A geometric transformation is a rule that sends every point of the plane somewhere else. The familiar ones fall into a short list: slide the whole picture without turning it, spin it about a fixed point, flip it across a line, blow it up or shrink it, and push it sideways so that squares become slanted parallelograms. The first three keep every distance exactly as it was and are called rigid motions — a figure moved by one of them is congruent to the original, which is precisely what school geometry means when it says two triangles are the same. Enlarging keeps every angle and every ratio but not the actual sizes, giving similar rather than congruent figures.
Two facts turn this from a list of tricks into a subject. The first is that carrying out one transformation and then another always gives a transformation of the same kind, so they combine like numbers under multiplication — two reflections in intersecting lines make a rotation, two in parallel lines make a slide, and a rotation followed by an enlargement is again a rotation-and-enlargement. The second is that the order matters: turn a figure and then stretch it and you will not land where you would by stretching first and then turning. Felix Klein built his 1872 programme on exactly this idea, defining a geometry as the study of whatever survives a chosen family of transformations — which is how Euclidean, affine and projective geometry became one subject seen from different heights.
The practical payoff is that every one of these moves can be written as a small table of numbers and applied to millions of points at once. That is what a graphics processor does for a living: each frame of an animation, each pan of a map, each rotation of a model in a CAD program is one such table applied to a mesh of coordinates. Adding an extra row and column so that shifts can be folded into the same table — the trick called homogeneous coordinates — is what lets a whole chain of moves collapse into a single multiplication, and it is the reason robotics, computer vision and 3D graphics all speak the same algebraic language.