Isometry

An isometry is a transformation that preserves distance.

This means the size and shape stay exactly the same, and is called a rigid transformation.

It's in the name: "iso" means equal, and "metry" means measure, so isometry says "equal measure".

In fact the new shape is congruent to the original.

Domino tiles on a wooden table being slid, rotated, and flipped to show rigid motion

Think of it like moving a physical object around on a table
We can slide it, turn it, or flip it over, but we can't stretch or squash it.

The Four Isometries

In the plane, there are only four types of isometries:

Translation:

a "slide" by a vector

Rotation:

a "turn" by angle θ around a center point

Reflection:

a "flip" across a line

Glide Reflection:

a reflection and translation combined

And now ... with Functions!

We can use functions to describe each transformation, unlocking the massive power of algebra. Any point P gets mapped to a new point P':

We use a transformation rule (like T, R, or G) to mean "a rule that takes a point P at (x, y) and finds a new position P':

Translation:

T(P) = (x+a, y+b) where P = (x, y)

Rotation:

R_θ(P) = P'

Reflection:

Ref_L(P) = P'

Glide Reflection:

G(P) = T(Ref_L(P))

Examples

Triangle reflected across a vertical line L, showing identical size and shape

Reflection
across line L

Asymmetrical shape reflected across a diagonal line

Also a Reflection

Curved shape rotated around a single fixed pivot point

Rotation
(one point is invariant)

How to Identify an Isometry

We look for clues!

Orientation

Does the shape still face the same way?

Label the vertices in clockwise order A, B, C and so on:

Direct (Preserved): A, B, C, ... stay clockwise. (Translation, Rotation)
Opposite (Reversed): A, B, C, ... become counter-clockwise. (Reflection, Glide Reflection)

2. Invariant Points

Are there invariant points (that stay fixed).

Translation: No invariant points (unless the slide is zero)
Rotation: Exactly one invariant point (the center of rotation)
Reflection: An entire line of invariant points (the axis)
Glide Reflection: No invariant points

3. Parallel

Does a line segment stay parallel to its original version?

Translation: Every line stays parallel to its original
Rotation: No lines stay parallel (except for one or more 180° turns)
Reflection/Glide: Generally no, except for lines parallel or perpendicular to the axis

Example

Shape reflected across a horizontal axis and then translated right, showing glide reflection

The orientation is reversed and there are no invariant points (all points are in different locations), so we are looking at a Glide Reflection.

In Summary:

Transformation	Orientation	Invariant Points
Translation	Preserved	None
Rotation	Preserved	One point (center)
Reflection	Reversed	A whole line
Glide Reflection	Reversed	None

Justifying an Isometry

To justify that a transformation is an isometry, we must prove that the distance between any two points A and B stays the same.

So for any two points A and B we have the same distance before as after:

distance = √(x_B − x_A)² + (y_B − y_A)²