curvature curve

There are four derived curves that are related to curvature:

• evolute
• involute
• radial
• the curvature itself

Given a curve C₁, the evolute is the curve C₂ defined by the loci of the centers of curvature of C₁. In other words: construct in each point P of curve C₁ a circle that is a tangent to C₁ in P; then the center of the circle belongs to C₂.

When C₁ is given by (x, y) = (f(t), g(t)), then C₂ has the form:



If a line l rolls without slipping as a tangent along a curve C₁, then the path of a point P on l forms a new curve C₂, the involute of C₁. Involution is the reverse operation of evolution: if C₂ is the involute of C₁, then C₁ is the evolute of C₂.
You might ask yourself whether there exists a curve whose involute is exactly the same curve. Well, there are two curves with this property:

• cycloid
• logarithmic spiral

Besides, there are some curves whose involute is the same curve, but not equal in position or magnitude:

• epicycloid
• hypocycloid
  

Some other involute-evolute couples are:

In fact, the evolute of a curve is the same as the envelope of its normal.

The radial is a variation on the evolute: draw, from a fixed point, lines parallel to the radii of curvature, with the same length as the radii. The set of end points is the radial. The logarithmic spiral is the curve whose radial is the curve itself.

Radials of some other curves are:

Given a curve, the curvature κ is defined as the inclination per arc length: κ(s) = dφ/ds.

This curvature can be expressed for a curve y = f (x) as follows:
 

If the curvature is positive (>= 0), we speak of a convex curve.
If the curvature is strictly positive (>0),  we speak of a strictly convex curve.

If the curvature is negative (<= 0), we speak of a concave curve.
If the curvature is strictly negative (<0),  we speak of a strictly concave curve.


Given the curvature as function of the arc length, you can look for the representing curve.
Some example curves are the following:

The radius of curvature R is the reciprocal of the absolute value of the curvature κ, so that R = 1/κ.
The Cesaró equation writes a curve in terms of a radius of curvature R and an arc length s.