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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 a non-rigid transformation the image can be similar to the original.
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.
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 can be thought of as half of something matching the other half. This is one example of line symmetry.
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.
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.
The following figures do not have point symmetry although they might have line symmetry.
Line symmetry and point symmetry can be formally defined as follows:
In other words, the perpendicular line segment is bisected by the line passing through the figure.
The definition for point symmetry can be described as follows.
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.
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 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 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 can be defined as follows:
A simple way to think of a translation is to think of an object being moved in any direction without being turned or rotated.
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.
Translation can be defined as follows:
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.
Another way of looking at rotation is with the reflection of the figure across two intersecting lines and moved back.
There is a postulate associated with rotation:
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.
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