Examples of Non-Rigid Transformations

What are Non-Rigid Transformations?

Non-Rigid Transformations are transformations that are not rigid. Ok, yeah, that's the simplest definition, so let's dive a little deeper.

If you remember what rigid transformations aim to do, it is to take one fixed object, then perform some operations to it to get that same fixed object in a different position. Whether that be translation, rotation, or reflection, you are not changing the shape's original form in any way, you are just changing its position in space.

Non-Rigid Transformations actually change the structure of our original object. For example, it can make our object bigger or smaller using scaling. Or, it can be "pushed" up or to the side using shearing. Basically, non-rigid transformations allows the object to increase or decrease in size.

Let's take a look at both Scaling and Shearing.

Scaling as a Transformation

Scaling is the size transformation of a shape by a certain scale factor that is the same in all directions.

Note the picture above. The square on the left is the original square. If we want to scale it by $2$, that means each side of the new square will be equivalent to $2$ of the old square, as shown on the right. This also means that if the sides are doubled, then the area will be quadrupled. You can easily see this since there are $4$ of the original squares located in the perimeter.

Scaling does not always have to be increasing the size of an object. It works for decreasing as well. If we wanted to scale the square on the right by $\frac{1}{2}$, then we would get the square on the left by halving each side of the right square.

NOTE: Whatever number you scale the sides by, that means that you are squaring that scaled number when determining what the area is going to be scaled by. For example, if you scale the sides by $3$, then your area is scaled by $9$. Or, if you scale the sides by $\frac{1}{4}$, then the area is scaled by $\frac{1}{16}$

Shearing as a Transformation

Shearing makes it look like you are "pushing" one part of the original shape, while keeping the bottom cemented to the floor.

Note the image above. We start with a normal square. If we were to push the top of the square to the right while keeping the bottom cemented, then it would look like the shape on the right. This is a very basic, yet effective, example of shearing.