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:

$$v_2 = S v_1$$

Where $v_1$ is the vector to be scaled, $S$ is the scale matrix, and $v_2$ is the scaled vector.

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

$$v_3 = R v_2$$

Where $v_3$ is the final vector and $R$ is the rotation matrix.

In full, we could write:

$$v_3 = R (S v_1)$$

However, we can also compute the composite matrix $R * S$, then apply this to $v_1$ and get the same result.

$$v_3 = (R * S) v_1$$

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 $R S v_1$ is different from $S R v_2$

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 $M$ that transform a point in the following form:

$$x\prime = m_{1,1} * x + m_{1,2} * y$$ $$y\prime = m_{2,1} * x + m_{2,2} * y$$

But this kind of transform cannot allow us to move locations. As an example, you can’t multiply the point ${0, 0}$ with any number to move it. To translate something, we need a transform of the form:

$$x\prime = x + x_t$$ $$y\prime = y + y_t$$

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:

$$A = \begin{bmatrix}1 & shear\\0 & 1\end{bmatrix}$$

A 3D shear looks like:

$$A = \begin{bmatrix}1 & 0 & x_s\\0 & 1 & y_s\\0 & 0 & 1\end{bmatrix}$$

Applied to some 3D vector:

$$\begin{bmatrix}1 & 0 & x_s\\0 & 1 & y_s\\0 & 0 & 1\end{bmatrix} \begin{bmatrix}x\\y\\z\end{bmatrix} = \begin{bmatrix}x+x_s*z\\y+y_s*z\\z\end{bmatrix}$$

So we set the third or $z$ component of our vector to $1$:

$$\begin{bmatrix}1 & 0 & x_t\\0 & 1 & y_t\\0 & 0 & 1\end{bmatrix} \begin{bmatrix}x\\y\\1\end{bmatrix} = \begin{bmatrix}x+x_t\\y+y_t\\1\end{bmatrix}$$

Thus by using a 3D transformation matrix and a 3D vector with $1$ in the $z$ 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 ${t_x, t_y}$ and then rotate by an angle of $\theta$ by using the following composite matrices:

$$\begin{bmatrix}cos\theta & -sin\theta & 0\\sin\theta & cos\theta & 0\\0 & 0 & 1\end{bmatrix} \begin{bmatrix}1 & 0 & x_t\\0 & 1 & y_t\\0 & 0 & 1\end{bmatrix} \begin{bmatrix}x\\y\\1\end{bmatrix}$$

Notice that we’ve added a column of $0, 0, 1$ and a row of $0, 0, 1$ to the rotation matrix from before. This holds true for other 2D transforms. For example, scale:

$$\begin{bmatrix}s_x & 0 & 0\\0 & s_y & 0\\0 & 0 & 1\end{bmatrix}$$

and shear-x:

$$\begin{bmatrix}1 & shear_x & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix}$$

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 $0$ in the $z$ component instead of a $1$.

Thus, for vectors:

$$\begin{bmatrix}1 & 0 & x_t\\0 & 1 & y_t\\0 & 0 & 1\end{bmatrix} \begin{bmatrix}x\\y\\0\end{bmatrix} = \begin{bmatrix}x\\y\\0\end{bmatrix}$$

This extra component is called the homogeneous coordinate. This is because it is typically either $0$ or $1$.

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:

$$\begin{bmatrix}s_x & 0 & 0 & 0\\0 & s_y & 0 & 0\\0 & 0 & s_z & 0\\0 & 0 & 0 & 1\end{bmatrix}$$

3D rotate (around Z axis):

$$\begin{bmatrix}cos\theta & -sin\theta & 0 & 0\\sin\theta & cos\theta & 0 & 0\\0 & 0 & 1 & 0\\0 & 0 & 0 & 1\end{bmatrix}$$