# Rigid Motions

## Rigid Motion

A rigid motion of the plane is a transformation that does not change the distance between any two points.

Any shapes in the plane may end up in a different location, but they will not be distorted.

## Searching for Rigid Motion

Recall that we had several types of linear transformations:
• Stretch
• Rotate
• Reflect
• Shear
• Project

Plus, the affine transformation:
• Translate

To decide which of these transformations are rigid motions, we just have to check whether or not they preserve distances.

## Searching for Rigid Motion

When we stretch the plane, the distance between two points can get larger. This means
stretching is NOT a rigid motion.

## Searching for Rigid Motion

When we rotate the plane, the distance between two points will stay the same.
Rotation is a rigid motion.

## Searching for Rigid Motion

When we skew the plane, the distance between two points can change.
Shear/skew is NOT a rigid motion.

## Searching for Rigid Motion

Reflections will change orientation, but not distance.
Reflection is a rigid motion.

## Searching for Rigid Motion

Finally, translations only involve moving the plane, not changing it.
Translations are rigid motions.

Rotations

Translations

Reflections

## Describing Rigid Motions

Rotations are described by their center and angle of rotation.

## Describing Rigid Motions

Reflections are described by their axis, the line we "flip" the image over.

## Describing Rigid Motions

Translations are described by a direction and distance to move. They can be described by vectors.

## Fixed Points

Fixed Points of a rigid motion are the points of the plane (or image in it) that are unchanged by the motion.

Reflections and rotations have fixed points, translations do not.

## Fixed Points

The fixed points of a reflection lie along its reflection axis, the line it's reflected across. This line always goes through the center of the image.
Rotations have only one fixed point, in the center of the image.

## Orientation

A rigid motion is called orientation preserving if the points of the image are not "reversed" by it. Orientation can be thought of as handedness - we'd have to change orientation in order to turn a right handed glove into a left one.

Rotations and translations are orientation preserving, reflections are not.

## One More Motion

Another named motion is really a combination of two others: a glide reflection.

This is a reflection combined with a translation (in either order).

## Summary of Rigid Motions

Motion Described By Fixed Points Orientation Preserving?
Rotations Center, Angle One Point, Center Yes
Translations Distance, Direction None Yes
Reflections Axis of Reflection Along axis No
Glide Reflection Axis, Distance None No