Understanding Isometries | Preserving Distances in Mathematics

Isometry

In mathematics, an isometry refers to a transformation between two metric spaces that preserves distances

In mathematics, an isometry refers to a transformation between two metric spaces that preserves distances. In simpler terms, it is a function or a mapping that maintains the distances between points in a space.

Formally, let (X, d_X) and (Y, d_Y) be metric spaces, where X and Y are sets and d_X and d_Y are distance functions that assign a non-negative real number to pairs of points in X and Y, respectively. Then, a function f: X -> Y is an isometry if it satisfies the following condition:

For any two points x1 and x2 in X, the distance between f(x1) and f(x2) in Y is equal to the distance between x1 and x2 in X, i.e., d_Y(f(x1), f(x2)) = d_X(x1, x2).

In simpler terms, an isometry preserves the distances between any two points in the given metric spaces. This means that if you measure the distance between two points in the original space, and then apply the isometry to those points and measure the distance between their images in the new space, the distances will be the same.

Isometries are often encountered in geometry, where they play a significant role in shapes and transformations. They preserve geometric properties such as lengths, angles, and shapes. Some common examples of isometries include translations, rotations, and reflections.

Overall, isometries are important in mathematics as they provide a way to analyze and understand the relationships between different metric spaces while preserving the underlying structure and distances.

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