Conchoid (mathematics)

A conchoid is a curve derived from a fixed point O, another curve, and a length d. It was invented by the ancient Greek mathematician Nicomedes.

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 the given curve and a circle of radius d and center O. 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. If

expresses the given curve, then

expresses the conchoid.

If the curve is a line, then the conchoid is the conchoid of Nicomedes.

For instance, if the curve is the line ${\displaystyle x=a}$, then the line's polar form is ${\displaystyle r=a\sec \theta }$ and therefore the conchoid can be expressed parametrically as

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

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