Inversion and stereographic projection
Overview
Important
Inversion is a transformation of the plane (or space) that 'flips' points with respect to a fixed circle (or sphere), called the center of inversion. Stereographic projection is a way to map points from a sphere onto a plane, often used to study properties of circles and spheres.
Important properties
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Inversion with respect to a circle maps points inside the circle to points outside, and vice versa.
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Circles passing through the center of inversion become straight lines, and vice versa.
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Stereographic projection preserves angles (it is conformal).
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Stereographic projection maps circles on the sphere to circles or lines on the plane.