Isometry in geometry is a type of transformation that preserves the size and shape of a geometric figure. It is essentially a rigid transformation that moves an object or image on a plane without altering its fundamental characteristics.
Understanding Isometry as a Rigid Transformation
An isometry is best understood as a movement that changes a figure's position or orientation, but never its shape or size. When a figure undergoes an isometry, it is as if the figure is picked up and moved without being stretched, shrunk, or bent. This means all lengths and angle measures within the figure remain precisely the same before and after the transformation.
Types of Isometric Transformations
Several common geometric movements are classified as isometries:
- Translation (Sliding): This transformation moves every point of a figure by the same distance in a specified direction. The figure slides to a new location without rotating or flipping.
- Reflection (Flipping): A reflection creates a mirror image of a figure by flipping it across a line, known as the line of reflection. While the shape and size are preserved, the orientation of the figure is reversed.
- Rotation (Rotating): In a rotation, a figure is turned around a fixed point, called the center of rotation, by a certain angle. The figure pivots while maintaining its original dimensions.
- Glide Reflection (Gliding): This is a combination of two isometries: a translation and a reflection performed sequentially. The translation is parallel to the line of reflection, resulting in a unique combined movement.
Key Characteristics of Isometries
The defining feature of all isometries is their ability to preserve core geometric properties:
- Preservation of Shape: The overall form, angles, and curvature of the object remain identical.
- Preservation of Size: The lengths of all sides and the area of the object do not change.
- Preservation of Distance: The distance between any two points on the original figure is the same as the distance between their corresponding points on the transformed figure.
- Change in Position or Orientation: Although shape and size are constant, an isometry always results in a change in the figure's location or its alignment in space.
