There are four derived curves that are related to curvature:
Given a curve C_{1}, the **evolute**
is the curve C_{2} defined by the loci of the centers of curvature of C_{1}. In other
words:
construct in each point P of curve C_{1} a circle that is a tangent to C_{1} in P; then the
center of the circle belongs to C_{2}.
When C_{1} is given by (x, y) = (f(t), g(t)), then C_{2} has the form:
If a line l rolls without slipping as a tangent along a curve C_{1},
then the path of a point P on l forms a new curve C_{2}, the **involute**
of C_{1}. Involution is the reverse operation of evolution: if C_{2} is the involute of
C_{1}, then C_{1} is the *evolute* of C_{2}.
You might ask yourself whether there exists a curve whose involute is exactly the same
curve. Well, there are two curves with this property:
Besides, there are some curves whose involute is the same curve, but not equal in
position or magnitude:
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. |