What Is Shear Transform?

Shearing is an affine transformation \(f: P \rightarrow f(P)\) such that, for any point \(P\), the line through \(P\) and \(f(P)\) is parallel to a fixed line, say \(\lambda\), while the distance from \(P\) to \(f(P)\) is proportional to the distance from \(P\) to \(\lambda\). All points on \(\lambda\) itself remain fixed: \(f(P) = P\), \(P\in\lambda\). Shearing transform relates to Euclid propositions I.35 - I.38 that assert preservation of areas of parallelograms and triangles with fixed base and other vertices moved parallel to it. Shearing is the main tool in several proofs of the Pythagorean theorem.

In a Cartesian system of coordinates where λ serves as the x-axis, the shearing transform has the matrix form

\( \begin{pmatrix} u \\ v \end{pmatrix} = \begin{pmatrix} 1 & k \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} \)

so that \(v = y\) and \(u = x + ky\).

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