ISOMETRIC TRANSFORMATIONS: COMPOSITION, TYPES AND EXAMPLES - MATHS - 2024

The isometric transformations are changes of position or orientation of a given figure which do not alter the form or the size of this. These transformations are classified into three types: translation, rotation and reflection (isometry). In general, geometric transformations allow you to create a new figure from a given one.

A transformation into a geometric figure means that, in some way, it has undergone some change; that is, it was altered. According to the sense of the original and the similar in the plane, geometric transformations can be classified into three types: isometric, isomorphic and anamorphic.

characteristics

Isometric transformations occur when the magnitudes of the segments and the angles between the original figure and the transformed figure are preserved.

In this type of transformation, neither the shape nor the size of the figure is altered (they are congruent), it is only a change in its position, either in orientation or direction. In this way, the initial and final figures will be similar and geometrically congruent.

Isometry refers to equality; in other words, geometric figures will be isometric if they have the same shape and size.

In isometric transformations, the only thing that can be observed is a change of position in the plane, a rigid movement occurs thanks to which the figure goes from an initial position to a final one. This figure is called homologous (similar) of the original.

There are three types of movements that classify an isometric transformation: translation, rotation, and reflection or symmetry.

Types

By translation

They are those isometries that allow all points of the plane to be moved in a straight line in a given direction and distance.

When a figure is transformed by translation, it does not change its orientation in relation to the initial position, nor does it lose its internal measures, the measures of its angles and sides. This type of displacement is defined by three parameters:

- One direction, which can be horizontal, vertical or oblique.

- One direction, which can be to the left, right, up or down.

- Distance or magnitude, which is the length from the initial position to the end of any point that moves.

For an isometric transformation by translation to be fulfilled, the following conditions must be met:

- The figure must always keep all its dimensions, both linear and angular.

- The figure does not change its position with respect to the horizontal axis; that is, its angle never varies.

- Translations will always be summarized in one, regardless of the number of translations made.

In a plane where the center is a point O, with coordinates (0,0), the translation is defined by a vector T (a, b), which indicates the displacement of the initial point. That is to say:

P (x, y) + T (a, b) = P '(x + a, y + b)

For example, if a translation T (-4, 7) is applied to the coordinate point P (8, -2), we obtain:

P (8, -2) + T (-4, 7) = P '= P' (4, 5)

In the following image (left) it can be seen how point C moved to coincide with D. It did so in a vertical direction, the direction was upwards and the distance or magnitude CD was 8 meters. In the right image the translation of a triangle is observed:

By rotation

They are those isometries that allow the figure to rotate all the points of a plane. Each point rotates following an arc that has a constant angle and a fixed point (center of rotation) determined.

That is, all rotation will be defined by its center of rotation and angle of rotation. When a figure is transformed by rotation, it keeps the measure of its angles and sides.

The rotation occurs in a certain direction, it is positive when the rotation is counterclockwise (the opposite direction to how the hands of the clock turn) and negative when its rotation is clockwise.

If a point (x, y) is rotated with respect to the origin - that is, its center of rotation is (0,0) -, at an angle of 90 ^or 360 ^or the coordinates of the points will be:

In the case where the rotation does not have a center at the origin, the origin of the coordinate system must be transferred to the new given origin, in order to be able to rotate the figure with the origin as the center.

For example, if P (-5,2) point is applied a rotation of 90 ^or, around the origin and positively its new coordinates are (-2.5).

By reflection or symmetry

They are those transformations that invert the points and figures of the plane. This inversion can be with respect to a point or it can also be with respect to a line.

In other words, in this type of transformation each point of the original figure is associated with another point (image) of the homologous figure, in such a way that the point and its image are at the same distance from a line called the axis of symmetry..

Thus, the left part of the figure will be a reflection of the right part, without changing its shape or dimensions. Symmetry transforms a figure into another equal but in the opposite direction, as can be seen in the following image:

Symmetry is present in many aspects, such as in some plants (sunflowers), animals (peacock) and natural phenomena (snowflakes). The human being reflects it on her face, which is considered a factor of beauty. Reflection or symmetry can be of two types:

Central symmetry

It is that transformation that occurs with respect to a point, in which the figure can change its orientation. Each point of the original figure and its image are at the same distance from a point O, called the center of symmetry. Symmetry is central when:

- Both the point and its image and center belong to the same line.

- With a rotation of 180 ^o of center O, a figure equal to the original is obtained.

- The lines of the initial figure are parallel with the lines of the formed figure.

- The sense of the figure does not change, it will always be clockwise.

Composition of a rotation

The composition of two turns with the same center results in another turn, which has the same center and whose amplitude will be the sum of the amplitudes of the two turns.

If the center of the turns has a different center, the cut of the bisector of two segments of similar points will be the center of turn.

Composition of a symmetry

In this case, the composition will depend on how it is applied:

- If the same symmetry is applied twice, the result will be an identity.

- If two symmetries are applied with respect to two parallel axes, the result will be a translation, and its displacement is twice the distance of those axes:

- If two symmetries are applied with respect to two axes that intersect at point O (center), a rotation with center at O ​​will be obtained and its angle will be twice the angle formed by the axes:

References

1. V Bourgeois, JF (1988). Materials for construction of geometry. Madrid: Synthesis.
2. Cesar Calavera, IJ (2013). Technical Drawing II. Paraninfo SA: Editions of the Tower.
3. Coxeter, H. (1971). Fundamentals of Geometry. Mexico: Limusa-Wiley.
4. Coxford, A. (1971). Geometry A Transformation Approach. USA: Laidlaw Brothers.
5. Liliana Siñeriz, RS (2005). Induction and formalization in the teaching of rigid transformations in the CABRI environment.
6. , PJ (1996). The group of isometries of the plane. Madrid: Synthesis.
7. Suárez, AC (2010). Transformations in the plane. Gurabo, Puerto Rico: AMCT.