What Is an Isometry?
What Is an Isometry? In geometry, some transformations distort shapes by stretching, squashing, or bending them. Others preserve the shape perfectly. These distortion-free transformations are called isometries . An isometry keeps every distance between points exactly the same. The object may move, rotate, or flip, but its size and structure remain unchanged. 1. The Core Idea A transformation f is an isometry if: distance( f(x), f(y) ) = distance( x, y ) for every pair of points x and y. This is the mathematical way of saying: nothing is stretched, compressed, or distorted. 2. A Simple Real-Life Analogy Place a phone on a desk. Slide it. Rotate it. Flip it. It remains the same phone—same size, same shape, same geometry. Each of those motions is an isometry . 3. What Isometries Never Do An isometry does not : stretch or shrink a shape, shear it diagonally, bend or curve it, change angles or proportions. It behaves exactly like movi...