What Are Rigid Motions?

Simply put, rigid motions are transformations that preserve the side lengths and angles in a shape. You can see any of these transformations by moving an object you may have around you right now. Grab your notebook or phone and slide it across the desk you're working on. Now turn it so it's facing sideways. Now flip it over on one of its edges so you're looking at the back side of it. All of these motions are rigid motions.

Let's get into some more formal definitions with some wonderfully drawn MS Paint pictures.

A translation is a transformation that pushes or slides an object. On the picture above, the triangle has just moved to the right a little bit, while keeping its shape. The slide could be in any direction on the plane as long as it keeps its shape, and stays facing the same way.

A rotation is exactly what it sounds like. When the shape turns, you have a rotation. In the picture above, the triangle is turning $90^{\circ}$ clockwise. You can rotate a shape clockwise or counter clockwise, and for any number of degrees. Once again, the shape is preserved, but now it has a different corner on the bottom, and is facing a different direction. Think of turning a doorknob, when you turn it to one direction, the doorknob still remains the same shape.

A reflection is when a shape flips. In this case, the triangle has flipped to the right over an imaginary line in the center. You can flip a shape in any direction and over any part of itself, but this can be a little harder to visualize on your own, so here is a picture showing some other types of flips.

As you can see, horizontal reflections result in the same shape, and vertical reflections result in the same shape. What has changed between them is what point is being flipped on. For the upwards reflection, the shape is being flipped on the top corner of the triangle, while the downwards reflection is being flipped on the bottom side. The same goes for the horizontal reflections. The right reflection is being flipped on the right most point, and the left reflection is being flipped on the left side.

Let's think about these rigid transformations in a way that is a little big more familiar by using a screen shot from a level in Super Mario Maker.

While this screenshot doesn't have many objects in it, they can demonstrate every rigid transformation.

The arrows here show the translations within the stage. The saws can move up and down on their tracks, while your character can move in any direction that you want them to. While they move, they retain their shape, otherwise the game wouldn't work as intended. The only slight difference here is that your character may have a slightly different sprite while jumping, but that can be ignored for the sake of visualizing the translations.

The rotations are shown in this level by the saws. While the saws move up and down on their tracks, they rotate along their center, showing the player that they will in fact cut you up if you land on them.

Each time your character moves right or left, they will always face the direction they are moving. In the above picture, let's say you notice that you're going to hit the saw, so you choose to start moving back towards the platform that you jumped from. Your character flips horizontally so they are facing the direction they are moving, while still staying the same exact size and the same position they were in before moving.

To review, the rigid motions are translations (slides), rotations (spins/turns), and reflections (flips). All of these types of motions occur without changing the shape of the object or figure being moved.

A rigid motion is a transformation (of the plane) that "preserves distance". In other words, if $A$ is sent/mapped/transformed to $A'$ and $B$ is sent to $B'$, then the distance between $A$ and $B$ (the length of segment $AB$) is the same as the distance between $A'$ and $B'$ (the length of segment $A'B'$).

Intuitively, you can think of rigid motions as (the results of) "movements" that preserve the shapes (and sizes) of objects.

For example, here are the basic types of rigid motions you'll encounter later on:

1. A translation is basically the result of "sliding/pushing" an object (for some distance, and in some direction).

2. A rotation is basically the result of "spinning/turning" an object about some center (for example, turning a door knob rotates the door knob).

3. A reflection is basically the result of "flipping". For example, when you turn a page in a book (flat on a desk), you are essentially reflecting the page over the spine of the book. Note that this is only a "continuous movement" in space, since you're not only moving in the plane of the book/desk itself. But the intuition for reflection as a "movement" is still there—the "shape" of the page should certainly stay the same.

(Food for thought: by this definition, the reflection you see in your mirror should also be a "flip", right? But a "flip" through what?)

Caveat: We won't actually "prove" that these are rigid motions, because it's hard to do so without defining all of geometry in terms of coordinates. Instead, these three facts be some of our most important "assumptions/postulates/axioms" throughout this map. But Matt and I think they're more satisfying "assumptions" than Euclid's postulates.

Suppose we have some figure drawn on an infinite two-dimensional transparent piece of paper. Let's cut the figure out using a pair of scissors. We can move the cut-out figure around on the paper—these movements are called transformations.

Now we can slide the figure around on the piece of paper. The figure itself looks the same, but now it's on a different part of the paper than it was before. This movement is called a translation.

We can also rotate the figure, so perhaps now it looks like it is in a different orientation (perhaps it is upside down). Not surprisingly, this movement is called a rotation.

The third possible movement we could perform is lifting our cut-out figure off the piece of paper and flipping it over, so that the other side of the paper is now face-up. Because the paper is transparent, we now see the reflection of the figure.

We define a rigid motion as a combination of translation, rotation, and reflection. Importantly, a rigid motion preserves the original shape and size of the figure—so the new figure after the rigid motion and the old figure before it would be congruent.

Note: Rigid motions are also called isometries, or congruence transformations.

Examples and Images: Rigid Motions $\\$

Rigid motions are transformations that preserve lengths and angles.

If an object or shape is identical (or congruent) before and after the transformations, these transformations are rigid motions.

For example:

The triangle drawn with dashed lines is a reflection across the x-axis of the one in solid lines.

The two triangles are identical: they have the same lengths and angles.

These two triangles are also identical, they are transformed via rotation.

Rigid motions are also called isometries or congruence transformations.

Translations, rotations, and reflections are rigid motions.

These transformations (rigid motions) are also known as congruence transformations or isometries. As implied by the name "congruence transformations", these transformations produce congruent figures, preserving area, distance, angle measure, betweenness, collinearity, and the like.