Geometry
Transformations – Moving and Resizing Shapes
A geometric transformation is a rule that moves, flips, turns, or resizes a shape. The original shape is called the object and the result is the image. Transformations are used in computer graphics, design, robotics, and map projections.
The Dutch artist M.C. Escher became famous in the 20th century for artworks built almost entirely from translations, rotations, and reflections of a single repeating shape — most famously his tessellating lizards, fish, and birds that tile the page with no gaps, inspired partly by a visit to the Moorish tilework of the Alhambra palace in Spain. The same four transformations you are learning here are exactly what power modern computer animation and video games: every time a character walks (translation), a wheel spins (rotation), a sprite is mirrored to face the other way (reflection), or the camera zooms in (enlargement), the software is applying one of these transformation rules to a set of coordinates many times per second.
The Four Main Transformations
| Transformation | What It Does | Preserves Size? | Preserves Shape? |
|---|---|---|---|
| Translation | Slides the shape in a given direction | Yes | Yes |
| Rotation | Turns the shape around a fixed point | Yes | Yes |
| Reflection | Flips the shape over a mirror line | Yes | Yes |
| Enlargement | Scales the shape by a scale factor | No | Yes |
Translation
A translation moves every point the same distance in the same direction. It is described by a column vector (a over b), meaning move a units right (negative for left) and b units up (negative for down).
New position: (3 + 4, 2 + (−1)) = (7, 1).
Rotation
A rotation turns a shape about a fixed centre of rotation by a given angle (clockwise or anti-clockwise). You must specify: angle, direction, and centre.
Rule for 90° anti-clockwise: (x, y) → (−y, x). So (3, 1) → (−1, 3).
Rotation Rules About the Origin
| Rotation | Transformation Rule |
|---|---|
| 90° clockwise | (x, y) → (y, −x) |
| 90° anti-clockwise | (x, y) → (−y, x) |
| 180° (either direction) | (x, y) → (−x, −y) |
Reflection
A reflection flips a shape over a mirror line. Every point maps to its mirror image, equidistant from the line on the other side.
| Mirror Line | Rule |
|---|---|
| x-axis (y = 0) | (x, y) → (x, −y) |
| y-axis (x = 0) | (x, y) → (−x, y) |
| y = x | (x, y) → (y, x) |
| y = −x | (x, y) → (−y, −x) |
Enlargement
An enlargement scales a shape by a scale factor from a centre of enlargement. Scale factor k > 1 makes it bigger; 0 < k < 1 makes it smaller; negative k flips and scales it.
Multiply each coordinate by 2: (2, 2), (6, 2), (2, 6). The image is twice the size.
Key Takeaways
- Translation: describe with a column vector. Image is congruent to object.
- Rotation: specify angle, direction, and centre of rotation.
- Reflection: specify the mirror line. Image is congruent, but orientation reverses.
- Enlargement: specify scale factor and centre. Image is similar (not congruent unless k = 1 or −1).
Practice: Find the Image
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