Editing Transformation matrix (section)

==Examples in 2 dimensions==

Most common [[geometric transformation]]s that keep the origin fixed are linear, including rotation, scaling, shearing, reflection, and orthogonal projection; if an affine transformation is not a pure translation it keeps some point fixed, and that point can be chosen as origin to make the transformation linear.  In two dimensions, linear transformations can be represented using a 2×2 transformation matrix.

===Stretching===

A stretch in the ''xy''-plane is a linear transformation which enlarges all distances in a particular direction by a constant factor but does not affect distances in the perpendicular direction. We only consider stretches along the x-axis and y-axis. A stretch along the x-axis has the form {{math|1=<var>x'</var> = <var>kx</var>}}; {{math|1=<var>y'</var> = <var>y</var>}} for some positive constant {{mvar|k}}. (Note that if {{math|1=<var>k</var> &gt; 1}}, then this really is a "stretch"; if {{math|1=<var>k</var> &lt; 1}}, it is technically a "compression", but we still call it a stretch. Also, if {{math|1=<var>k</var> = 1}}, then the transformation is an identity, i.e. it has no effect.)

The matrix associated with a stretch by a factor {{mvar|k}} along the x-axis is given by:
<math display="block">\begin{bmatrix} k &  0 \\ 0 & 1 \end{bmatrix} </math>

Similarly, a stretch by a factor <var>k</var> along the y-axis has the form {{math|1=<var>x'</var> = <var>x</var>}}; {{math|1=<var>y'</var> = <var>ky</var>}}, so the matrix associated with this transformation is
<math display="block">\begin{bmatrix} 1 &  0 \\ 0 & k \end{bmatrix} </math>

===Squeezing===

If the two stretches above are combined with reciprocal values, then the transformation matrix represents a [[squeeze mapping]]:
<math display="block">\begin{bmatrix} k &  0 \\ 0 & 1/k \end{bmatrix} .</math>
A square with sides parallel to the axes is transformed to a rectangle that has the same area as the square. The reciprocal stretch and compression leave the area invariant.

===Rotation===

For [[coordinate rotation|rotation]] by an angle θ '''counterclockwise''' (positive direction) about the origin the functional form is <math>x' = x \cos \theta - y \sin \theta</math> and <math>y' =  x \sin \theta + y \cos \theta</math>.  Written in matrix form, this becomes:<ref>{{Cite web | url=http://ocw.mit.edu/courses/aeronautics-and-astronautics/16-07-dynamics-fall-2009/lecture-notes/MIT16_07F09_Lec03.pdf | title=Lecture Notes | website=ocw.mit.edu | access-date=2024-07-28}}</ref>
<math display="block">\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta & -\sin\theta \\ \sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}</math>

Similarly, for a rotation '''clockwise''' (negative direction) about the origin, the functional form is <math>x' = x \cos \theta + y \sin \theta</math> and <math>y' = -x \sin \theta + y \cos \theta</math> the matrix form is:
<math display="block">\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} \cos \theta &   \sin\theta \\ -\sin \theta & \cos \theta \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}</math>

These formulae assume that the ''x'' axis points right and the ''y'' axis points up.

===Shearing===

For [[shear mapping]] (visually similar to slanting), there are two possibilities.

A shear parallel to the ''x'' axis has <math>x' = x + ky</math> and <math>y' = y</math>. Written in matrix form, this becomes:
<math display="block">\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}</math>

A shear parallel to the ''y'' axis has <math>x' = x</math> and <math>y' = y + kx</math>, which has matrix form:
<math display="block">
\begin{bmatrix} x' \\ y' \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ k & 1 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix}
</math>

===Reflection===
{{main|Householder transformation}}

For reflection about a line that goes through the origin, let <math>\mathbf{l} = (l_x, l_y)</math> be a [[vector (geometric)|vector]] in the direction of the line. Then the transformation matrix is:
<math display="block">\mathbf{A} = \frac{1}{\lVert\mathbf{l}\rVert^2} \begin{bmatrix} l_x^2 - l_y^2 & 2 l_x l_y \\ 2 l_x l_y & l_y^2 - l_x^2 \end{bmatrix}</math>

===Orthogonal projection===
{{further|Orthogonal projection}}

To project a vector orthogonally onto a line that goes through the origin, let <math>\mathbf{u} = (u_x, u_y)</math> be a [[vector (geometric)|vector]] in the direction of the line.  Then the transformation matrix is:
<math display="block">\mathbf{A} = \frac{1}{\lVert\mathbf{u}\rVert^2} \begin{bmatrix} u_x^2 & u_x u_y \\ u_x u_y & u_y^2 \end{bmatrix}</math>

As with reflections, the orthogonal projection onto a line that does not pass through the origin is an affine, not linear, transformation.

[[Projection (linear algebra)|Parallel projections]] are also linear transformations and can be represented simply by a matrix.  However, perspective projections are not, and to represent these with a matrix, [[Homogeneous coordinates#Use in computer graphics and computer vision|homogeneous coordinates]] can be used.