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. For instance, if the curve is the straight line x = a, then r = a⁄cos(θ) and therefore the conchoid can be expressed as x = a ± d.cos(θ) and y = a.tan(θ) ± d.sin(θ).
All conchoids are cissoids with a circle centered on O as one of the curves.
A limaçon is a conchoid with a circle as the given curve.
- 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
Media related to Conchoid at Wikimedia Commons
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