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16.02.01 - Non-Rigid Transformations (Geometry)

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If a transformation changes the shape and/or size of a figure, then this transformation is non-rigid. There are many possible non-rigid transformations. In quarter 1, we considered vertical and horizontal stretches, skews and dilations. There are others; the non-rigid transformations we considered preserved certain properties we were interested in.

A vertical stretch is one in which a figure is stretched upward or downward. A vertical stretch preserves horizontal distances and scales vertical distances. The image shows how triangle LMN has been stretched vertically to form triangle L'MN. In quarter 1, we only performed stretches on figures, as it didn't make much sense to perform stretches on points or lines. However, as we have moved all our objects to a grid, even a point is associated with a distance. In particular, each point has a vertical distance to the x-axis, and a horizontal distance to the y-axis. Therefore, it is no longer unreasonable to stretch a point or a line horizontally.

A horizontal stretch is one in which a figure is stretched to the left or the right. A horizontal stretch preserves vertical distances, and scales horizontal distances. In the image, triangle TUV has undergone a horizontal stretch to the right forming triangle TU'V. As with vertical stretches, in quarter 1 we only performed horizontal stretches on figures, but in this quarter we will be able to extend this to other objects.

We already mentioned skews when discussing the areas of parallelograms. A skew is important in geometry as it preserves area. In the image, triangle ABC is skewed to become A'BC. We will not actually be performing any skews on the Cartesian grid in this class.

The remaining non-rigid transformation we previously considered is a dilation. In a dilation, the shape of a figure is constant, but the size changes. The image shows that figure FGH has been dilated from point G to form triangle F'GH'. Dilating a figure on the Cartesian plan is reasonably straight-forward. Slopes will remain the same, and lengths will be scaled. We extensively considered dilated figures in each previous quarter. We introduced dilations in quarter 1, then used dilations in quarters 2 and 3 to prove similar figures.

In the previous lesson, we mentioned that analytical geometry lends itself to certain transformations. Horizontal and vertical stretches are very important transformations on the Cartesian grid. To perform a dilation on the Cartesian grid, perform a horizontal stretch followed by a vertical stretch, where the scaling factor is the same for both.