Editing Transformation matrix (section)

==Examples in 3D computer graphics==

===Rotation===

The [[rotation matrix#General 3D rotations|matrix to rotate]] an angle ''θ'' about any axis defined by [[unit vector]] (''x'',''y'',''z'') is<ref>{{cite book |page = 154 |title = Basic Mathematics for Electronic Engineers:Models and Applications |first = John E. |last = Szymanski |publisher = Taylor & Francis |year = 1989 |isbn = 0278000681 }}</ref>
<math display="block">\begin{bmatrix}
xx(1-\cos \theta)+\cos\theta & yx(1-\cos\theta)-z\sin\theta & zx(1-\cos\theta)+y\sin\theta\\
xy(1-\cos\theta)+z\sin\theta & yy(1-\cos\theta)+\cos\theta & zy(1-\cos\theta)-x\sin\theta \\
xz(1-\cos\theta)-y\sin\theta & yz(1-\cos\theta)+x\sin\theta & zz(1-\cos\theta)+\cos\theta
\end{bmatrix}.</math>

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

To reflect a point through a plane <math>ax + by + cz = 0</math> (which goes through the origin), one can use <math>\mathbf{A} = \mathbf{I} - 2\mathbf{NN}^\mathrm{T} </math>, where <math>\mathbf{I}</math> is the 3×3 identity matrix and <math>\mathbf{N}</math> is the three-dimensional [[unit vector]] for the vector normal of the plane.  If the [[L2 norm|''L''<sup>2</sup> norm]] of <math>a</math>, <math>b</math>, and <math>c</math> is unity, the transformation matrix can be expressed as:
<math display="block">\mathbf{A} = \begin{bmatrix} 1 - 2 a^2  & - 2 a b & - 2 a c \\ - 2 a b  & 1 - 2 b^2 & - 2 b c  \\ - 2 a c & - 2 b c & 1 - 2c^2 \end{bmatrix}</math>

Note that these are particular cases of a [[Householder reflection]] in two and three dimensions.  A reflection about a line or plane that does not go through the origin is not a linear transformation — it is an [[affine transformation]] — as a 4×4 affine transformation matrix, it can be expressed as follows (assuming the normal is a unit vector):
<math display="block">\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \begin{bmatrix} 1 - 2 a^2  & - 2 a b & - 2 a c & - 2 a d \\ - 2 a b  & 1 - 2 b^2 & - 2 b c & - 2 b d \\ - 2 a c & - 2 b c & 1 - 2c^2 & - 2 c d \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix} </math>
where <math>d = -\mathbf{p} \cdot \mathbf{N}</math> for some point <math>\mathbf{p}</math> on the plane, or equivalently, <math>ax + by + cz + d = 0</math>.

If the 4th component of the vector is 0 instead of 1, then only the vector's direction is reflected and its magnitude remains unchanged, as if it were mirrored through a parallel plane that passes through the origin. This is a useful property as it allows the transformation of both positional vectors and normal vectors with the same matrix.  See [[homogeneous coordinates]] and [[#Other kinds of transformations|affine transformations]] below for further explanation.