A rigid transformation, also known as a Euclidean transformation or Euclidean isometry, is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points. In simpler terms, a rigid transformation is a transformation that does not change the size or shape of an object or pre-image when returning the final image. The three most common types of rigid transformations are:
- Reflection: A transformation that flips a shape over a line, called the line of reflection.
- Rotation: A transformation that turns a figure about a fixed point, called the center of rotation.
- Translation: A transformation that simply moves a shape without changing its orientation or size.
Rigid transformations have the following properties:
- They preserve side lengths, angle measures, perimeter, and area.
- They may not keep the same coordinates or relationships to lines outside the figure.
In some cases, reflections are excluded from the definition of a rigid transformation by requiring that the transformation also preserve the handedness of objects in the Euclidean space. A transformation that preserves handedness is known as a proper rigid transformation or rototranslation. Any proper rigid transformation can be decomposed into a rotation followed by a translation, while any improper rigid transformation can be decomposed into an improper rotation followed by a translation or into a sequence of reflections.
