Isometry

Isometry in Geometry

Congruence

Rotation and Transformation

Isometry is a concept of geometry. Isometry means that one shape can be transformed into another, but metrics such as the arrangement of the points in relation to each other stay the same. An isometry is a special way of changing things that makes sure the distance between every two points stays the same in both places. An isometry is a special kind of function that moves points from one location in space to some other location. Some of these ways include turning objects, reflecting them across an axis, translating them, or not moving them at all.

Most transformation that repeats itself after every two moves can either be a mirror image or a rotation by half. Every movement in a flat surface can be made by reflecting it at most three times. Every small group of transformations has at least one point that doesn't move. To be an isometry, a function needs to keep the distances between all points in both places the same. The distance from x to y is the same as the distance from f(x) to f(y).

If you draw something and then move it according to an isometry, the distances between the points on the first drawing will be the same as on the new drawing. Isometries are useful tools in many areas of mathematics, such as studying shapes, patterns, and how numbers behave. Metric spaces are helpful in geometry (and other branches of mathematics such as topology) because they help us study concepts such as symmetry, congruence, and transformations.