Definition:Harmonic Range

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


Let $A$ and $B$ be points on a straight line.

Let $P$ and $Q$ lie on $AB$ such that $P$ is on the line segment $AB$ while $Q$ is outside the line segment $AB$.


Let $P$ and $Q$ be positioned such that the cross-ratio $\set {A, B; P, Q}$ forms a harmonic ratio:

$\dfrac {AP} {PB} = -\dfrac {AQ} {QB}$

Then $\tuple {AB, PQ}$ are said to be a harmonic range.

Also known as

A harmonic range is also known as a harmonic set.


Ratio of Unity

Let $\tuple {AB, PQ}$ be a harmonic range such that $P$ is the midpoint of $AB$.

Then $Q$ is the point at infinity.

Also see

  • Results about harmonic ranges can be found here.