conchoid

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) (x² + y²)(x - 1) = a² x²