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Geometry: Symmetry and Transformations





 



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Figures in geometry such as polygons and circles can be changed or rearranged by a process called transformation. It can be thought of as creating a new geometric object from an old one or moving it to a new position. The transformed object is called an image of the original. The original is also called the preimage. There are two main types of transformation and each type can have different transformations. The two main types are rigid and non-rigid. In a rigid transformation the image is congruent to the original.

In rigid transformation, the image and preimage are congruent.

In a non-rigid transformation the image can be similar to the original.

In non-rigid transformation, the image and preimage are similar.

Mapping

A transformation can also be thought of as a mapping process. A mapping in geometry is when a point in the original object or preimage is relocated to exactly one other point in the image. Since one point is only mapped to another single point, a transformation is also called a one-to-one mapping.

Each point in the preimage corresponds to one point in the image in a one-to-one mapping.

Another name for rigid transformation is isometry and there are three types: reflection, translation and rotation. A type of non-rigid transformation is also referred to as dilation.

Symmetry

Symmetry can be thought of as half of something matching the other half. This is one example of line symmetry.

A geometric object with line symmetry has a part that is a reflection of another part across a dividing line.

It is similar to holding half an object up to a mirror and seeing the other half. If an object does not have this property it is asymmetrical.

An asymmetrical figure in geometry appears to have incongruent parts.

Another type of symmetry is called point symmetry. Point symmetry exists when there is a point in the center that is an equal distance away from a point in the figure and from a point on the opposite side of the geometric figure.

A geometric object has point symmetry if part of it appears reflected from the opposite side of a point.

The following figures do not have point symmetry although they might have line symmetry.

A geometric figure may have line symmetry but not point symmetry.

Line symmetry and point symmetry can be formally defined as follows:

Line Symmetry
If there is a line of symmetry through a geometric figure then for every point in the figure there is a line segment perpendicular to the line with the endpoints in the figure at equal distances from the line.


In other words, the perpendicular line segment is bisected by the line passing through the figure.

The line of symmetry through a figure in geometry contains the midpoints of perpendicular line segments whose endpoints are contained in the opposite sides of the figure.

The definition for point symmetry can be described as follows.

Point Symmetry
A geometric figure has point symmetry about a point of symmetry if the point is a midpoint for any point in the figure and some other point also in the figure.


In point symmetry, the point of symmetry is the midpoint of all line segments passing through it and connecting opposite sides of the object.

Isometry

The type of transformation that is rigid transformation is when the image matches the original and is called isometry. There are three kinds of transformation that can be done with isometry. One is reflection which is related to symmetry. The next would be translation where the object changes position but not orientation. The third would be rotation where the object is rotated about a point.

Isometry is also called congruence transformation since the resulting image is congruent to the original. Non-rigid transformation is also called similarity transformation since the image is similar to the original object. Similarity transformation is also called dilation.

Isometry involves three types of rigid transformation: reflection, rotation and translation. A similary transformation can include dilation or scaling in addition to isometry.

Some of the important characteristics of isometry are angle and distance. The measures of the angles in the image of a geometric figure are preserved. The distance is preserved also. All the pairs of points in the new figure have the same distance as the old one.

Reflection

Reflection is related to symmetry because a set of points is mapped or projected across a line to create a new set of points with congruence to the original set. In the case of reflection, a complete geometric figure is recreated on the other side of the line of reflection so that each corresponding pair of points, one from the image and one from the original, are the same distance from the line.

In geometry, reflection is similar to symmetry in that the points of an object are projected at equal distances across a line of reflection.

In the case of symmetry, part of a geometric figure is reflected about a line of symmetry to create a new object. Where reflection creates two figures, symmetry creates one.

Reflection creates an additional figure while symmetry is related to the figure as a whole.

Reflection can be defined as follows:

Reflection
The projection of a point or set of points to create another point or set of points across a point or line that contains the midpoint of a line segment connecting the image point to the point in the pre-image.


Corresponding points in the image and preimage can be connected by a line segment with its midpoint on the line or point of symmetry.

Point reflection involves projecting a figure through a point to create a new figure with points the same distance away from the point of reflection as the original figure.

In line reflection, a geometric figure is reflected across a line with points in the new figure at the same distance from the line of reflection as the original.

Translations

A simple way to think of a translation is to think of an object being moved in any direction without being turned or rotated.

A translated object in geometry appears unchanged except for its new location.

Another way of looking at it is with the idea that an object is reflected across a line and then reflected again across a different line.

An object reflected twice each time across two parallel lines is equivalent to translation.

Translation can be defined as follows:

Translation
A composite reflection across two parallel lines.

Rotation

The rotation of an object in geometry usually involves changing its angle without changing its overall position. The object is turned, in other words, so that it points in a different direction.

Rotation involves turning an object so it has a new orientation without changing its relative position.

Another way of looking at rotation is with the reflection of the figure across two intersecting lines and moved back.

The effect of rotation is equivalent to reflecting a geometric figure across two intesecting lines.
Rotation
A composite reflection of a geometric object across two intersecting lines.


There is a postulate associated with rotation:

Rotation Angle
The angle of rotation is twice the measure of the angle between the intersecting lines.


Dilation

As mentioned earlier, dilation is a non-rigid transformation related to similarity. It is basically taking a figure in geometry and either enlarging it or shrinking it while maintaining proportion. In other words, its size changes but not its shape. Dilation is also known as scaling.

Dilation alters the size of an object but not its shape.
 




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