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(of the plane)

Let l be a line in the plane. Then the mirror-image of a point P is the point P′ such that PP′ is perpendicular to l and l cuts PP′ at its midpoint. The reflection of the plane in the line l is the transformation of the plane that maps each point P to its mirror-image P′. Suppose that the line l passes through the origin O and makes an angle α with the x-axis. If P has polar coordinates (r, θ), its mirror-image P′ has polar coordinates (r, 2α−θ). In terms of Cartesian coordinates, reflection in the line l maps P with coordinates (x, y) to P′ with coordinates (x′, y′), where

x′ = x cos 2α + y sin 2α,

y′ = x sin 2α − y cos 2α.

Subjects: Mathematics.

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