: There being given, in any places, the velocity with which a body describes a given figure, by means of forces directed to some common centre: to find that centre.Imagine a car moving along a curved road on a vast flat plane.

Suddenly, at one point along the road, the steering wheel locks in its present position.

Thereafter, the car moves in a circle that "kisses" the road at the point of locking.

The curvature of the circle is equal to that of the road at that point.

Let γ(s) be a regular parametric plane curve, where s is the arc length (the natural parameter).

This determines the unit tangent vector T(s), the unit normal vector N(s), the signed curvature k(s) and the radius of curvature R(s) at each point for which s is composed:

The corresponding center of curvature is the point Q at distance R along N, in the same direction if k is positive and in the opposite direction if k is negative.

The circle with center at Q and with radius R is called the osculating circle to the curve γ at the point P. If C is a regular space curve then the osculating circle is defined in a similar way, using the principal normal vector N. It lies in the osculating plane, the plane spanned by the tangent and principal normal vectors T and N at the point P. The plane curve can also be given in a different regular parametrization

Then the formulas for the signed curvature k(t), the normal unit vector N(t), the radius of curvature R(t), and the center Q(t) of the osculating circle are

We can obtain the center of the osculating circle in Cartesian coordinates if we substitute t = x and y = f(x) for some function f. If we do the calculations the results for the X and Y coordinates of the center of the osculating circle are:

are obtained through solving a linear system of two equations:

is orthogonal to the unit tangent vector at

and the radius of the osculating circle is precisely the inverse of the curvature.

Developing the trigonometric functions to the second order in

, we find the results of the preceding section.

For a curve C given by a sufficiently smooth parametric equations (twice continuously differentiable), the osculating circle may be obtained by a limiting procedure: it is the limit of the circles passing through three distinct points on C as these points approach P.[3] This is entirely analogous to the construction of the tangent to a curve as a limit of the secant lines through pairs of distinct points on C approaching P. The osculating circle S to a plane curve C at a regular point P can be characterized by the following properties: This is usually expressed as "the curve and its osculating circle have the second or higher order contact" at P. Loosely speaking, the vector functions representing C and S agree together with their first and second derivatives at P. If the derivative of the curvature with respect to s is nonzero at P then the osculating circle crosses the curve C at P. Points P at which the derivative of the curvature is zero are called vertices.

If P is a vertex then C and its osculating circle have contact of order at least three.

If, moreover, the curvature has a non-zero local maximum or minimum at P then the osculating circle touches the curve C at P but does not cross it.

The curve C may be obtained as the envelope of the one-parameter family of its osculating circles.

Their centers, i.e. the centers of curvature, form another curve, called the evolute of C. Vertices of C correspond to singular points on its evolute.

Within any arc of a curve C within which the curvature is monotonic (that is, away from any vertex of the curve), the osculating circles are all disjoint and nested within each other.

the radius of curvature equals R(0) = 0.5 (see figure).

The parabola has fourth order contact with its osculating circle there.

For large t the radius of curvature increases ~ t3, that is, the curve straightens more and more.

A Lissajous curve with ratio of frequencies (3:2) can be parametrized as follows It has signed curvature k(t), normal unit vector N(t) and radius of curvature R(t) given by

with respect to the arc length s. A cycloid with radius r can be parametrized as follows:

For some historical notes on the study of curvature, see For application to maneuvering vehicles see