Sławomir Czerwiński

Working with OpenGL and 3D graphics would be impossible without affine transformations, which in general can be represented as:

\[\vec{y} = \boldsymbol{A} \vec{x} + \vec{a}\]

where \(\vec{a}\) is a translation vector, and \(\boldsymbol{A}\) is a matrix representation of a linear mapping.

The same transformation might also be written as a single multiplication of an augmented matrix (also called an affine transformation matrix) and an augmented vector:

\[\begin{bmatrix}\vec{y}\\1\end{bmatrix} = \left[\begin{array}{ccc|c} & \boldsymbol{A} & & \vec{a} \\ 0 & … & 0 & 1 \end{array}\right] \begin{bmatrix}\vec{x}\\1\end{bmatrix}\]

In 3D space, an augmented vector consists of 4 coordinates: \((x, y, z, 1)\), while the size of an augmented matrix is 4×4.

Using this representation, we can compose affine transformations by simply multiplying augmented matrices:

\[\begin{bmatrix}\vec{y}\\1\end{bmatrix} = \left[\begin{array}{ccc|c} & \boldsymbol{A} & & \vec{a} \\ 0 & … & 0 & 1 \end{array}\right] \left[\begin{array}{ccc|c} & \boldsymbol{B} & & \vec{b} \\ 0 & … & 0 & 1 \end{array}\right] \begin{bmatrix}\vec{x}\\1\end{bmatrix}\]

The problem: Normal matrix

While matrix multiplication is a relatively simple operation, computing a so called normal matrix proves to be quite challenging.

When a 3D model is transformed using an augmented matrix \(\boldsymbol{A}\), the normals should be transformed using a normal matrix \(\boldsymbol{N}\) defined as follows:

\[\boldsymbol{N} = \big(\boldsymbol{A}^{-1}\big)^\mathrm{T} = \big(\boldsymbol{A}^\mathrm{T}\big)^{-1}\]

This calculation could be done in a single line of a vertex shader:

mat4 normalMatrix = transpose(inverse(modelViewMatrix));

However, this would mean that the GPU should perform the same operation for each vertex. Hence, it is much better to calculate normal matrix only once per each frame with the CPU instead.

Unfortunately, a typical implementation of matrix inversion is long and hard to understand.

Functional implementation of a square matrix

After studying Wikipedia, I have come up with a nice and clear algorithm for matrix inversion. Since multiple matrices are used in the course of this calculation, it is better to avoid copying the entries every time a new matrix is created. Thus I have decided to use a functional approach to the problem.

Let’s start with creating a class representing a square matrix of a specific size n×n, the elements of which are defined by a function:

\[f: \{1, 2 \ldots n\}^2 \to \Bbb{R}\]

We may implement such a matrix as:

class SquareMatrix(val size: Int, private val elements: (Int, Int) -> Float) {
    operator fun get(row: Int, col: Int): Float {
        require(row in 0..size - 1) { "Row ${row} out of bounds: 0..${size - 1}" }
        require(col in 0..size - 1) { "Column ${col} out of bounds: 0..${size - 1}" }
        return elements(row, col)

The elements function, provided in the constructor, returns a matrix entry in the specified row and column, e.g. identity matrix \(\boldsymbol{I}_n\) of a requested size can be defined as:

fun identity(size: Int) = SquareMatrix(size) { row, col ->
    if (row == col) 1f else 0f

As you may have noticed, both row and col are required to fit into range of 0 to n−1. That’s because, by convention, in computer programming indices start with 0. It is not a problem though, as this shift does not influence any of our calculations.

Now, let’s try to define some operations on the matrix.

Matrix transposition

The transpose of the matrix can be created by simply swapping indices of the rows and the columns:

\[\left[\boldsymbol{A}^\mathrm{T}\right]_{i,j} = \left[\boldsymbol{A}\right]_{j,i}\]

So the method responsible for this operation can be defined as follows:

fun transpose(): SquareMatrix = SquareMatrix(size) { row, col -> this[col, row] }

Note that the transpose does not copy entries from the original matrix, but rather refers to them.


Before we can go any further, we need to calculate the determinant of the matrix.

Let’s start with the simplest case of a (rather degenerated) 1×1 matrix. Wikipedia does not describe, how to perform the calculation in this case, but we may try to infer the formula from determinant’s properties.

For n×n matrices, the following statements are always true:

\[\mathrm{det}(\boldsymbol{I}_n) = 1\] \[\mathrm{det}(c\boldsymbol{A}) = c^n \mathrm{det}(\boldsymbol{A})\]

On the other hand, for any 1×1 matrix \(\boldsymbol{A}\) we can say:

\[\boldsymbol{A} = a_{1,1} \boldsymbol{I}_1\]

Therefore, we can calculate the determinant of a 1×1 matrix as:

\[\mathrm{det}(\boldsymbol{A}) = \mathrm{det}(a_{1,1} \boldsymbol{I}_1) = \left(a_{1,1}\right)^1 \mathrm{det}(\boldsymbol{I}_1) = a_{1,1} \cdot 1 = a_{1,1}\]

For any larger n×n matrix, the determinant can be defined as a sum of all elements in the first row multiplied by respective cofactors:

\[\mathrm{det}(\boldsymbol{A}) = \sum_{j=1}^n a_{1,j} C_{1,j}\]

So, taking both cases into account, the implementation should look like this:

val det: Float by lazy {
    if (size == 1) this[0, 0]
    else (0..size - 1).map { item -> this[0, item] * comatrix[0, item] }.sum()

Lazily initialized value is used instead of a function to prevent the same determinant from being calculated multiple times.

Cofactors and comatrix

To calculate the determinant, we’ve used a comatrix or, in other words, a matrix of cofactors:

private val comatrix: SquareMatrix by lazy {
    SquareMatrix(size) { row, col -> cofactor(row, col) }

A cofactor is a first minor of the matrix (\(M_{i,j}\)) multiplied by \((-1)^{i+j}\), where \(i\) and \(j\) are indices of the row and the column of the matrix:

\[C_{i,j} = (-1)^{i+j} M_{i,j}\]
private fun cofactor(row: Int, col: Int): Float =
        minor(row, col) * if ((row + col) % 2 == 0) 1f else -1f

Even though the indices have been shifted, it happened both for row and for column. So the parity of the sum row + col is still preserved.

First minor

A first minor \(M_{i,j}\) is the determinant of the submatrix created by removing i-th row and j-th column from the original matrix.

private fun minor(row: Int, col: Int): Float = sub(row, col).det
private fun sub(delRow: Int, delCol: Int) = SquareMatrix(size - 1) { row, col ->
    this[if (row < delRow) row else row + 1, if (col < delCol) col else col + 1]

As you may have noticed, the algorithm for finding the determinant used in SquareMatrix is recursive. This fact may cause risks related to stack space. However, since in 3D graphics only 4×4 matrices are used, we may be sure that nested method calls will not lead to stack overflow.

Adjugate matrix

Now, we can find the adjugate matrix, which is simply a transpose of the comatrix:

\[\mathrm{adj}(\boldsymbol{A}) = \boldsymbol{C}^\mathrm{T}\]
val adj: SquareMatrix by lazy { comatrix.transpose() }

Matrix inversion

With the determinant and the adjugate matrix, we have all the elements needed to finally perform matrix inversion:

\[\boldsymbol{A}^{-1} = \dfrac{1}{\mathrm{det}(\boldsymbol{A})} \mathrm{adj}(\boldsymbol{A})\]
fun inverse(): SquareMatrix = SquareMatrix(size) { row, col -> adj[row, col] / det }


The matrix inversion algorithm is ready. Now, let’s try to use it.

As an example, we will use a diagonal matrix, which should give some predictable results:

val diag = SquareMatrix(4) { row, col ->
    if (row == col) (row + 1).toFloat() else 0f

The matrix diag should look like this:

\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 4 \\ \end{bmatrix}\]

Now, let’s create an inverse of this matrix:

val inv = diag.inverse()

The result should be as follows:

\[\begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & \frac{1}{2} & 0 & 0 \\ 0 & 0 & \frac{1}{3} & 0 \\ 0 & 0 & 0 & \frac{1}{4} \\ \end{bmatrix}\]

Let’s verify that by calling:

println(inv[2, 2])

In the console, we should receive approximately \(\frac{1}{3}\):


Full implementation of SquareMatrix class can be found at GitHub Gist.