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[parent] conchoid of Nicomedes (Topic)

The conchoid of Nicomedes is the locus of the endpoints of a line segment (length = $ 2b$), the midpoint of which is on a given line ($ l$) and which lies on a line through a given point ($ O$, with distance $ a$ from $ l$).

The curve was invented by the Greek matematician Nicomedes in about 200 BCE. With his conchoid he solved two classical problems of constructibility, viz. doubling the cube and trisecting the angle. The name of the curve is derived from Greek $ \varkappa\acute{o}\gamma\chi\eta$ `mussel', $ \varepsilon\iota\delta{o}\varsigma$ `form, kind, type'.


\begin{pspicture}(-5.5,-4.5)(5.5,4) \psaxes[Dx=9,Dy=9]{->}(0,0)(-1.2,-5)(4.8,5) ... ...linecolor=blue](1,1.337)(3,4.005) \psdot[linecolor=red](2,2.667) \end{pspicture}

Choosing for $ O$ the origin and $ l$ vertical, we have in the polar coordinates $ r,\, \varphi$ for the conchoid of Nicomedes the expression

$\displaystyle r = \frac{a}{\cos\varphi}\pm{b}$ (1)

or
$\displaystyle \left(r-\frac{a}{\cos\varphi}\right)^{\!2} = b^2.$
Her we may substitute $ \cos\varphi = \frac{x}{r}$ and then $ r^2 = x^2\!+\!y^2$, when the equation of the curve can be simplified to
$\displaystyle (x^2\!+\!y^2)(x\!-\!a)^2 = b^2x^2.$ (2)

The form
$\displaystyle y = \pm\frac{x}{x\!-\!a}\sqrt{b^2\!-\!(x\!-\!a)^2}$ (3)

of the equation tells that the curve has as an asymptote the line $ x = a$.

It's not hard to derive the following parametric presentation of the conchoid of Nicomedes:

$\displaystyle x = a\pm\frac{ab}{\sqrt{a^2+h^2}},\quad y = h\pm\frac{bh}{\sqrt{a^2+h^2}}$ (4)

The shape of the conchoid depends on the ratio $ a\!:\!b$. Below in the picture there are three cases where $ a = 2$ and $ b$ has the values $ 1.5$ (green), $ 2$ (blue) and $ 2.5$ (cyan).



\begin{pspicture}(-4,-6)(9,6.3) \psaxes[Dx=1,Dy=1]{->}(0,0)(-0.9,-5.9)(5,6) \rpu... ...x mul sub 4 x mul add sqrt mul} \par \rput(7,3){Three conchoids} \end{pspicture}



"conchoid of Nicomedes" is owned by pahio.
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Other names:  conchoid of line, conchoid

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Cross-references: ratio, parametric presentation, asymptote, equation, expression, polar coordinates, origin, trisecting the angle, doubling the cube, viz, classical problems of constructibility, curve, distance, point, lies on, line, midpoint, line segment, endpoints, locus
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This is version 14 of conchoid of Nicomedes, born on 2007-09-15, modified 2008-06-25.
Object id is 9940, canonical name is ConchoidOfNicomedes.
Accessed 936 times total.

Classification:
AMS MSC: 51-00 (Geometry :: General reference works )
 51N20 (Geometry :: Analytic and descriptive geometry :: Euclidean analytic geometry)
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