A plane quartic curve consisting of two separate branches either side of and asymptotic to a central straight line (the asymptote), such that if a line is drawn from a fixed point (the pole) to intersect both branches, the part of the line falling between the two branches is of constant length and is exactly bisected by the asymptote.
Such curves are represented by the general equation (x − a)²(x² + y²) = b²x², where a is the distance between the pole and the asymptote, and b is the constant length. The branch on the same side of the asymptote as the pole typically has a cusp or loop‘Nicomedes is famous for his treatise On conchoid lines which contain his discovery of the curve known as the conchoid of Nicomedes.’
- ‘The ruler has a fixed distance marked on it and one mark is kept on a given line while the other traces the conchoid curve.’
- ‘Nicomedes, who was highly critical of Eratosthenes' mechanical solution, gave a construction which used the conchoid curve which he also used to solve the problem of trisection of an angle.’
- ‘The conchoid has x = b as an asymptote and the area between either branch and the asymptote is infinite.’
- ‘Dürer calls the curve ‘ein muschellini’ which means a conchoid, but since it is not a true conchoid we have called it Dürer's shell curve (muschellini = conchoid = shell).’
- ‘Pappus tells us about the conchoid of Nicomedes in his Mathematical collection.’