Projecting the ARKit face tracking 3D mesh to 2D image coordinates

Maybe you can use the projectPoint function of the SCNSceneRenderer.

extension ARFaceAnchor{
    // struct to store the 3d vertex and the 2d projection point
    struct VerticesAndProjection {
        var vertex: SIMD3<Float>
        var projected: CGPoint
    }
    
    // return a struct with vertices and projection
    func verticeAndProjection(to view: ARSCNView) -> [VerticesAndProjection]{
        
        let points = geometry.vertices.compactMap({ (vertex) -> VerticesAndProjection? in

            let col = SIMD4<Float>(SCNVector4())
            let pos = SIMD4<Float>(SCNVector4(vertex.x, vertex.y, vertex.z, 1))
            
            let pworld = transform * simd_float4x4(col, col, col, pos)
            
            let vect = view.projectPoint(SCNVector3(pworld.position.x, pworld.position.y, pworld.position.z))

            let p = CGPoint(x: CGFloat(vect.x), y: CGFloat(vect.y))
            return VerticesAndProjection(vertex:vertex, projected: p)
            })
        
        return points
    }
}

Here is a convenient way to get the position:

extension matrix_float4x4 {
    
    /// Get the position of the transform matrix.
    public var position: SCNVector3 {
        get{
            return SCNVector3(self[3][0], self[3][1], self[3][2])
        }
    }
}

If you want to check that the projection is ok, add a debug subview to the ARSCNView instance, then, with a couple of others extensions to draw the 2d points on a view such as:

extension UIView{
    
    private struct drawCircleProperty{
        static let circleFillColor = UIColor.green
        static let circleStrokeColor = UIColor.black
        static let circleRadius: CGFloat = 3.0
    }
    
    func drawCircle(point: CGPoint) {
    
        let circlePath = UIBezierPath(arcCenter: point, radius: drawCircleProperty.circleRadius, startAngle: CGFloat(0), endAngle: CGFloat(Double.pi * 2.0), clockwise: true)
        let shapeLayer = CAShapeLayer()
        shapeLayer.path = circlePath.cgPath
        shapeLayer.fillColor = drawCircleProperty.circleFillColor.cgColor
        shapeLayer.strokeColor = drawCircleProperty.circleStrokeColor.cgColor
        
        self.layer.addSublayer(shapeLayer)
    }
    
    func drawCircles(points: [CGPoint]){
        
        self.clearLayers()
        
        for point in points{
            self.drawCircle(point: point)
        }
    }
    
    func clearLayers(){
        if let subLayers = self.layer.sublayers {
            for subLayer in subLayers {
                subLayer.removeFromSuperlayer()
            }
        }
    }

You can compute the projection, and draw the points with:

let points:[ARFaceAnchor.VerticesAndProjection] = faceAnchor.verticeAndProjection(to: sceneView)
     
// keep only the projected points
let projected = points.map{ $0.projected}
// draw the points !
self.debugView?.drawCircles(points: projected)

I can see all the 3d vertices projected on the 2d screen (picture generated by https://thispersondoesnotexist.com).

I added this code to the Apple demo project, available here https://github.com/hugoliv/projectvertices.git