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Transformed images of pictures and drawings provide compelling and revealing ways to view and understand geometric and mathematical transformations. Sketchpad provides a number of ways of transforming pictures and creating objects based on pictures.
If an original picture is independent or attached to a single point, you can transform the original in various ways.
Use Sketchpad’s built-in transformations to create a transformed image of a picture.
Select the picture and choose Translate, Rotate, Reflect, or Dilate from the Transform menu. Depending on the transformation you want, you may need to mark various objects or values to serve as the center, distance, angle, ratio, or mirror.
These four transformations, and any transformation that combines them, are called similarity transformations because the transformed image is similar to the original: angles and ratios of distances are preserved.
Use Sketchpad’s custom transformations to create an arbitrary transformed image of a picture.
Use two points, one of which depends on the other, to define a custom transformation.
Select the picture you want to transform.
Choose the custom transformation you want from the bottom of the Transform menu.
If the two points that define the custom transformation are related using only Sketchpad’s built-in transformations, the result is a normal transformed picture, just as if you had applied the various transformations directly to the original picture.
Such a normal transformed picture can be displayed more quickly and with less distortion than a sampled picture.
If the two points that define the custom transformation are related in some other way, the result is a sampled transformed picture.
A sampled transformed picture is a custom transformed image of a picture in which the custom transformation doesn’t rely strictly on Sketchpad’s built-in transformations of translation, rotation, reflection, and dilation.
Transformed images of pictures constructed using the built-in transformations retain the essential shapes of their pre-images, though their size and orientation may change.
A rectangular area in the pre-image always corresponds to a similar rectangular area in the image.
There is no guarantee of similarity for a transformed image of a picture constructed in other ways. The resulting image must be displayed using a sampling process: The transformed picture is displayed by sampling rectangles within the pre-image and then transforming each sample to create the transformed image. The smoothness and level of detail of the transformed path depend on the number of samples used. To change the number of samples and the allowable distortion, select the transformed picture and choose Edit | Properties | Plot.