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[parent] quadratrix (Definition)

\begin{pspicture}(-1,-1)(4,3) \psaxes[Dx=9,Dy=9]{->}(0,0)(-0.5,-0.5)(4,3) \rput(... ... \rput(0.7,0.2){$t$} \rput(3.5,1){$kt$} \rput(3.4,2.25){$(x,y)$} \end{pspicture}

Let the polar angle $ \theta$ and the ordinate of the point $ (x,\,y)$ of the plane be proportional to a parametre $ t$, e.g. such that $ \theta = t$, $ y = kt$. The above diagram shows that $ \displaystyle\frac{x}{kt} = \cot{t}$. Into this we can substitute $ \displaystyle t = \frac{y}{k}$, whence we get an equation

$\displaystyle x = y\cot\frac{y}{k}$ (1)

between $ x$ and $ y$. The given proportionalities as locus condition, the point $ (x,\,y)$ draws a plane curve called quadratrix. If we change the $ x$ and $ y$ coordinates, the equation of the quadratrix is
$\displaystyle y = x\cot\frac{x}{k}.$ (2)

The equation (2) defines an even function $ x \mapsto y$, where for $ x$ is allowed all real values at which the right hand side of (2) is defined, thus $ x \neq n\pi k$ ( $ n \in \mathbb{Z}$).

\begin{pspicture}(-5.5,-6.5)(5.5,5) \psaxes[Dx=9,Dy=9]{->}(0,0)(-1.5,-5.5)(3.5,5... ...){$x$} \rput(-0.24,5.4){$y$} \rput(-5.5,-6.5){.} \rput(5.5,5){.} \end{pspicture}

The quadratrix was used by the ancient Greek geometers for squaring the circle, the name comes from the Latin quadratrix = `a feminine squarer'.

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Other names:  quadratrix of Hippias, quadratrix of Dinostratus

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Cross-references: squaring the circle, right hand side, real, even function, coordinates, plane curve, locus, equation, diagram, parametre, plane, point, ordinate, polar angle
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This is version 8 of quadratrix, born on 2008-05-27, modified 2008-09-09.
Object id is 10630, canonical name is Quadratrix.
Accessed 699 times total.

Classification:
AMS MSC53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space)

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