Editing Shear mapping (section)

===Horizontal and vertical shear of the plane===
{{refimprove section|date=April 2025}}
[[File:SVG skewX.svg|thumb|250px|Horizontal shear of a square into parallelograms with factors <math>\cot(60^\circ) = \tan(30^\circ) \approx 0.58</math> and <math>\cot(45^\circ) = \tan(45^\circ) = 1</math>]]
In the plane <math>\R^2 = \R\times\R</math>, a '''horizontal shear''' (or '''shear parallel''' to the {{mvar|x}}-axis) is a function that takes a generic point with coordinates <math>(x,y)</math> to the point <math>(x + m y,y)</math>; where {{mvar|m}} is a fixed parameter, called the '''shear factor'''.

The effect of this mapping is to displace every point horizontally by an amount proportionally to its {{mvar|y}}-coordinate.  Any point above the {{mvar|x}}-axis is displaced to the right (increasing {{mvar|x}}) if {{math|''m'' > 0}}, and to the left if {{math|''m'' < 0}}. Points below the {{mvar|x}}-axis move in the opposite direction, while points on the axis stay fixed.

Straight lines parallel to the {{mvar|x}}-axis remain where they are, while all other lines are turned (by various angles) about the point where they cross the {{mvar|x}}-axis. Vertical lines, in particular, become [[Angle#Types of angles|oblique]] lines with [[slope]] <math>\tfrac 1 m.</math> Therefore, the shear factor {{mvar|m}} is the [[cotangent]] of the '''shear angle''' <math>\varphi</math> between the former verticals and the {{mvar|x}}-axis.{{fact|date=April 2025}} In the example on the right the square is tilted by 30°, so the shear angle is 60°.

If the coordinates of a point are written as a [[column vector]] (a 2×1 [[matrix (mathematics)|matrix]]), the shear mapping can be written as [[matrix product|multiplication]] by a 2×2 matrix:
: <math>
  \begin{pmatrix}x^\prime \\y^\prime \end{pmatrix}  =
  \begin{pmatrix}x + m y \\y \end{pmatrix} =
  \begin{pmatrix}1 & m\\0 & 1\end{pmatrix} 
    \begin{pmatrix}x \\y \end{pmatrix}.
</math>

A '''vertical shear''' (or shear parallel to the {{mvar|y}}-axis) of lines is similar, except that the roles of {{mvar|x}} and {{mvar|y}} are swapped. It corresponds to multiplying the coordinate vector by the [[transpose of a matrix|transposed matrix]]:

:<math>
  \begin{pmatrix}x^\prime \\y^\prime \end{pmatrix}  = 
  \begin{pmatrix}x \\ m x + y \end{pmatrix} = 
  \begin{pmatrix}1 & 0\\m & 1\end{pmatrix} 
    \begin{pmatrix}x \\y \end{pmatrix}.
</math>

The vertical shear displaces points to the right of the {{mvar|y}}-axis up or down, depending on the sign of {{mvar|m}}.  It leaves vertical lines invariant, but tilts all other lines about the point where they meet the {{mvar|y}}-axis.  Horizontal lines, in particular, get tilted by the shear angle <math>\varphi</math> to become lines with slope {{mvar|m}}.

====Composition====
Two or more shear transformations can be combined.

If two shear matrices are <math display="inline">\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}</math> and <math display="inline">\begin{pmatrix} 1 & 0 \\ \mu & 1 \end{pmatrix}</math>

then their composition matrix is
<math display="block">\begin{pmatrix} 1 & \lambda \\ 0 & 1 \end{pmatrix}\begin{pmatrix} 1 & 0 \\ \mu & 1\end{pmatrix} = \begin{pmatrix} 1 + \lambda\mu & \lambda \\ \mu & 1 \end{pmatrix},</math>
which also has determinant 1, so that area is preserved.

In particular, if <math>\lambda=\mu</math>, we have

<math display="block">\begin{pmatrix} 1 + \lambda^2 & \lambda \\ \lambda & 1 \end{pmatrix},</math>

which is a [[positive definite matrix]].