# Linear transformations

## General affine mappings

The transformation $P.'=T(P.)$represented by the system of linear equations

$x'=ax+byy'=cx+dy$

is referred to as affine mapping with a fixed coordinate origin in two dimensions. The coefficients $a,b,c,d$ are elements of an invertible matrix

$T=abcd.$

If one also takes into account a translation (shift) around a constant vector$t=ef$, the general affine mapping in two dimensions reads:

$x'y'=abcdxy+ef$

or in vector form:

$r'=Tr+t.$

In 3D space, an affine mapping is expressed by:

$x'=a11x+a12y+a13z+b1y'=a21x+a22y+a23z+b2z'=a31x+a32y+a33z+b3.$

Affine mapping is characterized in that collinearity is preserved, i.e.

• all points lying on a segment remain on the segment after the transformation, and
• the ratio of the lengths of lines is retained after the transformation, e.g. the midpoint of a straight line remains the midpoint after the transformation.

In contrast, i. In general, the length of a segment and the angle between two straight lines are not obtained.

An isometric mapping or isometry is subject to the restriction that the matrix $T$is orthogonal, i.e.$TTT=TTT=I.$. Lengths (and angles) are preserved.

Put in the matrix $T$ $b=c=0$ , the following transformation types are possible:

• Unit transformation (isometry)
• Scaling
• Reflection (isometry)

Removal of the restriction $b=c=0$ results in:

• Shear $b≠0 and c=0$ or $b=0 and c≠0$
• Rotation (isometry) $b≠0 and c≠0$

In the animation, two-dimensional affine images can be visualized on a rectangle. The Elements $a,b,c,d$ the transformation matrix can be freely selected and with the help of the parameter$s$ apply to the figure:

$T(s=0)=1001andT(s=1)=abcd.$

Here are some cases:

Tab. 1
Affine images
matrixRemarks
$-100-1$Orthogonal matrix. Isometry. Reflection in $y=x$ and then in $y=-x$. Equivalent to turn around $π$.
$1-101$No isometry. $x$-Shear.
$0100$Not invertible. No affine mapping.

Note: When producing the map, the surface of the globe is mapped onto a plane. But this map is not an isometry, which Gauss proved (known as Theorema Egregium). In an isometric illustration, the Gaussian curvature is retained, that for a sphere (radius $r$) $1r2$ and one level is equal to 0.