geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 may refer to: *Nicomedes (mathematician), ancient Greek mathematician who discovered the conchoid *Nicomedes of Sparta, regent during the youth of King Pleistoanax, commanded the Spartan army at the Battle of Tanagra (457 BC) *Saint Nicom ...

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 In geometry, a cissoid (() is a plane curve generated from two given curves , and a point (the pole). Let be a variable line passing through and intersecting at and at . Let be the point on so that \overline = \overline. (There are actua ...

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 Conch () is a common name of a number of different medium-to-large-sized sea snails. Conch shells typically have a high spire and a noticeable siphonal canal (in other words, the shell comes to a noticeable point at both ends). In North Ame ...

. 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 Line most often refers to: * Line (geometry), object with zero thickness and curvature that stretches to infinity * Telephone line, a single-user circuit on a telephone communication system Line, lines, The Line, or LINE may also refer to: Art ...

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

References

External links

Plane curves {{geometry-stub

Description

See also

References

External links