by Jongsuk Keum

Students learn the definition of an isometry, products, and enjoy the isometry game using GSP.

Definition and examples

Definition: A transformation of the plane is an

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.

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.

Products (Composition)

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 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.

The Classification Theorem: Every isometry is one of the following: the identity, a translation, a rotation, a

reflection, or a glide reflection.

(1) Player A, Jin, sets up the problem:

First Jin draws a figure using Geometer's Sketchpad, and she moves it using one or more of the transformations translation, rotation, reflection, or glide reflection.

Then Jin hides the defining data for the transformations, leaving only the original figure and its transformed image.

( 2) Player B, Sion, then has to solve the problem:

Sion tries to find transformations which move the original figure to its image.

*
Beginner's level:* Sion moves the figure in several steps, using
trial and error. (He finds defining data for the transformations approximately.)

*
Intermediate level:* Sion moves the figure in several steps,
but finds an exact solution. (He constructs defining data for the transformations
exactly.)

*
Advanced level:* Sion moves the figure in one step, and finds
an exact solution. (Note that a glide reflection counts as one step, even
though it must be defined in two steps using Geometer's Sketchpad.)

*
Note: The theory of isometries implies that Sion can solve the problem
in one step, even if Jin used lots of steps to set up the problem!*

Some Examples

1.

2.