Understanding Isometries | Preserving Distances in Mathematics

isometry

In mathematics, an isometry refers to a transformation that preserves distances between points

In mathematics, an isometry refers to a transformation that preserves distances between points. Put simply, it is a function that maintains the same lengths of segment distances before and after the transformation.

More formally, let’s consider two metric spaces, (X, d_X) and (Y, d_Y), where X and Y are sets and d_X and d_Y are distance functions defined on X and Y respectively. An isometry is a function f: X -> Y that satisfies the property:

d_Y(f(x), f(y)) = d_X(x, y)

for all points x and y in X. In other words, the distance between the images of two points under an isometry is equal to the distance between the original points. This property holds for all points in X.

Isometries can be thought of as “distance-preserving transformations” since they preserve the distances between points. Examples of isometries include translations, rotations, and reflections. These transformations intuitively don’t change the lengths of any line segments or the distances between points.

Isometries are significant in various mathematical fields, such as geometry, as they preserve important geometric properties. For instance, if two geometric figures are congruent (i.e., they have equal lengths, angles, and shapes), applying an isometry to one figure will result in another congruent figure.

It’s important to note that isometries can exist in both two- and three-dimensional spaces, as well as higher-dimensional spaces. They play a fundamental role in understanding geometric transformations and studying the symmetry of objects.

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