Conchoid (mathematics)

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Conchoids of line with common center.
The fixed point O is the red dot, the black line is the given curve, and each pair of coloured curves is length d from the intersection with the line that a ray through O makes. In the blue case d is greater than O's distance from the line, so the upper blue curve loops back on itself. In the green case d is the same, and in the red case it's less.

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. He called them conchoids because the shape of their outer branches resemble conch shells.

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

[edit] References

• J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36,49–51,113,137. ISBN 0-486-60288-5. 

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