GXWeb Conchoid of Nicomedes

Saltire Software, home of Geometry Expressions and GXWeb

GXWeb Curve Construction Collection (like this one!)

About the Conchoid (from Wolfram MathWorld)

\(a\)
	0	0	5
\(b\)
	0	0	5
\(\theta\)
	-1.57	0	1.57

App generated by GXWeb

A Conchoid is a curve derived from a fixed point O, another curve, and a length \(b\). It was invented by the ancient Greek mathematician Nicomedes

For every line through O that intersects the given curve at A the two points on the line which are \(b\) from A are on the conchoid. The conchoid is, therefore, the Cissoid of the given curve and a circle of radius \(b\) and center O. They are called conchoids because the shape of their outer branches resembles conch shells.

If the curve is a line, then the conchoid is the conchoid of Nicomedes.

A Limaçon is a conchoid with a circle as the given curve.

Explore the model here and observe connections between the conchoid and both the Cissoid and the Limaçon.

The Conchoid constructed here is the locus of points B and C where

\(a = Length(OD)\)

\(b = Length(AB) = Length(AC)\)

and \(angle(AOD) = \theta\).

Cartesian equation: \[X^4+X^2·Y^2-2·X^3·a-2·X·Y^2·a\]\[+Y^2·a^2+X^2·(a^2-b^2)\]

GXWeb Conchoid of Nicomedes

Construct your own Model with GXWeb

WolframAlpha: CAS+

Behind the Scenes