swift - ARKit – What do the different columns in Transform Matrix represent?

Question

An ARAnchor's 4x4 matrix has 4 columns. The fourth column of the matrix contains 3 translate values for x, y, and z coordinates.

I was wondering what the other 3 columns represent?

Answer 1

ARKit, RealityKit and SceneKit frameworks use 4 x 4 Transformation Matrices to translate, rotate, scale and shear 3D objects (just like simd_float4x4 matrix type). Let's see how these matrices look like.

In 3D Graphics we often use a 4x4 Matrix with 16 useful elements. The Identity 4x4 Matrix is as following:

Between those sixteen elements there are 6 different shearing coefficients:

shear XY
shear XZ
shear YX
shear YZ
shear ZX
shear ZY

In Shear Matrix they are as followings:

Because there are no Rotation coefficients at all in this Matrix, six Shear coefficients along with three Scale coefficients allow you rotate 3D objects about X, Y, and Z axis using magical trigonometry (sin and cos).

Here's an example how to rotate 3D object (CCW) about its Z axis using Shear and Scale elements:

Look at 3 different Rotation patterns using Shear and Scale elements:

And, of course, 3 elements for translation (tx,ty,tz) in the 4x4 Matrix are located on the last column:

 ┌               ┐
 |  1  0  0  tx  |
 |  0  1  0  ty  |
 |  0  0  1  tz  |
 |  0  0  0  1   |
 └               ┘

The columns' indices in ARKit, SceneKit and RealityKit are: 0, 1, 2, 3.

The fourth column (index 3) is for translation values:

var translation = matrix_identity_float4x4

translation.columns.3.x
translation.columns.3.y
translation.columns.3.z

You can read my illustrated story about MATRICES on Medium.

Perspective and orthographic projections

Values located in the bottom row of a 4x4 matrix is used for perspective projection.

And, of course, you need to see an example of how to setup an orthographic projection.