Xcode simd - issue with Translation and Rotation Matrix Example

Question

Not only is using column-major vs row-major counter-intuitive, Apple's documentation on "Working with Matrices" further exacerbates the confusion by their examples of "constructing" a "Translate Matrix" and a "Rotation Matrix" in 2D.

Translate Matrix Per Apple's Documentation ()

Translate A translate matrix takes the following form:

1  0  0
0  1  0 
tx ty 1

The simd library provides constants for identity matrices (matrices with ones along the diagonal, and zeros elsewhere). The 3 x 3 Float identity matrix is matrix_identity_float3x3.

The following function returns a simd_float3x3 matrix using the specified tx and ty translate values by setting the elements in an identity matrix:

func makeTranslationMatrix(tx: Float, ty: Float) -> simd_float3x3 {
    var matrix = matrix_identity_float3x3

    matrix[0, 2] = tx
    matrix[1, 2] = ty

    return matrix 
}

My Issue with it

The line of code matrix[0, 2] = tx sets the value of the first column and the third row to tx. let translationMatrix = makeTranslationMatrix(tx: 1, ty: 3) and printing out the 2nd column print(translationMatrix.columns.2) will yield float3(0.0, 0.0, 1.0). I am very confused regarding why it is the last row that contains the translation values, rather than the column. This convention is not used when using SCNMatrix4MakeTranslation and creating a simd_float4x4 out of the SCNMatrix4 object.

var A = SCNMatrix4MakeTranslation(1,2,3)
var Asimd = simd_float4x4(A)

A.m41 // 1
A.m42 // 2
A.m43 // 3
A.m44 // 1

Asimd.columns.3 // float4(1.0, 2.0, 3.0, 1.0)

Both SCNMatrix4 and simd_float4x4 follow the column major naming convention. In the 2D example from Apple, it is the last row that contains the translation values, whereas with SCNMatrix4 and converting to simd_float4x4, it is the last column that contains the translation values. Apple's example seems to be doing the same with the Rotation Matrices as well.

What am I missing?

Answer 1

This can be confusing, yes.

The documentation you mentions makes the following computation:

let translatedVector = positionVector * translationMatrix

Note that the matrix is on the right side of the multiplication. You are probably used to the notation b = M * a but if you take the transpose you get b' = a' * M' which is what the sample does.

In SIMD there's no way to differentiate a vector from its transpose (b from b') and the library allows you to make the multiplication in both ways:

static simd_float3 SIMD_CFUNC simd_mul(simd_float3x3 __x, simd_float3 __y);
static simd_float3 SIMD_CFUNC simd_mul(simd_float3 __x,  simd_float3x3 __y) { return simd_mul(simd_transpose(__y), __x); }