Wednesday, January 19

Conchoid Eric sends me a conchoid, pictured, as a fine example of his Mac's built in graphing-calculator capabilities.

For those of us not teaching maths at Harvard or writing Calculus books (and pinched from the Internets), a conchoid is a curve derived from a fixed point O, another curve, and a length d. For every line through O that intersects the given curve at A the two points on the line which are d from A are on the conchoid. The conchoid is, therefore, the cissoid of a circle with center O and the given curve. They are called conchoids because the shape of their outer branches resembles conch shells.

The simplest expression uses polar coordinates with O at the origin. Ifr = α(θ) expresses the given curve then expresses the conchoid. Parametrically, it can be expressed

asx = a + cos(t) and y = atan(t) + sin(t)

All conchoids are cissoids with a circle centered on O as one of the curves.

The prototype of this class is the conchoid of Nicomedes in which the given curve is a line.

A limaçon is a conchoid with a circle as the given curve.

The often-so-called conchoid of de Sluze and conchoid of Dürer do not fit this definition; the former is a strict cissoid and the latter a construction more general yet.