Osculating circle

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An osculating circle

A circle with 4-point contact at a vertex of a curve

In differential geometry, the osculating circle of a curve at a point is a circle which:

Touches the curve at that point
Has its unit tangent vector ${\hat{\mathbf{T}}}$ , equal to the unit tangent of the curve at that point
Has its derivative of unit tangent (with respect to arc length), $\frac{d \boldsymbol{\hat{T}}} {d s}$ , equal to that of the curve at that point.

Note that both the circle and the curve must be parameterised by arc length.

[edit] In lay terms

Imagine driving a car along a curved road on a vast flat plane. Suddenly lock the steering wheel in its present position at one point along the road. 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. That circle is the osculating circle of the road curve at that point.

Osculate literally means to kiss; the term is used because osculation is a more gentle form of contact than simple tangency.

[edit] Further mathematical details

These three conditions define a unique circle for each point on the curve, provided that the derivative of the unit tangent is not zero.

The length of the vector $\frac{d \boldsymbol{\hat{T}}} {d s}$ is called the curvature, and is denoted

$\kappa = \left\| \frac{d \boldsymbol{\hat{T}}} {d s} \right\|$

The radius of the osculating circle, r is called the radius of curvature and is given by

$r= \frac{1}{\kappa}$

The osculating circle of a mathematically defined curve shares location, first derivative, and second derivative with the curve, just as a tangent line shares location and first derivative.

The centers of the osculating circles form the evolute of the curve. Some points on the curve will form vertices where there is a higher degree of contact.