# 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.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**conchoid** - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 25992 \, 10498 \, 94873 \, 16476 \ldots$

- Weisstein, Eric W. "Conchoid of Nicomedes." From
*MathWorld*--A Wolfram Web Resource. https://mathworld.wolfram.com/ConchoidofNicomedes.html