2D transformations

Translation, rotation, scaling, reflection and shear of a figure; the centre of a rotation; composing transformations and the order that matters.

The outline, topic by topic

Each topic is weighted the way the exam weighs it.

Affine transformation: what is it?
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A transformation of the plane that keeps lines straight and parallels parallel: translation, rotation, scaling, reflection, shear and their compositions.
An affine transformation maps a segment to a segment, so a figure can be transformed by computing only the images of its vertices.
Which of these transformations keep all distances (they are isometries)?
- Translation, rotation and reflection
- Translation, rotation and scaling by a ratio of 2
- Rotation, reflection and shear
- Scaling by a ratio of 2 and shear
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Translation, rotation and reflection
A scaling of ratio 2 doubles distances; a shear distorts them. Translation, rotation and reflection move the figure without distorting it.
To turn a figure around its own centre C, you … C to the origin, … around the origin, then translate back the other way.
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To turn a figure around its own centre C, you translate C to the origin, rotate around the origin, then translate back the other way.
The rotation formula turns around the origin. The two translations first bring the wanted centre to the origin, then put it back.
Translation: what is it?
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A transformation that moves all points by the same vector (tx, ty).
(x, y) becomes (x + tx, y + ty): the figure slides without turning or changing size.