Overview: What Are Transformations?
Transformations in geometry occur when a figure (the preimage) is moved or changed into another figure (the new image, or image), such that all points in the preimage map onto the image, and all points in the image can be mapped back onto the preimage. The figures correspond to each other, but are at different places in the coordinate plane. In some types of transformations, the shape of the preimage and the image are the same, but in others the preimage may be split into parts to form the image, or the preimage might be bent out of shape entirely to create the image.
Translations
Translations are one type of transformation that occur when a figure slides to a different place in the coordinate plane. The preimage and image are the same size and shape and all the points correspond to each other. The relationship between the preimage and image can be measured by direction and magnitude. In order to measure magnitude, measure the distance between a point in the preimage with its corresponding point in the image. In order to measure direction, measure the ray between a point in the preimage with its corresponding point in the image.
Rotations
Rotations are the type of transformation that occur when the preimage and the image turn around a central point. The preimage and image are the same size and shape also, and all the points correspond to each other. However, the preimage and image are at different orientations around a central point. The magnitude of the rotation is the angle between the central point, a point on the preimage, and its corresponding point on the image. (Cosine and sine functions can be expressed in terms of rotations.)
Complex Transformations
With translations and rotations the preimage and its resulting image have the same size and shape, but there are more complex transformations that do not necessarily have the same size or exactly the same shape. Some of these transformations occur when the figure in the preimage is split into parts in the image, or when the preimage is distorted so that the image does not have the same shape. For example, the representation of an area on a map is not the same shape as its corresponding area on a globe, although the locations are the map and the globe are the same.
The Problems of Perspective
Geometric transformations are a mathematical expression of the way artists distort figures in perspective in order to represent three-dimensional objects on a two-dimensional flat canvas. Lines and angles are distorted and repeated in order to fool the eye into seeing distance where the only distances are an optical illusion. The difference between a two dimensional square and a box drawn to look like it is in three dimensions involve translations , rotations, and complex transformations.
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