- Conchoid (mathematics)
-
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. 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. If r = α(θ) expresses the given curve then r = α(θ) ± d expresses the conchoid. Parametrically, it can be expressed as x = a + cos(θ) and y = atan(θ) + sin(θ).
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.
References
- J. Dennis Lawrence (1972). A catalog of special plane curves. Dover Publications. pp. 36, 49–51, 113, 137. ISBN 0-486-60288-5.
- "Conchoïde" at Encyclopédie des Formes Mathématiques Remarquables
This geometry-related article is a stub. You can help Wikipedia by expanding it.
