Geometric Transformations
Pick a transformation type and adjust the parameters. See the original shape (blue) and its image (orange) with the mapping rule displayed live.
Transform
Properties
| Transform | — |
| Rule | — |
| Parameters | — |
| Preserves | — |
| Isometry | — |
The Math Behind It
Translation & Rotation
- Translation: (x, y) → (x + tx, y + ty)
- Every point shifts by the same vector (tx, ty)
- Preserves shape, size, and orientation
- Rotation by θ: (x, y) → (x cosθ - y sinθ, x sinθ + y cosθ)
- Center of rotation is the origin
- Preserves shape, size, and distances
Reflection & Dilation
- Reflect across x-axis: (x, y) → (x, -y)
- Reflect across y-axis: (x, y) → (-x, y)
- Reflect across y = x: (x, y) → (y, x)
- Reflections preserve size but reverse orientation
- Dilation by factor k: (x, y) → (kx, ky)
- Preserves shape and angles but scales distances by |k|
- Isometries (distance-preserving): translation, rotation, reflection. Not dilation.
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