Definition:Harmonic Ratio

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

Definition

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

Let the cross-ratio $\set {A, B; C, D}$ of $A$, $B$, $C$ and $D$ be equal to $-1$:

$\dfrac {AC / CB} {AD / DB} = -1$

that is:

$\dfrac {AC \cdot DB} {AD \cdot CB} = -1$

Then $\set {A, B; C, D}$ is known as a harmonic ratio.

Also see

• Definition:Harmonic Range

• Results about harmonic ratios can be found here.

Sources

• 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