Shear mapping

In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction. This type of mapping is also called shear transformation, transvection, or just shearing.

An example is the mapping that takes any point with coordinates ${\displaystyle (x,y)}$ to the point ${\displaystyle (x+2y,y)}$. In this case, the displacement is horizontal, the fixed line is the ${\displaystyle x}$-axis, and the signed distance is the ${\displaystyle y}$ coordinate. Note that points on opposite sides of the reference line are displaced in opposite directions.

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