A rigid motion is a transformation which moves an object in a plane to a new position while keeping its shape and size constant. A transformation is generally defined by a rule. That is, given an object S (or set of points) in the coordinate plane, a transformation of S, denoted by T, is a rule that moves (takes) every point in S to another object (set of points) S′ in the plane. In this case we may write,

T(S) = S′;

and say that S′ is the image of S under T.

Also, if P∈S and P′∈S′ such that T(P) = P′, then P′ is said to be the image of P under
T. Now, we state the formal definition of a rigid motion.

Definition

Rigid Motion
A transformation T is said to be a rigid motion (rigid transformation) if it preserves distance. That is, for every two points A and B, if T(A)=A′ and T(B) = B′, then the distance between A′ and B′ is equal to the distance between A and B (A′ B′ = AB).

Consequently, a rigid motion preserves also angles. For example, an identity transformation is a rigid motion. A transformation is said to be an identity transformation, if the image of every point is itself.