Scaling, like translation is is a simple transformation which just scales the coordinates of an object. It is specified either by working directly with the local coordinates, or by expressing the coordinates in terms of Frames
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Development of the Transformation via Scaled Frames
Given a frame , a scaled frame would be one that is given by - that is, we just expand (or contract) the lengths of the vectors defining the frame. It is fairly easy to see that we can write the frame in terms of the frame by
Applying the Transformation Directly to the Local Coordinates
Given a frame and a point that has local coordinates in , if we apply the transformation to the local coordinates of the point, we obtain
Scaling about Points other than the Origin
It is difficult to see the origin of the scaling operation when working only with coordinates - so for example, consider the eight vertices of a cube centered at the origin in the Cartesian frame.
We note that this operation scales about the origin of the coordinate system. If the center of the object is not at the origin, this operation will move the object away from the origin of the frame. If we consider a cube with the following coordinates at its corners
If the desired scaling point is not at the origin of the frame, we must utilize a combination of transformations to get an object to scale correctly. If the scaling point is at in the frame, we can utilize the translation to first move the point to the origin of the frame, then scale the object, and finally use the translation to move the point back to the origin of the scaling. These transformations are all represented by matrices:
The scaling transformation can be represented by a simple matrix whose only entries are on the diagonal. This transformation, when applied to an object multiplies each of the local coordinates of the object by a factor - effectively scaling the object about the origin. Scaling about other points can be done by combining the scaling transformation with two translation transformations.
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