Editing Cartesian coordinate system (section)

==Cartesian formulae for the plane==
===Distance between two points===
The [[Euclidean distance]] between two points of the plane with Cartesian coordinates <math>(x_1, y_1)</math> and <math>(x_2, y_2)</math> is

<math display=block>d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}.</math>

This is the Cartesian version of [[Pythagoras's theorem]]. In three-dimensional space, the distance between points <math>(x_1,y_1,z_1)</math> and <math>(x_2,y_2,z_2)</math> is

<math display=block>d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+ (z_2-z_1)^2} ,</math>

which can be obtained by two consecutive applications of Pythagoras' theorem.<ref>{{harvnb|Hughes-Hallett|McCallum|Gleason|2013}}</ref>

===Euclidean transformations===
The [[Euclidean plane isometry|Euclidean transformations]] or '''Euclidean motions''' are the ([[bijective]]) mappings of points of the [[Euclidean plane]] to themselves which preserve distances between points. There are four types of these mappings (also called isometries): [[Translation (geometry)|translations]], [[Rotation (mathematics)|rotations]], [[Reflection (mathematics)|reflections]] and [[glide reflection]]s.<ref>{{harvnb|Smart|1998|loc=Ch. 2}}.</ref>

====Translation====
[[Translation (geometry)|Translating]] a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers {{nowrap|(''a'', ''b'')}} to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are {{nowrap|(''x'', ''y'')}}, after the translation they will be

<math display=block>(x', y') = (x + a, y + b) .</math>

====Rotation====
To [[rotation (geometry)|rotate]] a figure [[clockwise|counterclockwise]] around the origin by some angle <math>\theta</math> is equivalent to replacing every point with coordinates (''x'',''y'') by the point with coordinates (''x<nowiki>'</nowiki>'',''y<nowiki>'</nowiki>''), where

<math display=block>
\begin{align}
x' &= x \cos \theta - y \sin \theta \\
y' &= x \sin \theta + y \cos \theta .
\end{align}
</math>

Thus:

<math display="block">(x',y') = ((x \cos \theta - y \sin \theta\,) , (x \sin \theta + y \cos \theta\,)) .</math>

====Reflection====
If {{nowrap|(''x'', ''y'')}} are the Cartesian coordinates of a point, then {{nowrap|(−''x'', ''y'')}} are the coordinates of its [[coordinate rotations and reflections|reflection]] across the second coordinate axis (the y-axis), as if that line were a mirror. Likewise, {{nowrap|(''x'', −''y'')}} are the coordinates of its reflection across the first coordinate axis (the x-axis). In more generality, reflection across a line through the origin making an angle <math>\theta</math> with the x-axis, is equivalent to replacing every point with coordinates {{nowrap|(''x'', ''y'')}} by the point with coordinates {{nowrap|(''x''′,''y''′)}}, where

<math display=block>
\begin{align}
x' &= x \cos 2\theta + y \sin 2\theta \\
y' &= x \sin 2\theta - y \cos 2\theta .
\end{align}
</math>

Thus:
<math display="block">(x',y') = ((x \cos 2\theta + y \sin 2\theta\,) , (x \sin 2\theta - y \cos 2\theta\,)) .</math>

====Glide reflection====
A glide reflection is the composition of a reflection across a line followed by a translation in the direction of that line. It can be seen that the order of these operations does not matter (the translation can come first, followed by the reflection).

====General matrix form of the transformations====
All [[affine transformation]]s of the plane can be described in a uniform way by using matrices. For this purpose, the coordinates <math>(x,y)</math> of a point are commonly represented as the [[column matrix]] <math>\begin{pmatrix}x\\y\end{pmatrix}.</math> The result <math>(x', y')</math> of applying an affine transformation to a point <math>(x,y)</math> is given by the formula
<math display=block>\begin{pmatrix}x'\\y'\end{pmatrix} = A \begin{pmatrix}x\\y\end{pmatrix} + b,</math>
where
<math display=block>A = \begin{pmatrix} A_{1,1} & A_{1,2} \\ A_{2,1} & A_{2,2} \end{pmatrix}</math>
is a 2×2 [[Square matrix|matrix]] and <math>b=\begin{pmatrix}b_1\\b_2\end{pmatrix}</math> is a column matrix.<ref>{{harvnb|Brannan|Esplen|Gray|1998|p=49}}.</ref> That is,
<math display=block>
\begin{align}
x' &= x A_{1,1} + y A_{1,1} + b_{1} \\
y' &= x A_{2,1} + y A_{2, 2} + b_{2}.
\end{align}
</math>

Among the affine transformations, the [[Euclidean transformation]]s are characterized by the fact that the matrix <math>A</math> is [[orthogonal matrix|orthogonal]]; that is, its columns are [[orthogonal vectors]] of [[Euclidean norm]] one, or, explicitly,
<math display=block>A_{1,1} A_{1, 2} + A_{2,1} A_{2, 2} = 0</math>
and
<math display=block>A_{1, 1}^2 + A_{2,1}^2 = A_{1,2}^2 + A_{2, 2}^2 = 1.</math>

This is equivalent to saying that {{math|''A''}} times its [[transpose]] is the [[identity matrix]]. If these conditions do not hold, the formula describes a more general [[affine transformation]].

The transformation is a translation [[if and only if]] {{math|''A''}} is the [[identity matrix]]. The transformation is a rotation around some point if and only if {{math|''A''}} is a [[rotation matrix]], meaning that it is orthogonal and
<math display=block> A_{1, 1} A_{2, 2} - A_{2, 1} A_{1, 2} = 1 .</math>

A reflection or glide reflection is obtained when,
<math display=block> A_{1, 1} A_{2, 2} - A_{2, 1} A_{1, 2} = -1 .</math>

Assuming that translations are not used (that is, <math>b_1=b_2=0</math>) transformations can be [[function composition|composed]] by simply multiplying the associated transformation matrices. In the general case, it is useful to use the [[augmented matrix]] of the transformation; that is, to rewrite the transformation formula 
<math display=block>\begin{pmatrix}x'\\y'\\1\end{pmatrix} = A' \begin{pmatrix}x\\y\\1\end{pmatrix},</math>
where 
<math display=block>A' = \begin{pmatrix} A_{1,1} & A_{1,2}&b_1 \\ A_{2,1} & A_{2,2}&b_2\\0&0&1 \end{pmatrix}.</math>
With this trick, the composition of affine transformations is obtained by multiplying the augmented matrices.

===Affine transformation===
[[File:2D_affine_transformation_matrix.svg|thumb|Effect of applying various 2D affine transformation matrices on a unit square (reflections are special cases of scaling)]]

[[Affine transformation]]s of the [[Euclidean plane]] are transformations that map lines to lines, but may change distances and angles. As said in the preceding section, they can be represented with augmented matrices:
<math display=block>\begin{pmatrix} A_{1,1} & A_{2,1} & b_{1} \\ A_{1,2} & A_{2,2} & b_{2} \\ 0 & 0 & 1 \end{pmatrix}
\begin{pmatrix} x \\ y \\ 1 \end{pmatrix}
=
\begin{pmatrix} x' \\ y' \\ 1 \end{pmatrix}.</math>

The Euclidean transformations are the affine transformations such that the 2×2 matrix of the <math>A_{i,j}</math> is [[orthogonal matrix|orthogonal]].

The augmented matrix that represents the [[function composition|composition]] of two affine transformations is obtained by multiplying their augmented matrices.

Some affine transformations that are not Euclidean transformations have received specific names.

====Scaling====
An example of an affine transformation which is not Euclidean is given by scaling. To make a figure larger or smaller is equivalent to multiplying the Cartesian coordinates of every point by the same positive number ''m''. If {{nowrap|(''x'', ''y'')}} are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates

<math display=block>(x',y') = (m x, m y).</math>

If ''m'' is greater than 1, the figure becomes larger; if ''m'' is between 0 and 1, it becomes smaller.

====Shearing====
A [[shear mapping|shearing transformation]] will push the top of a square sideways to form a [[parallelogram]]. Horizontal shearing is defined by:

<math display=block>(x',y') = (x+y s, y)</math>

Shearing can also be applied vertically:

<math display=block>(x',y') = (x, x s+y)</math>