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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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