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# quadratrix

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 $x\neq n\pi k$ ($n\in\mathbb{Z}$).

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

## Mathematics Subject Classification

53A04*no label found*

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