The inverse of an isometry is also an isometry.
An isometry is a geometric transformation that preserves distances between points. In simpler terms, it's a movement of a geometric figure where its size and shape remain unchanged. Because isometries preserve these fundamental properties, they are inherently invertible transformations.
Understanding Isometries
Isometries are fundamental transformations in geometry, often referred to as rigid motions. They ensure that the distance between any two points in a figure remains the same before and after the transformation. This preservation of distance is what makes them unique and ensures their invertibility.
Common types of isometries include:
- Translations: Shifting a figure from one position to another without changing its orientation.
- Rotations: Turning a figure around a fixed point (the center of rotation) by a certain angle.
- Reflections: Flipping a figure across a line (the line of reflection), creating a mirror image.
- Glide Reflections: A combination of a translation and a reflection along a line parallel to the direction of translation.
The Inverse of an Isometry
The unique property of an isometry is that its inverse is also an isometry. This means that for any distance-preserving transformation, there is another distance-preserving transformation that "undoes" the original one, returning the figure to its initial position and orientation without altering its shape or size. For instance, in a 2D plane, if an isometry performs a specific action, its inverse is uniquely determined to reverse that action, ensuring the transformation remains an isometry.
Let's look at the inverse for each type of isometry:
| Isometry Type | Inverse Isometry | Explanation |
|---|---|---|
| Translation | Translation in the opposite direction | If you translate an object 5 units to the right, its inverse is a translation of 5 units to the left. The distance moved is preserved, just the direction is reversed. |
| Rotation | Rotation by the same angle in the opposite direction | If you rotate an object 90 degrees clockwise, its inverse is a rotation of 90 degrees counter-clockwise around the same center point. The rotational effect is undone, and distances remain constant. |
| Reflection | The same reflection | Reflecting an object across a line and then reflecting it again across the same line returns the object to its original position. A reflection is its own inverse because applying it twice results in the identity transformation (no change). |
| Glide Reflection | Inverse translation followed by the same reflection | A glide reflection is a translation followed by a reflection. To reverse it, you first apply the inverse of the translation (a translation in the opposite direction) and then perform the same reflection. The combination ensures the figure returns to its original state while maintaining its distance-preserving properties. |
In essence, because an isometry fundamentally preserves distances, its reversal must also preserve distances, making the inverse operation itself an isometry.
