The isometric transformations They are changes in the position or orientation of a certain figure that do not alter its shape or size. 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.
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Isometric transformations occur when the magnitudes of the segments and the angles between the original figure and the transform 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 in direction. In this way, the initial and final figures will be similar and geometrically congruent..
Isometry refers to equality; that is, 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.
They are those isometries that allow all the points of the plane to be displaced 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.
- The translations will always be summarized in a single one, regardless of the number of translations that are 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. Namely:
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 '[(8 + (-4)), ((-2) + 7)] = 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:
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 (counterclockwise) 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) -, by an angle of 90or to 360or the coordinates of the points will be:
In the case in which 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 the point P (-5,2) is rotated 90or, around the origin and in a positive direction its new coordinates will be (-2,5).
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 his face, which is considered a factor of beauty. Reflection or symmetry can be of two types:
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 180or from center O, a figure equal to the original is obtained.
- The strokes of the initial figure are parallel with the strokes of the formed figure.
- The sense of the figure does not change, it will always be clockwise.
This transformation occurs with respect to the axis of symmetry, where each point of the initial figure is associated with another point in the image and these are at the same distance from the axis of symmetry. Symmetry is axial when:
- The segment that joins a point with its image is perpendicular to its axis of symmetry.
- Figures change direction with respect to rotation or clockwise.
- When dividing the figure with a central line (axis of symmetry), one of the resulting halves completely coincides with another of the halves.
A composition of isometric transformations refers to the successive application of isometric transformations on the same figure.
The composition of two translations results in another translation. When performed on the plane, on the horizontal axis (x) only the coordinates of that axis change, while the coordinates of the vertical axis (y) remain the same, and vice versa.
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.
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:
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