Isometry

Isometry is a geometric transformation that preserves the distances between points.

In simpler terms, if two points have a certain distance between them before the transformation, that distance remains unchanged afterward.

This means that every isometry transforms a shape into a congruent shape.

For example, the first triangle is rotated 90° clockwise and shifted to the right. The distances between points A, B, and C remain the same.

Therefore, rotation and translation are both examples of isometric transformations.

Additionally, both triangles are congruent because their side lengths stay the same.

The term 'isometric' is often used interchangeably with 'congruent'. The word has Greek origins, where 'isos' means 'same' and 'metros' means 'measure'. For example, the two triangles mentioned earlier are congruent because their sides and angles remain congruent even after rotation and translation.

Transformations that result in an isometry are called isometric transformations or rigid transformations (or rigid motions) because they do not alter the shape or size of objects.

Shapes that remain unchanged under isometry are called isometric figures, as they retain the same measurements.

Types of Isometries

The main types of isometric transformations are:

• Translation
  Every point is moved by a fixed amount in a specific direction.
  
• Rotation
  Every point rotates around a fixed point known as the center of rotation.
• Reflection
  Every point is reflected across a line, called the axis of reflection.
• Central Symmetry
  Every point is mapped to its diametrically opposite point relative to a central point.

Compositions of Isometries

When you apply two or more isometric transformations in a specific sequence to a geometric figure, the final result can be described as a composition of isometries.

For example, you can translate a triangle 10 cm along a horizontal line (translation) and then rotate it 180° (rotation).

It's important to note that the composition of two isometries is still an isometry.

Since each isometry individually preserves distances and shapes, a composition of isometries will also retain these properties.

In other words, a figure subjected to a composition of isometries maintains its original shape and size.

Note: However, the composition of isometries is not always commutative. This means the order in which you perform the geometric transformations can affect the final result. For example, a rotation followed by a translation yields a different result than a translation followed by a rotation.

Invariants in Isometry

In an isometry, certain characteristics of a shape may change, such as its position, while others remain unchanged.

The features that do not change are known as "invariants" in isometry.

The main invariant properties during an isometry are:

• Distance between points
  In isometries, the distance between points remains unchanged. This means that, for example, if the distance between two points A and B is 5 cm in the original figure, it will remain the same in the resulting figure after an isometric transformation.
• Length of segments
  The length of segments does not change between the original figure and its image. For instance, the segments AB, BC, and AB have the same length in both triangles.
• Angle measures
  The measure of the angles remains constant after an isometry. For example, the angles α, β, and γ of the triangle maintain the same measure after the isometric transformation.
• Area of the figure
  The total area of the figure does not change. For example, the area of the triangle remains the same after the isometric transformation.
• Shape and size
  In an isometry, the shape and size of the geometric figure do not change. These properties are invariant.
• Alignment of points
  If two or more points are aligned in the original figure, they will also be aligned in its image. For example, points A and B are aligned in both the original triangle and the resulting triangle after the transformation.
• Parallelism of lines
  Lines that are parallel in the original figure remain parallel in the image.
• Perpendicularity of lines
  Lines that are perpendicular in the original figure maintain this characteristic in their image as well. For example, segments AB and BC are perpendicular in both the first triangle (original) and the second triangle (isometric image).

Additional Observations

Here are some additional observations and side notes on isometries:

• Isometries are congruences
  Isometries are geometric transformations that produce congruent figures because they preserve segment lengths and angle measures. As a result, the transformed figures are point-for-point superimposable onto the original figures after some rigid motions, fulfilling the definition of congruence in Euclidean geometry.
• Isometries are a specific type of similarity
  Isometries are similarities with a similarity ratio of 1. They satisfy all the properties of similarities:
  • they preserve the parallelism between segments,
  • corresponding angles are congruent, and
  • corresponding segments are congruent.
  Therefore, in an isometry, the shape of the figure does not change, which is an invariant property (as in similarities), because the angles remain congruent. Additionally, the corresponding segments have the same length, meaning they are proportional with a ratio of 1.

And so on.