Definition:Conchoid

Definition

Let $\CC$ be a curve.

Let $P$ be a fixed point which may or may not lie on $\CC$ itself.

A conchoid is the locus of points $p$ such that:

each $p$ is on a straight line $\LL$ that passes through $P$ and intersects $\CC$

the distance from $p$ to the point of intersection with $\CC$ is a specified constant.

As there are in general $2$ such points $p$ which satisfy the criteria, a conchoid has in most cases two branches.

Focus Point

Let $\KK$ be a conchoid.

Let $P$ be the fixed point on which the moving straight line $\LL$ is constrained to lie.

$P$ is known as the focus point of $\KK$.

Directrix

Let $\KK$ be a conchoid.

Let $\CC$ be the curve with respect to which the conchoid is constructed.

$\CC$ is known as the directrix of $\KK$.

Branch

Let $\KK$ be a conchoid.

The two parts of $\KK$ corresponding to the loci of the points at either endpoint of the generating line segment are referred to as the branches of $\KK$.

Also see

• Definition:Conchoid of Nicomedes, where $\CC$ is a straight line

• Results about conchoids can be found here.

Linguistic Note

The word conchoid derives from the Latin concha, which means mussel.

Ultimately the word derives from the Ancient Greek κόγχη (kónkhē) plus -oid, or directly from Ancient Greek κογχοειδής (konkhoeidḗs), which refers to anything with the general shape of a mussel shell.

It is properly pronounced kon-khoid, where the kh sound is the one found in the Scots loch or German ich.

However, it is commonplace to use the pronunciation kon-koid.

Note that the pronunciations kon-tshoid and kon-shoid are technically incorrect.

Sources

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