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Composite Transformations

In the previous sections, we laid the groundwork for a visual understanding of transformations. We learned how these transformations, due to their linearity (which we currently understand as preserving the origin, and lines staying parallel and evenly spaced), can be represented by matrices. We walked through the process of applying these transformations to vectors, and we saw how the matrix-vector product is a way to apply a transformation to a vector.

In this section, we will explore how we can combine multiple transformations into a single transformation.

Table of Contents​

Combining Transformations​

Suppose we have two transformations, and , that rotate and scale vectors, respectively. If we apply to a vector, and then apply to the result, we will rotate the vector and then scale the result.

We can represent this process as a single transformation, , that is the result of applying to the result of applying . This transformation is called the composite transformation of and .

Recall that when we apply a transformation to a vector, we multiply the transformation matrix by the vector. Suppose our transformation matrices are:

Then, applying to a vector is equivalent to multiplying the matrix by :

And then applying to the result is equivalent to multiplying the matrix by the result:

We can combine these two matrices into a single matrix, , that represents the composite transformation of and . This matrix is the result of matrix multiplication, a new operation that we will explore in this section.

Deriving the Composite Transformation Matrix​

To summarize the previous section, we have two transformations, and , with matrices and . To find the composite transformation matrix , we multiply the matrices and :

Let's zoom in on the matrix multiplication step:

Recall that each column in the matrix represents the result of applying the transformation to the basis vectors. So we just need to apply the transformation to the basis vectors and fill in the resulting columns of the matrix.

When we apply the rotation matrix, , we get that and (since the columns represent the transformed basis vectors). Then, we can apply the scaling matrix, , to these vectors to get the columns of the composite transformation matrix:

Then, these new vectors are the columns of the composite transformation matrix:

Generalizing this process, let's say we have two transformations, and , with matrices and as:

Let's say we apply first, and then apply . Applying to the basis vectors gives us and . Then, applying to these vectors gives us the columns of the composite transformation matrix:

Therefore, the composite transformation matrix is:

Just as a memory aid, going from left to right, top to bottom, the composite transformation matrix can be calculated as:

  • Row 1 (of left matrix), Column 1 (of right matrix):
  • Row 1, Column 2:
  • Row 2, Column 1:
  • Row 2, Column 2:

Essentially, go through the rows of the left matrix, and for each row, go through the columns of the right matrix, multiplying the corresponding elements and summing them up.

However, understanding the process intuitively through the composition of linear transformations is far more important and insightful than rote memorization of the formula.

Deriving the 3D Composite Transformation Matrix​

The process of deriving the composite transformation matrix for 3D transformations is similar to the 2D case.

Suppose we have two transformations, and , with matrices and as:

Let's say we apply first, and then apply .

Since is applied first, the columns of represent the basis vectors after the first transformation. Applying to these vectors gives us the columns of the composite transformation matrix:

Therefore, the composite transformation matrix is:

This process can be generalized to any number of dimensions, but the intuition remains the same: the columns of the composite transformation matrix are the result of applying the second transformation to the basis vectors after applying the first transformation.

As it is hopefully clear by now, multiplying matrices can get quite tedious, especially for larger matrices. Therefore, it is usually more practical to use computers to perform these calculations. Then, we can focus on understanding the intuition behind the process and the geometric implications of combining transformations.

Summary and Next Steps​

In this section, we explored how we can combine multiple transformations into a single transformation, known as the composite transformation.

Here are the key points to remember:

  • The composite transformation of two transformations, and , is the result of applying to the result of applying .

  • The composite transformation matrix is found by multiplying the matrices of the individual transformations.

  • The columns of the composite transformation matrix are the result of applying the second transformation to the basis vectors after applying the first transformation.

  • The process can be generalized to any number of dimensions, but the intuition remains the same.

  • For the 2D case, it looks like this:

In the next section, we will introduce a new concept known as the determinant of a matrix.