Curvature and its environs

Just as of all lines that pass through a point P of a curve, the tangent is the one that best approximates the curve, of all the circles that pass through P there is one which best adapts to the curve's shape near P. This circle, whose centre is on the line perpendicular to the curve (or to its tangent, which is the same thing), is called the osculatory circle.

We can thus measure the curvature of a curve. The tangent allows us to determine the direction of a curve C. If we imagine a point moving along C, we can think that in every moment the point moves in the same direction than the tangent. Analogously, the curvature of C will be given by that of its osculatory circle, and since a circle is more curved the smaller its radius, we can measure C's curvature through the inverse of the osculatory circle's radius, also called curvature radius.

As point P varies on the curve, the curvature centres (the centres of the osculatory circles) describe a second curve, which is called the evolute of the first. This curve is also the envelope of the lines perpendicular to the given curve.

Reciprocally, the first curve is the involute of the second. The involute of a curve can be materially obtained by attaching a piece of string to the profile of the curve, then slowly detaching it, being careful to keep the detached part always taut. The free extremity of the string will then describe the involute. Thus it is possible to draw the involute of the circle.