Definition:Harmonic Range

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

Definition

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.

Examples

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

• Definition:Harmonic Conjugates

• Results about harmonic ranges 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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): cross-ratio
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): harmonic range or ratio
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): cross-ratio
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): harmonic range or ratio
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): harmonic range