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Plane isometries

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:

  1. $|TT’|=|v|$, the same lenght as the vector $v$
  2. the same direction as the vector $v$
  3. 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 $\bigtriangleup A”B”C”$, .