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[parent] cissoid of Diocles (Definition)

Let $ c$ be a circle with diameter $ OA = a$. Set a tangent line $ t$ of the circle at the point $ A$. For any point $ C$ of $ c$ let $ P$ be the intersection point of the secant line $ OC$ and the tangent line $ t$. Determine on the secant line between $ O$ and $ P$ the point $ Q$ such that

$\displaystyle PQ = OC.$
Then the locus of the point $ Q$ is the cissoid of Diocles. The name is derived from Greek $ \varkappa\iota\sigma\sigma\acute{o}\varsigma$ (kissos) `ivy', $ \varepsilon\iota\delta{o}\varsigma$ (eidos) `form, kind, type'.

The cissoid is symmetric with regard to the line $ OA$, having at $ O$ a cusp. The line $ t$ is the asymptote of the curve.

For deriving the equation of the cissoid, chose the ray $ OA$ for the positive $ x$-axis. Let $ \varphi$ be the slope angle (polar angle) of any $ C$ on $ c$. From the triangle $ OAP$ we see that $ \displaystyle OP = \frac{a}{\cos\varphi}$. Since $ \angle ACO$ is a right angle, we have $ OC = a\cos\varphi$. It follows that $ \displaystyle OQ = r = \frac{a}{\cos\varphi}-a\cos\varphi$, that is,

$\displaystyle r = a\sin\varphi\tan\varphi.$ (1)

For obtaining the equation in rectangular coordinates, we may write (1) as $ r^2 = (r\sin\varphi)\,a\tan\varphi$, i.e. $ \displaystyle x^2\!+\!y^2 = ya\cdot\frac{y}{x}$, whence $ (x^2\!+\!y^2)x = ay^2$, or
$\displaystyle y \,=\, \pm{x}\,\sqrt{\frac{x}{a-x}}.$ (2)

The cissoid has also the parametric presentation
$\displaystyle x = \frac{au^2}{1+u^2}, \quad y = \frac{au^3}{1+u^2}.$ (3)

Note. If we apply the inversion formulae

$\displaystyle x \mapsto \frac{x}{x^2+y^2},\;\; y \mapsto \frac{y}{x^2+y^2}$
(where $ r = 1$) to the parabola $ y = \pm\sqrt{x}$, we get as the image of the parabola the cissoid $ \displaystyle{y} = \pm{x}\,\sqrt{\frac{x}{1-x}}$; correspondingly the image of this cissoid is that parabola.

\begin{pspicture}(-5.5,-6.5)(5.5,5) \psaxes[Dx=1,Dy=1]{->}(0,0)(-1.5,-5.5)(3.5,5... ...put(1,-6){$\mbox{The cissoid\,\,}y = \pm x\sqrt{\frac{x}{2-x}}$} \end{pspicture}
The form of the cissoid of Diocles resembles the tractrix.



"cissoid of Diocles" is owned by pahio.
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See Also: parameter, analytic geometry

Other names:  cissoid, ivy curve

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Cross-references: tractrix, image, parabola, inversion formulae, parametric presentation, rectangular coordinates, right angle, triangle, polar angle, slope angle, positive, ray, equation, curve, asymptote, cusp, line, symmetric, locus, secant line, intersection, point, tangent line, diameter, circle
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This is version 12 of cissoid of Diocles, born on 2007-06-20, modified 2007-12-15.
Object id is 9632, canonical name is CissoidOfDiocles.
Accessed 1756 times total.

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