Definition:Quadratrix of Hippias

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Definition

Definition 1

The quadratrix of Hippias is the plane curve defined in Cartesian coordinates as:

$y = x \cot \left({\dfrac {\pi x }{2 a} }\right)$

for some real constant $a \in \R$.


QuadratrixOfHippias.png


The above diagram illustrates the quadratrix of Hippias.


Definition 2

The quadratrix of Hippias is the plane curve be generated as follows.

Let $\Box ABCD$ be a square of side length $a$.

Let a quarter circle be inscribed in $\Box ABCD$ with the center at $A$, the arc going from $B$ to $D$.

Let $E$ be a point travelling around the arc $BD$ at a constant angular velocity.

Let $F$ be a point travelling along the side of $\Box ABCD$ at a constant velocity such that $E$ and $F$ take the same time to travel from the top $CD$ to the bottom $AB$ of $\Box ABCD$.

Let $S$ be the point at which the line $AE$ intersects the line through $F$ parallel to $AB$.


QuadratrixOfHippias2.png


The path traced out by $S$ is the quadratrix of Hippias.


Also known as

The quadratrix of Hippias is also known as the trisectrix of Hippias, as it can be used both for quadrature of the circle and trisection of a general angle.

It is also known as the quadratrix of Dinostratus.


Source of Name

This entry was named for Hippias of Elis.


Historical Note

The quadratrix of Hippias was invented by Hippias of Elis for the purpose of solving the problem of Trisecting the Angle.

It was subsequently used by Dinostratus for Squaring the Circle.

Its use for both trisection and quadrature (that is, finding area) explains the multiple nature of its names.


Sources