||last updated: 2004-12-04
The curve has the form of a snail 1), so that
it is also called the snail curve. It's the epitrochoid for which the rolling circle and the
rolled circle have the same radius. The curve can also be defined as a conchoid.
The curve is called the limašon of Pascal
(or snail of Pascal).
It is not named to the famous mathematician Blaise Pascal, but to his father, Etienne
Pascal. He was a correspondent for Mersenne, a mathematician who made a large
effort to mediate new knowledge (in writing) between the great mathematicians of that era.
The name of the curve was given by Roberval when he used the curve
drawing tangents to for differentiation.
But before Pascal, DŘrer had already discovered the curve,
since he gave a method for drawing the limašon, in 'Underweysung der Messung'
Sometimes the limašon is confined to values b < 1. We might call this curve an
ordinary limašon. It is a transitional
form between the circle (b=0) and
the cardioid (b=1).
When we extend the
curve to values for b > 1, a noose appears.
For b = 2, the curve is called the trisectrix
or the limašon trisectrix.
This curve has as alternative equation: r = sin (f/3)
The name of the trisectrix 2) is because of the
quality that angle BAP is 3 times angle APO.
An alternative name for the limašon is arachnid 3)
or spider curve.
Some fine properties of the curve are:
The limašon is an anallagmatic
For b unequal zero, the curve is a quartic, in Cartesian coordinates
it can be written as a fourth degree equation 4).
1) limax (Lat.) = snail
2) Tri = three, sectrix = angle.
3) from arakne (Gr.) = spider.
In French: arachnÚe
4) equation: (x2 + y2 - by)2 = x2