Definition:Harmonic Conjugates/Harmonic Pencil

This page is about Harmonic Conjugates. For other uses, see Harmonic.

Definition

Let $AB$ and $PQ$ be line segments on a straight line such that $\tuple {AB, PQ}$ is a harmonic range.

Let $O$ be a point which is not on the straight line $AB$.

Let $\map O {AB, PQ}$ be the harmonic pencil formed from $O$ and $\tuple {AB, PQ}$.

The rays $OP$ and $OQ$ are said to be harmonic conjugates with respect to $OA$ and $OB$.

Also known as

Two pairs of:

points $\set {A, B}$ and $\set {P, Q}$

rays $\set {OA, OB}$ and $\set {OP, OQ}$

which are harmonic conjugates are also known as a conjugate pair.

Harmonic conjugates can also be said to be apolar.

Also see

• Definition:Harmonic Pencil

• Results about harmonic conjugates can be found here.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {II}$. The Straight Line: $19$. Harmonic ranges and pencils