geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a conchoid is a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

derived from a fixed point , another curve, and a length . It was invented by the ancient Greek mathematician

Nicomedes Nicomedes is a Greek given name ). Notable people with the name include: *Nicomedes (mathematician), ancient Greek mathematician who discovered the conchoid named after him *Nicomedes of Sparta, regent during the youth of King Pleistoanax, commanded ...

Description

For every line through that intersects the given curve at the two points on the line which are from are on the conchoid. The conchoid is, therefore, the cissoid of the given curve and a circle of radius and center . They are called ''conchoids'' because the shape of their outer branches resembles conch shells. The simplest expression uses polar coordinates with at the origin. If :

r=\alpha(\theta)

expresses the given curve, then :

r=\alpha(\theta)\pm d

expresses the conchoid. If the curve is a line, then the conchoid is the ''conchoid of

''. For instance, if the curve is the line , then the line's polar form is and therefore the conchoid can be expressed

parametrically A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...

as :

x=a \pm d \cos \theta,\, y=a \tan \theta \pm d \sin \theta.

limaçon In geometry, a limaçon or limacon , also known as a limaçon of Pascal or Pascal's Snail, is defined as a roulette curve formed by the path of a point fixed to a circle when that circle rolls around the outside of a circle of equal radius. I ...

is a conchoid with a circle as the given curve. The so-called conchoid of de Sluze and

conchoid of Dürer Conchoid can refer to: * Conchoid (mathematics), an equation of a curve discovered by the mathematician Nicomedes * Conchoidal fracture A conchoidal fracture is a break or fracture of a brittle material that does not follow any natural planes ...

are not actually conchoids. The former is a strict cissoid and the latter a construction more general yet.

References

External links

conchoid with conic sections
- interactive illustration *
conchoid
at mathcurves.com Plane curves Greek mathematics {{geometry-stub

Description

See also

References

External links