SceneKit – What is the lowest row of elements in SCNMatrix4 for?

Question

In SCNMatrix4 struct, the last m44 element is used for Homogeneous Coordinates.

I'd like to know what are m14, m24 and m34 elements in SCNMatrix4? And what exactly I can use these three elements for?

init(m11: Float, m12: Float, m13: Float, m14: Float, 
     m21: Float, m22: Float, m23: Float, m24: Float, 
     m31: Float, m32: Float, m33: Float, m34: Float, 
     m41: Float, m42: Float, m43: Float, m44: Float)

Answer 1

The bottom row is used for translations, and for projections, such as perspective, or a camera viewing frustum. Although not Apple code, the following source (Licensed under the MIT license), taken from https://referencesource.microsoft.com/#System.Numerics/System/Numerics/Matrix4x4.cs,78a107de58a17946 (starting from line 864), shows "[how to create] a perspective projection matrix based on a field of view, aspect ratio, and near and far view plane distances."

    /// <summary>
    /// Creates a perspective projection matrix based on a field of view, aspect ratio, and near and far view plane distances. 
    /// </summary>
    /// <param name="fieldOfView">Field of view in the y direction, in radians.</param>
    /// <param name="aspectRatio">Aspect ratio, defined as view space width divided by height.</param>
    /// <param name="nearPlaneDistance">Distance to the near view plane.</param>
    /// <param name="farPlaneDistance">Distance to the far view plane.</param>
    /// <returns>The perspective projection matrix.</returns>
    public static Matrix4x4 CreatePerspectiveFieldOfView(float fieldOfView, float aspectRatio, float nearPlaneDistance, float farPlaneDistance)
    {
        if (fieldOfView <= 0.0f || fieldOfView >= Math.PI)
            throw new ArgumentOutOfRangeException("fieldOfView");

        if (nearPlaneDistance <= 0.0f)
            throw new ArgumentOutOfRangeException("nearPlaneDistance");

        if (farPlaneDistance <= 0.0f)
            throw new ArgumentOutOfRangeException("farPlaneDistance");

        if (nearPlaneDistance >= farPlaneDistance)
            throw new ArgumentOutOfRangeException("nearPlaneDistance");

        float yScale = 1.0f / (float)Math.Tan(fieldOfView * 0.5f);
        float xScale = yScale / aspectRatio;

        Matrix4x4 result;

        result.M11 = xScale;
        result.M12 = result.M13 = result.M14 = 0.0f;

        result.M22 = yScale;
        result.M21 = result.M23 = result.M24 = 0.0f;

        result.M31 = result.M32 = 0.0f;
        result.M33 = farPlaneDistance / (nearPlaneDistance - farPlaneDistance);
        result.M34 = -1.0f;

        result.M41 = result.M42 = result.M44 = 0.0f;
        result.M43 = nearPlaneDistance * farPlaneDistance / (nearPlaneDistance - farPlaneDistance);

        return result;
    }

This snippet is ARKit - it gets the position of the camera... https://gist.github.com/fisherds/a2a75c914c293213f594daf5fd940d3d

@IBAction func pressedAddEarth(_ sender: Any) {

    guard let pointOfView = sceneView.pointOfView else {return}
    let transform = pointOfView.transform
    let orientation = SCNVector3(-transform.m31, -transform.m32, -transform.m33)
    let location = SCNVector3(transform.m41, transform.m42, transform.m43)
    let currentPositionOfCamera = orientation + location

    // other stuff   
}    

func +(left: SCNVector3, right: SCNVector3) -> SCNVector3 {
  return SCNVector3Make(left.x + right.x, left.y + right.y, left.z + right.z)
}

Answer 2

Based on @BlackMirrorz's answer.

The obvious practical benefit of using m41, m42, and m43 elements in SCNMatrix4 appears when employing SceneKit's camera projection.

Answer 3

I could be wrong but I think m41, m42, and m43 can be used to get positional data and is essentially the same as using result.worldTransform.columns.3when performing a hitTest.

As such when placing an SCNNode via performing an ARSCNHitTest you could use either:

let hitTestTransform = SCNMatrix4(result.worldTransform)
let positionFromMatrix4 = SCNVector3(hitTestTransform.m41, hitTestTransform.m42, hitTestTransform.m43)
let positionFromColumns =  SCNVector3(result.worldTransform.columns.3.x, result.worldTransform.columns.3.y, result.worldTransform.columns.3.z) 

The below example should help clarify things:

/// Places Our Model At The Position Of An Existining ARPlaneAnchor
///
/// - Parameter gesture: UITapGestureRecognizer
@IBAction func placeModel(_ gesture: UITapGestureRecognizer){

    //1. Get The Touch Location
    let touchLocation = gesture.location(in: self.augmentedRealityView)

    //2. Perform An ARSCNHitTest For Any Existing Planes
    guard let result = self.augmentedRealityView.hitTest(touchLocation, types: [.existingPlane, .existingPlaneUsingExtent]).first else { return }

        //3. Get The World Transform
        let hitTestTransform = SCNMatrix4(result.worldTransform)

        //4. Initialize Our Position Either From .m41, .m42, .m43 Or From Columns.3
        let positionFromMatrix4 = SCNVector3(hitTestTransform.m41, hitTestTransform.m42, hitTestTransform.m43)
        let positionFromColumns =  SCNVector3(result.worldTransform.columns.3.x, result.worldTransform.columns.3.y, result.worldTransform.columns.3.z)

        //5. Log Them To Check I'm Not Being A Moron
        print(
            """
            Position From Matrix 4 == \(positionFromMatrix4)
            Position From Columns == \(positionFromColumns)
            """)

        /*
        Position From Matrix 4 == SCNVector3(x: -0.39050543, y: -0.004766479, z: 0.08107365)
        Position From Columns == SCNVector3(x: -0.39050543, y: -0.004766479, z: 0.08107365)
        */

        //6. Add A Node At The Position & Add It To The Hierachy
        let boxNode = SCNNode(geometry: SCNBox(width: 1, height: 1, length: 1, chamferRadius: 0))
        boxNode.geometry?.firstMaterial?.diffuse.contents = UIColor.cyan
        boxNode.position = positionFromMatrix4
        self.augmentedRealityView.scene.rootNode.addChildNode(boxNode)
}

Hope it helps...