# Definition:Conchoid of Nicomedes

## Definition

The conchoid of Nicomedes is the plane curve defined in Cartesian coordinates as:

$\paren {x - a}^2 \paren {x^2 + y^2} = b^2 x^2$

or in polar coordinates as:

$r = b + a \sec \theta$

for some real constants $a \in \R$, $b \in \R_{> 0}$.

The above diagram illustrates the conchoid of Nicomedes for $b = 1$ and various values of $a$ from $0$ to $3$.

## Also known as

Some sources suggest that the conchoid of Nicomedes can also be referred to as a cochloid. However, this usage can be confused easily with the cochleoid which some dictionaries give cochloid as an alternative for.

Others refer to it as merely a conchoid, but that term is best used to refer to the more general object of which the conchoid of Nicomedes is an example.

## Source of Name

This entry was named for Nicomedes.

## Historical Note

Nicomedes designed the curve now known as the conchoid of Nicomedes specifically for solving the problem of Doubling the Cube.

It can also be used for Trisecting the Angle.