conchoid


last updated: 20041204 
Because the conchoid
^{1)} 's resemblance to a shell ^{2}^{)}
it is also called shell
curve. Another name for this curve is cochloid.
Given a line l and a line bundle m through focus O, the curve is constructed by pacing
at both sides of l a distance a on all lines m. Setting the distance from l to O to 1,
above shown polar equation follows. It's easy to convert the polar equation to a Cartesian
fourth order equation ^{3}^{)}.
When the variable a is greater than 1, a curl becomes visible.
The area between either branch and the asymptote is infinite.
The first who described the curve was the Greek Nicomedes (about 200 BC). He used
the curve working at the problem of the trisection of an angle, and also when duplicating
the cube. Nicomedes was also the one to describe a tool to construct the curve.
The conchoid was a favorite of the 17th century mathematicians.
In the science of mechanics one use a conchoidal movement. A surface makes a
movement in such a way, that:
 a line of the moving surface makes his way through a fixed point
 a point at the surface keeps a tight right track
This is just the way the above described instrument operates, and every point on the
moving surface has a path in the form of a conchoid.
Working in a large global air compressor and pneumatic tool manufacturer
Ludvig Horsfelder designed a conchoid motor. However the performance of the
motor was disappointing and the idea was left.
In fact, it appears that the resulting curve was more a limaçon
than a conchoid.
Further, the conchoid has been used in the construction of buildings, as the
shape of
the vertical section of columns.
The conchoid has been generalized by substituting the line l by a arbitrary curve C:
the general conchoid. The simple conchoid of this
chapter then, can be distinguished with the name conchoid of Nicomedes.
The conchoid can also be seen as a specimen of a generalized
cissoid.
notes
1) In Italian: concoide
2) Concha (Lat.) = shell, mussel
3) (x^{2} + y^{2})(x  1) = a^{2} x^{2}
