This geometric transformation associates with each point M other than the origin the point M' collinear with O and M such that OM × OM' = 1. Its main virtue is that it leaves invariant the set of circles and lines in the plane. Its complex expression is simply
Inversion can be used to prove the Mohr–Mascheroni theorem, which states that any straightedge-and-compass construction can be carried out with compass alone. From a more practical standpoint, it is inversion that underlies the Peaucellier inversor, a device that takes its name from its inventor and that converts circular motion into rectilinear motion.
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Image of a line under inversion
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Let Δ be a line with equation y = ax. Every point on this line has an affix z of the form z = x + iax. We deduce that
