Isometry

Academ Reflections with parallel axis on wallpaper

Isometry refers to a mathematical concept in the field of geometry and metric spaces that denotes a transformation preserving distances between points. An isometry is a function between two metric spaces that conserves the distance, meaning that the length of the path between any two points in the original space is the same as the length of the path between their images in the target space. This concept is fundamental in various areas of mathematics, including geometry, algebra, and topology, as it helps in understanding the inherent properties of spaces that are invariant under certain transformations.

Definition

Formally, an isometry is a mapping \(f: X \rightarrow Y\) between two metric spaces \((X, d_X)\) and \((Y, d_Y)\), such that for any two points \(a, b \in X\), the following condition holds: \[d_Y(f(a), f(b)) = d_X(a, b)\] This definition ensures that the function \(f\) preserves distances exactly, making it a distance-preserving transformation.

Types of Isometries

Isometries can be classified into several types depending on the nature of the transformation and the properties of the spaces involved. Some common types include:

- Translations: Moving every point in a space by the same distance in a given direction. - Rotations: Turning a space around a fixed point without changing the distance between points. - Reflections: Flipping a space over a specified axis or plane. - Glide reflections: A combination of a reflection and a translation along the axis of reflection.

Properties

Isometries have several important properties: - They preserve angles between vectors, making them congruence transformations. - They are always injective functions, meaning they map distinct points in the original space to distinct points in the target space. - In Euclidean spaces, isometries are also bijective functions, implying they have inverses that are also isometries.

Applications

Isometries are used in various fields of study: - In geometry, they are crucial for understanding geometric figures and their properties under transformations. - In topology, isometries help in studying the properties of spaces that are preserved under continuous deformations. - In physics, the concept of isometry is used in the theory of relativity and in the study of space-time symmetries.

See Also

- Metric space - Euclidean geometry - Transformation geometry - Rigid motion