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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
quadratrix
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(Definition)
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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
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(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
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(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 $x \neq n\pi k$ ($n \in \mathbb{Z}$ ).
The quadratrix was used by the ancient Greek geometers for squaring the circle, the name comes from the Latin quadratrix = `a feminine squarer'.
Wiki
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"quadratrix" is owned by pahio.
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(view preamble | get metadata)
| Other names: |
quadratrix of Hippias, quadratrix of Dinostratus |
This object's parent.
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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
There is 1 reference to this entry.
This is version 8 of quadratrix, born on 2008-05-27, modified 2008-09-09.
Object id is 10630, canonical name is Quadratrix.
Accessed 1555 times total.
Classification:
| AMS MSC: | 53A04 (Differential geometry :: Classical differential geometry :: Curves in Euclidean space) |
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Pending Errata and Addenda
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![\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}](http://images.planetmath.org:8080/cache/objects/10630/js/img1.png "\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}")
![\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}](http://images.planetmath.org:8080/cache/objects/10630/js/img4.png "\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}")