# Lecture 12: Geometric Transforms

## Composite Transformations

Often we want to apply more than one transform. If we first wanted to scale an object, we could write:

Where is the vector to be scaled, is the scale matrix, and is the scaled vector.

Then, if we wanted to rotate the object, we would write:

Where is the final vector and is the rotation matrix.

In full, we could write:

However, we can also compute the composite matrix , then apply this to and get the same result.

This concept of a **composite** matrix is essential in graphics applications as it means
we can represent the work of two matrices in a single composite matrix.

Thus we can send a single matrix as a uniform for each draw call which can apply multiple transformations to an object.

Note that these transforms are applied right side first!

Thus is different from

## Translation

So far we have covered rotation, scale, and shear transformations.
There is one type of transformation conspiciously missing from this list: **translation** (or movement).

This is because we have been looking at matrices that transform a point in the following form:

But this kind of transform cannot allow us to *move* locations.
As an example, you can’t multiply the point with any number to move it.
To translate something, we need a transform of the form:

There is no way to do this with a 2x2 matrix!

What’s the solution? We could keep track of a separate translation… but the goal here is to represent everything with a matrix. Then we can easily composite transformations.

The solution is to move to a higher dimension.

Recall that a 2D shear looked like:

A 3D shear looks like:

Applied to some 3D vector:

So we set the third or component of our vector to :

Thus by using a 3D transformation matrix and a 3D vector with in the component, we can use a matrix to represent a 2D translation.

This video shows visually how a shear in 3D space works out to be a translate in 2D space:

## Composite with Translation

For example, we can translate a coordinate by and then rotate by an angle of by using the following composite matrices:

Notice that we’ve added a column of and a row of to the rotation matrix from before. This holds true for other 2D transforms. For example, scale:

and shear-x:

### Positions vs. Vectors

This also gives us a tool for distinguishing between positions and vectors.

We want positions to me affected by translation matrices, of course. But if we have something which is geometrically a vector (e.g. a surface normal), we want that to be scaled and rotated by not translated. We can accomplish this by putting a in the component instead of a .

Thus, for vectors:

This extra component is called the **homogeneous** coordinate.
This is because it is typically either or .

## What about 3D?

Since we used a third component to represent translations in our 2D coordinate system, it should come as no surprise that we used a fourth component to represent translations in a 3D coordinate system:

3D scale:

3D rotate (around Z axis):