Definition and examples
Definition: A transformation of the plane is an isometry if, for all points X and Y, the distance between the image points X' and Y' equals the distance between X and Y. In other words, an isometry is a transformation which preserves distance. ("Iso" means "same" and "metry" means "measurement," as in "geometry.") The shorthand for the isometry condition is X'Y' = XY.
Here's a way to say the same thing using function notation: Let P be the Euclidean plane with distance d. A function F: P → P is an isometry if, for all points X and Y of P, d(F(X),F(Y)) = d(X,Y).
The following types of transformations are isometries: translation, rotation, reflection, glide reflection.
The identity transformation is the function F defined by F(X) = X for all X. In other words, for all points X the transformed point X' equals X. A translation with translation vector 0 is the identity. A rotation with rotation angle 0 is the identity.
Every isometry is a bijection, i.e. it is injective ("one-to-one") and surjective ("onto").
Products
The product of two isometries is an isometry: For all transformations F and G, if F and G are isometries, then GF is an isometry. (Product means composition of functions: (GF)(X) = G(F(X)).)
The inverse of an isometry is an isometry: For all transformations F, if F is an isometry and G is its inverse, then G is an isometry. (G is the inverse of F if GF is the identity, i.e. G(F(X)) = X for all X.)
The product of isometries is associative: For all isometries F, G, H, (HG)F = H(GF).
The product of isometries is not commutative: There exist isometries F and G such that GF is not equal to FG. (For example, suppose F1 is a reflection with mirror m1 and F2 is a reflection with mirror m2, and suppose that m1 and m2 are not parallel. Let O be the intersection point of m1 and m2, and let a be the measure of the angle from m1 to m2. Then F2F1 is rotation with center O and angle measure 2a, but F1F2 is rotation with center O and angle measure -2a.)
Fixed points
Definition: A fixed point of a transformation F is a point X such that F(X) = X.
A translation T has no fixed points, unless T is the identity.
A rotation R has only one fixed point, unless R is the identity.
The set of fixed points of a reflection F is a line.
A glide reflection G has no fixed points, unless G is a reflection.
Orientation
Definition: An isometry F is orientation preserving if, for all noncollinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have the same sign. An isometry F is orientation reversing if, for all noncollinear points A, B, C, the proper angle measures of the angles ABC and A'B'C' have opposite signs.
(Note: "Proper angle measure" is the same thing as "directed degrees" in GSP.The proper angle measure of an angle is between -180 degrees and 180 degrees.)
In other words, an orientation preserving isometry takes counterclockwise angles to counterclockwise angles, and it takes clockwise angles to clockwise angles. An orientation reversing isometry takes counterclockwise angles to clockwise angles, and it takes clockwise angles to counterclockwise angles.
Translations and rotations are orientation preserving.
Reflections and glide reflections are orientation reversing.
Classification of isometries
The Triangle Theorem: If the two triangles ABC and A'B'C' are congruent, then there exists a unique isometry F such that F(A) = A', F(B) = B', and F(C) = C'. (We assume that A, B, C are noncollinear, that A', B', C' are noncollinear, and that the congruence takes A to A', B to B', and C to C'.)
The Classification Theorem: Every isometry is one of the following: the identity, a translation, a rotation, a reflection, or a glide reflection.