Given two curves C1 and C2 and a point P attached to curve
C2. Now let curve C2 roll along curve C1,
without slipping. Then P describes a roulette 1).
When P is on curve C2, the curve is called a point-roulette.
When P is on a line attached to C2, the curve is called a line-roulette.
When working in the complex plane and both curves C1 and C2 are
expressed as function of the arc length, the resulting curve z can be written as
a function of C1 and C2 2).
The first to describe these curves where Besant (1869) and Bernat (1869),
The following roulettes can be distinguished, for a curve C2 that is rolling
over a curve C1:
The point-roulettes for which a circle rolls on a line or on another circle,
are known as cycloidal curves.
Some authors name the roulette a spirographic curve.
The same curves can be defined as a glissette 3): as the locus of a point
or a envelope of a line which slides between two given curves C1 and
An wide-known example of a glissette is the astroid.
1) rouler (Fr.) = to roll.
In Dutch: rolkromme.
2) The formula follows from the isometric insight that C1(t)
- z(t) / C2(t) - z(0) = C1'(t) / C2'(t).
Example for the cycloid: C1(t) = t and C2(t)
= i - i e it where t is the arc length.
3) glisser (Fr.) = to glide.