Isometry is a transformation (the same as function) which preserves measurements, more specifically – it preserves distances between points. If $f$ is a transformation and $A$ and $B$ are points in the plane, then by the definition :
$|f(A),f(B)|=|A,B|$.
A transformation preserves distances so it is also bound to preserve angles, which follows from the SSS (side-side-side) theorem.
A plane isometry is a function that is defined for every point of the plane. If we consider plane figures as collections of points, then every such collection $S$ has an image $f(S)$ under the isometry $f$. The definition of isometry assures that relative positions of points in $S$ are preserved in $f(S)$.
Definition of isometry: A transformation $f : M \rightarrow M$, where $M$ is a plane, is an isometry if for any two points $A$, $B$ ∈ $M$, the Euclidean distance must be $|f(A) − f(B)| = |A − B|$.
There are only four types of isometries of the Euclidean plane – translations, rotations, reflections, and glide reflections, which together form a group under composition known as the Euclidean group of the plane.
Translation
Translation $t$ is a plane isometry, $t : M \rightarrow M$, where all points in a plane are moved for a fixed vector $v$. A vector is specified by its direction, length and orientation, and if one these is changed then it defines a different function.
The image of a point $T$ is the point $T’$ shuch that the directed line segment $|TT’|$ has:
- $|TT’|=|v|$, the same lenght as the vector $v$
- the same direction as the vector $v$
- the same orientation as the vector $v$
Important characteristic of translation:
- Translation preserves the orientation. For example, if a polygon is traversed clockwise, its translated image is also traversed clockwise.
- Translation is an isometry which means it preserves distances and angles.
- Translation maps parallel lines onto parallel lines and, moreover, a line and its image are also parallel.
- Except for the trivial translation by a zero vector, translation has no fixed points.
- Successive translations result in a translation.
- The order of translations does not matter: any two translations commute.
Rotation
- Rotation maps parallel lines onto parallel lines.
- Except for the trivial rotation through a zero angle which is identical, rotations have a single fixed point – the center of rotation. Except for the trivial case, rotations have no fixed lines. However, all circles centered at the center of rotation are fixed.
- Successive rotations result in a rotation or a translation.
- The product of rotations is not in general commutative. Two rotations with a common center commute as a matter of course.
Reflections
A transformation in which a geometric figure is reflected across a line, creating a mirror image. That line is called the axis of reflection.
How to draw an image of this transformation?
The plane is transformed one point at the time. Take the point $A$, through it draw a line perpendicular to the axis of reflection, let’s call it line $p$. Measure the lenght of the point $A$ from the intersection of the line $p$ and the aixs of reflection, let’s call this point $S$, the image of $A$, the point $A’$, must be on the line $p$ and $|SA|=|SA’|$. Repeat the same for the other two points and then connect the corresponding ones to get a triangle.
Glide reflections
A glide reflection is a composition of two transformations : reflection and translation.
For the $\bigtriangleup ABC$ we create its reflected image $\bigtriangleup A’B’C’$ with the same proces described above. We then translate vertices $A’$, $B’$ and $C’$ by a fixed vector by $v$ to get the $\bigtriangleup A”B”C”$, .
