There are four derived curves that are related to curvature:

• evolute
• involute
• radial
• thecurvature itself

Given a curve C₁, theevoluteis the curve C₂ defined by the loci of the centers of curvature of C₁. In otherwords:construct in each point P of curve C₁ a circle that is a tangent to C₁ in P; then thecenter 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, theinvoluteof C₁. Involution is the reverse operation of evolution: if C₂ is the involute ofC₁, then C₁ is theevolute of C₂.
You might ask yourself whether there exists a curve whose involute is exactly the samecurve. 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 inposition or magnitude:

• epicycloid
• hypocycloid
  

Some other involute-evolute couples are:

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

Theradial is a variation on theevolute:draw, from a fixed point, lines parallel to the radii of curvature, with the same lengthas the radii. The set of end points is the radial. Thelogarithmic spiral is the curve whose radial is the curve itself.

Radials of some other curves are:

Given a curve, thecurvature 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 thecurvature is positive (>= 0), we speak of aconvexcurve.
If the curvature is strictly positive (>0),  we speak of astrictlyconvex curve.

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


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

Theradius 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.