Osculating circle

The circle of curvature (also called Schmiegekreis or osculating circle ) to a certain point of a plane curve is the circle that best approximates the curve at that point. The center of curvature of the circle is called a center of curvature.

Its radius, the radius of curvature, the amount of the inverse of the curvature of the curve. Its tangent at that point coincides with the tangent of the curve.

Since the curvature of a curve generally varies locally, the curve is nestled in general only in an infinitesimally small neighborhood of the circle of curvature at.

Note: The example drawings suggest that the circle of curvature is always on one side of the curve. However, this is only the case when the curvature of the curve at the corresponding point has an extremum. Since the curvature of the circle of curvature itself is constant, a curve is of varying curvature usually on one side of the contact point within, on the other outside of their circle of curvature.

Determination of the circle of curvature

The center of the circle of curvature, the boundary position of the intersection point of the normal of the curve when the curve of the normal points strive successively:

, The curve presented in the parametric representation, as its radius, the radius of curvature is given by

The center of the circle of curvature has the coordinates

In this case, the amount of the radius to determine the center point must be omitted, so that the circle of curvature is on the right side of the curve? so

The way to describe the curvature of the circle center points are called the evolute of the curve.

Radius of curvature of a function graph

Also for the graph of a function can be used to specify a radius of curvature. Under the curve of the function at the point is defined as the curvature of the graph of the function at the point. The transformation and the function is transferred into a parametric representation and it is:

The derivatives are:

Thus applies to the radius of curvature of a function at the site of operation by insertion into (1):

Examples

Circle

The parametric representation of a circle is:

The derivatives are:

Used in (1) follows the radius of curvature of a unit circle of radius one of:

The animation shows the circle of radius 2, through a constant speed 1 clockwise. He has parametric representation

And constant curvature equal. Its radius of curvature is constant and equal to 2, that is equal to its radius. (The " acceleration vector " in this animation is the second derivative. )

Parabola

For the standard parabola applies:

Substituting into ( 4), it follows for the radius of curvature:

At the point x = 0 is the radius of curvature r = 0.5 ( see figure). For large x the radius of curvature ~ x3 grows, the curve is always straight.

Lissajous curve

The parametric representation of a Lissajous curve with frequency ratio 2:3 is

The first derivatives are:

The second derivatives are:

Substituting this in (1 ) and using the addition theorem for the sine and cosine, then it follows the radius of curvature of the Lissajous curve:

The figure shows an animation of the circle of curvature. The " acceleration vector" in this figure is the second derivative of by arc length.