Perspective transform from quadrilateral to quadrilateral in Swift
(For code that relies on SIMD for matrix operations and implements other improvements, see https://rethunk.medium.com/perspective-transform-from-quadrilateral-to-quadrilateral-in-swift-using-simd-for-matrix-operations-15dc3f090860)
My previous two posts provided Swift code to find the affine transform in 2D space from one triangle to another. One post presented the traditional method of finding the affine transform, and the other post introduced Simplex Affine Mapping. In this post I’ll provide Swift code to find the perspective transform from one quadrilateral to another based on a paper by Dave Eberly.
In 2D image processing, the affine transform is useful when a camera is pointed perpendicular to a flat surface. If the camera images a piece of paper on a table, and if we take photos of the paper in different positions on the table, then the affine transform maps how the paper translates and rotates: 100 pixels to the right, 40 degrees left, and so on. If the camera remained pointed perpendicular to the table while moving closer or farther away, then the affine transform can also provide accurate scaling: the paper looks half as big, or 1.37 times as big, and so on.
The same affine transform mapping the position, rotation, and scale of the whole piece of paper allows us to map any point on the paper, too. If the letter A is printed on the paper at some (x,y) point in the original image, then the affine transform allows us to predict accurately the (x,y) position of the same letter A in the new image.
What the affine transform can’t do is correct for perspective distortion. To the human sense of vision, and to a standard camera, objects farther away appear smaller. When the piece of paper from our example is viewed at a non-perpendicular angle by the camera, the far side of a piece of paper will appear smaller than the near side of the paper. In the image the rectangular paper will appear to be an irregular quadrilateral, no two sides of which may be the same length.
A perspective transform corrects for perspective distortion, allowing us to map one quadrilateral to another quadrilateral. One quadrilateral could be a rectangle such as a book cover imaged from a camera pointed perpendicular to it, or we could simply be calculating the perspective transform for images capture at two different non-perpendicular angles.
You’ll find perspective transforms in CoreImage, code in Swift on GitHub, and existing types such as CATransform3D, but the purpose here is to present the calculations in Swift code that I hope is clear and readable.
There’s plenty to be learned by reading up on 3D projections, homogeneous coordinates, and related subjects. For a comprehensive study of computational geometry, refer to Geometric Tools for Computer Graphics by Dave Eberly. Be sure to download up-to-date corrections from his books page.
The Swift code below follows Eberly’s paper “Perspective Mappings,” which is available as a free standalone PDF on his Geometric Tools website:
https://geometrictools.com/Documentation/PerspectiveMappings.pdf
Once again I’ll take the liberty of posting a screenshot of the author’s own work. The figure below shows the process of transforming each of two quadrilaterals to intermediate canonical forms.
The chain of calculations, described much better by Eberly but which I’ll summarize here, is as follows:
- Start with two quadrilaterals, p and q.
- Ensure the points of each quadrilateral are ordered counterclockwise, with point 00 at lower left. (My choice for a point ordering function was a simple convex hull algorithm.)
- Find the affine transform from each quadrilateral to its canonical form, with three points mapping to (0,0), (1,0), and (0,1). We’ll call those “canonical p” and “canonical q.”
- Calculate (a,b) for the transformed point p11 and (c,d) for the transformed point q11.
- Calculate the linear fractional transform from canonical p to canonical q.
- Calculate the perspective transform from p to q as the product of the transforms from p to canonical p, canonical p to canonical q, and canonical q to q.
It was more convenient to me to calculate the affine transforms from both p and q to their canonical forms, so in the Swift code the final calculation of the homography (perspective transform) H uses the inverses of the affine transforms found in Eberly’s paper.
Just like the Swift code in my posts about the affine transform, the code below is verbose. And that’s an understatement! Matrices and vectors and the operations necessary for calculations are defined from scratch. The goal was not to provide efficient, robust, production-ready code, but instead to get my hands dirty with the math while writing code in Swift. I also like presenting calculations normally hidden by calls to library functions. Thus you’ll find a few key implementations for matrix transpose, inverse, multiplication, subtraction, and so on. In the future I plan to rely on Apple’s built-in types available in the Accelerate framework’s SIMD (single instruction, multiple data).
So here you go, 600 lines of Swift code and comments to support one function to calculate the perspective transform. Copy and paste into an XCode 12 playground and then run it. A test function prints messages to the console.
(2021–01–05 This code includes a bug fix for the convexHull() function.)
(2021–03–13 The previous convexHull() function had a silly and rather obvious bug, so now the code below includes a convexHull() function from https://www.geeksforgeeks.org/convex-hull-set-1-jarviss-algorithm-or-wrapping/. )
/* Run the whole playground! The perspectiveTransform() function is displayed at top, but depends on types defined later. */
import CoreGraphics
import Foundation
/// Finds the perspective transform (the homography) from quadrilateral p to quadrilateral q.
/// Follows the method of Dave Eberly, but calculates affine transforms differently.
/// Algorithm based on PerspectiveMappings.pdf by Dave Eberly:
/// https://geometrictools.com/Documentation/PerspectiveMappings.pdf
/// For other code to improve this sample, please see Geometric Tools for Computer Graphics by Schneider & Eberly
/// Eberly's books: https://www.geometrictools.com/Books/Books.html
/// Code uses custom types, but could be rewritten to use matrix and vector types in the SIMD framework
func perspectiveTransform(p: Quadrilateral, q:Quadrilateral) -> Matrix3x3? {
//Eberly: H = Aq * F * Inv(Ap)
//here: H = Inv(Aq) * F * Ap
//affine transform of p to canonical quadrilateral - use all points but 3rd point (p11)
//NOTE: Eberly calculates affine transform for p in reverse direction!
let ptri = Triangle(point1: p.point1, point2: p.point2, point3: p.point4)
let canon = Triangle(x1: 0, y1: 0, x2: 1, y2: 0, x3: 0, y3: 1)
guard let Ap = affineTransform(from: ptri, to: canon) else {
print("Could not get affine transform for quadrilateral p")
return nil
}
//affine transform of 1 to canonical quadrilateral - use all points but 3rd point (111)
//deviating from Eberly, this is from q to canonical q; Eberly goes in the opposite direction
let qtri = Triangle(point1: q.point1, point2: q.point2, point3: q.point4)
guard let Aq = affineTransform(from: qtri, to: canon) else {
print("Could not get affine transform for quadrilateral q")
return nil
}
guard let InvAq = Aq.inverted() else {
print("Could not get inverse of affine transform for quadrilateral q")
return nil
}
// (a,b) is coordinate of p11 in canonical quadrilateral
// (c,d) is coordinate of q11 in canonical quadrilateral
let cp11 = p.p11.applying(Ap.toCGAffineTransform())
let cq11 = q.p11.applying(Aq.toCGAffineTransform())
let a = cp11.x
let b = cp11.y
let c = cq11.x
let d = cq11.y
let s = a + b - 1
let t = c + d - 1
//p convex if s > 0
//q convex if t > 0
//get fractional linear transformation
//from canonical p to canonical q
// F = | bcs 0 0 |
// | 0 ads 0 |
// | b(cs - at) a(ds - bt) abt |
// where
// s = a + b - 1
// t = c + d - 1
let F = Matrix3x3(
m11: b * c * s, m12: 0, m13: 0,
m21: 0, m22: a * d * s, m23: 0,
m31: b * (c * s - a * t), m32: a * (d * s - b * t), m33: a * b * t)
//"The 3 × 3 homography matrix H = Aq F Inv(Ap)"
//NOTE: We calculated the affine transforms differently than Eberly
return InvAq * F * Ap
}
/* TEST */
func printTransform(from: Quadrilateral, to: Quadrilateral) {
var p = from
var q = to
// ensure points are properly ordered
p.orderPoints(convexHull(_:))
q.orderPoints(convexHull(_:))
guard let H = perspectiveTransform(p: p, q: q) else {
print("Could not find transform of p and q")
return
}
print("homography\n\(H)")
// convert points in p to 1x3 matrices, apply homography, return as CGPoint
let hp00 = Matrix1x3(p.p00).applying(H).to2DPoint()
let hp10 = Matrix1x3(p.p10).applying(H).to2DPoint()
let hp11 = Matrix1x3(p.p11).applying(H).to2DPoint()
let hp01 = Matrix1x3(p.p01).applying(H).to2DPoint()
let f = NumberFormatter()
print()
print("H * p00 = \n\(f.string(hp00, 1)) calculated\n\(q.p00) expected ")
print()
print("H * p10 = \n\(f.string(hp10, 1)) calculated\n\(q.p10) expected ")
print()
print("H * p11 = \n\(f.string(hp11, 1)) calculated\n\(q.p11) expected ")
print()
print("H * p01 = \n\(f.string(hp01, 1)) calculated\n\(q.p01) expected")
}
func testPerspectiveTransform() {
let p = Quadrilateral(
point1: CGPoint(x: 2, y: 2),
point2: CGPoint(x: 5, y: 1),
point3: CGPoint(x: 6, y: 4),
point4: CGPoint(x: 1, y: 3))
let q = Quadrilateral(
point1: CGPoint(x: 0, y: 1),
point2: CGPoint(x: 6, y: 2),
point3: CGPoint(x: 5, y: 5),
point4: CGPoint(x: 2, y: 4))
print("*** p -> q ***")
printTransform(from: p, to: q)
print()
print("*** q -> p (swapped) ***")
printTransform(from: q, to: p)
}
testPerspectiveTransform()
/* SUPPORTING TYPES. Replace with SIMD types as desired. */
func degreesToRadians(_ degrees: CGFloat) -> CGFloat {
degrees * CGFloat.pi / 180.0
}
func radiansToDegrees(_ radians: CGFloat) -> CGFloat {
180.0 * radians / CGFloat.pi
}
extension CGVector {
init(_ point: CGPoint) {
self.init(dx: point.x, dy: point.y)
}
static func + (lhs: CGVector, rhs: CGVector) -> CGVector {
CGVector(dx: lhs.dx + rhs.dx, dy: lhs.dy + rhs.dy)
}
static func - (lhs: CGVector, rhs: CGVector) -> CGVector {
CGVector(dx: lhs.dx - rhs.dx, dy: lhs.dy - rhs.dy)
}
static func * (_ vector: CGVector, _ scalar: CGFloat) -> CGVector {
CGVector(dx: vector.dx * scalar, dy: vector.dy * scalar)
}
static func * (_ scalar: CGFloat, _ vector: CGVector) -> CGVector {
CGVector(dx: vector.dx * scalar, dy: vector.dy * scalar)
}
static func * (lhs: CGVector, rhs: CGVector) -> CGFloat {
lhs.dx * rhs.dx + lhs.dy * rhs.dy
}
static func dotProduct(v1: CGVector, v2: CGVector) -> CGFloat {
v1.dx * v2.dx + v1.dy * v2.dy
}
static func / (_ vector: CGVector, _ scalar: CGFloat) -> CGVector {
CGVector(dx: vector.dx / scalar, dy: vector.dy / scalar)
}
static func / (_ scalar: CGFloat, _ vector: CGVector) -> CGVector {
CGVector(dx: vector.dx / scalar, dy: vector.dy / scalar)
}
// Returns again between vectors in range 0 to 2 * pi [positive]
// a * b = ||a|| ||b|| cos(theta)
// theta = arc cos (a * b / ||a|| ||b||)
static func angleBetweenVectors(v1: CGVector, v2: CGVector) -> CGFloat {
acos( v1 * v2 / (v1.length() * v2.length()) )
}
func length() -> CGFloat {
sqrt(self.dx * self.dx + self.dy * self.dy)
}
}
extension NumberFormatter {
func string(_ value: Double, _ digits: Int, failText: String = "[?]") -> String {
minimumFractionDigits = max(0, digits)
maximumFractionDigits = minimumFractionDigits
guard let s = string(from: NSNumber(value: value)) else {
return failText
}
return s
}
func string(_ value: Float, _ digits: Int, failText: String = "[?]") -> String {
minimumFractionDigits = max(0, digits)
maximumFractionDigits = minimumFractionDigits
guard let s = string(from: NSNumber(value: value)) else {
return failText
}
return s
}
func string(_ value: CGFloat, _ digits: Int, failText: String = "[?]") -> String {
minimumFractionDigits = max(0, digits)
maximumFractionDigits = minimumFractionDigits
guard let s = string(from: NSNumber(value: Double(value))) else {
return failText
}
return s
}
func string(_ point: CGPoint, _ digits: Int = 1, failText: String = "[?]") -> String {
let sx = string(point.x, digits, failText: failText)
let sy = string(point.y, digits, failText: failText)
return "(\(sx), \(sy))"
}
func string(_ vector: CGVector, _ digits: Int = 1, failText: String = "[?]") -> String {
let sdx = string(vector.dx, digits, failText: failText)
let sdy = string(vector.dy, digits, failText: failText)
return "(\(sdx), \(sdy))"
}
func string(_ transform: CGAffineTransform, rotationDigits: Int = 2, translationDigits: Int = 1, failText: String = "[?]") -> String {
let sa = string(transform.a, rotationDigits)
let sb = string(transform.b, rotationDigits)
let sc = string(transform.c, rotationDigits)
let sd = string(transform.d, rotationDigits)
let stx = string(transform.tx, translationDigits)
let sty = string(transform.ty, translationDigits)
var s = "a: \(sa) b: \(sb) 0"
s += "\nc: \(sc) d: \(sd) 0"
s += "\ntx: \(stx) ty: \(sty) 1"
return s
}
}
struct Matrix1x3: CustomStringConvertible {
var m1: CGFloat
var m2: CGFloat
var m3: CGFloat
init(m1: CGFloat, m2: CGFloat, m3: CGFloat) {
self.m1 = m1
self.m2 = m2
self.m3 = m3
}
init(_ p: CGPoint) {
m1 = p.x
m2 = p.y
m3 = 1
}
var description: String {
let f = NumberFormatter()
return "\(f.string(m1, 2))"
+ "\n\(f.string(m2, 2))"
+ "\n\(f.string(m3, 2))"
}
/// | m11 m12 m13 | | m1 |
/// | m21 m22 m23 | * | m2 |
/// | m31 m32 m33 | | m3 |
func applying(_ x: Matrix3x3) -> Matrix1x3 {
Matrix1x3(
m1: x.m11 * m1 + x.m12 * m2 + x.m13 * m3,
m2: x.m21 * m1 + x.m22 * m2 + x.m23 * m3,
m3: x.m31 * m1 + x.m32 * m2 + x.m33 * m3)
}
func to2DPoint() -> CGPoint {
CGPoint(x: m1 / m3, y: m2 / m3)
}
}
/// Indices are m(row,column)
/// | m11 m12 m13 |
/// | m21 m22 m23 |
/// | m31 m32 m33 |
struct Matrix3x3: CustomStringConvertible {
var m11: CGFloat //row 1
var m12: CGFloat
var m13: CGFloat
var m21: CGFloat //row 2
var m22: CGFloat
var m23: CGFloat
var m31: CGFloat //row 3
var m32: CGFloat
var m33: CGFloat
var description: String {
let f = NumberFormatter()
var s = "\(f.string(m11, 2)) \(f.string(m12, 2)) \(f.string(m13, 2))"
s += "\n\(f.string(m21, 2)) \(f.string(m22, 2)) \(f.string(m23, 2))"
s += "\n\(f.string(m31, 2)) \(f.string(m32, 2)) \(f.string(m33, 2))"
return s
}
func inverted() -> Matrix3x3? {
let d = determinant()
//TODO pick some realistic near-zero number here
if abs(d) < 0.0000001 {
return nil
}
//transpose matrix first
let t = self.transpose()
//determinants of 2x2 minor matrices
let a11 = t.m22 * t.m33 - t.m32 * t.m23
let a12 = t.m21 * t.m33 - t.m31 * t.m23
let a13 = t.m21 * t.m32 - t.m31 * t.m22
let a21 = t.m12 * t.m33 - t.m32 * t.m13
let a22 = t.m11 * t.m33 - t.m31 * t.m13
let a23 = t.m11 * t.m32 - t.m31 * t.m12
let a31 = t.m12 * t.m23 - t.m22 * t.m13
let a32 = t.m11 * t.m23 - t.m21 * t.m13
let a33 = t.m11 * t.m22 - t.m21 * t.m12
//adjugate (adjoint) matrix: apply + - + ... pattern
let adj = Matrix3x3(
m11: a11, m12: -a12, m13: a13,
m21: -a21, m22: a22, m23: -a23,
m31: a31, m32: -a32, m33: a33)
return adj / d
}
func determinant() -> CGFloat {
m11 * (m22 * m33 - m32 * m23) - m12 * (m21 * m33 - m31 * m23) + m13 * (m21 * m32 - m31 * m22)
}
func transpose() -> Matrix3x3 {
Matrix3x3(m11: m11, m12: m21, m13: m31, m21: m12, m22: m22, m23: m32, m31: m13, m32: m23, m33: m33)
}
/// Converts the 3x3 matrix to a CGAffineTransform. Assumes that unused terms are zero.
func toCGAffineTransform() -> CGAffineTransform {
let CGM = self.transpose()
return CGAffineTransform(a: CGM.m11, b: CGM.m12, c: CGM.m21, d: CGM.m22, tx: CGM.m31, ty: CGM.m32)
}
/// | a11 a12 a13 | | b11 b12 b13 |
/// | a21 a22 a23 | * | b21 b22 b23 |
/// | a31 a32 a33 | | b31 b32 b33 |
static func * (_ a: Matrix3x3, _ b: Matrix3x3) -> Matrix3x3 {
return Matrix3x3(
m11: a.m11 * b.m11 + a.m12 * b.m21 + a.m13 * b.m31,
m12: a.m11 * b.m12 + a.m12 * b.m22 + a.m13 * b.m32,
m13: a.m11 * b.m13 + a.m12 * b.m23 + a.m13 * b.m33,
m21: a.m21 * b.m11 + a.m22 * b.m21 + a.m23 * b.m31,
m22: a.m21 * b.m12 + a.m22 * b.m22 + a.m23 * b.m32,
m23: a.m21 * b.m13 + a.m22 * b.m23 + a.m23 * b.m33,
m31: a.m31 * b.m11 + a.m32 * b.m21 + a.m33 * b.m31,
m32: a.m31 * b.m12 + a.m32 * b.m22 + a.m33 * b.m32,
m33: a.m31 * b.m13 + a.m32 * b.m23 + a.m33 * b.m33)
}
static func / (_ m: Matrix3x3, _ s: CGFloat) -> Matrix3x3 {
Matrix3x3(
m11: m.m11/s, m12: m.m12/s, m13: m.m13/s,
m21: m.m21/s, m22: m.m22/s, m23: m.m23/s,
m31: m.m31/s, m32: m.m32/s, m33: m.m33/s)
}
}
/// A representation of a 4x4 matrix used for calculation. The values are used to create
/// a CATransform3D.
struct Matrix4x4 {
var m11: CGFloat
var m12: CGFloat
var m13: CGFloat
var m14: CGFloat
var m21: CGFloat
var m22: CGFloat
var m23: CGFloat
var m24: CGFloat
var m31: CGFloat
var m32: CGFloat
var m33: CGFloat
var m34: CGFloat
var m41: CGFloat
var m42: CGFloat
var m43: CGFloat
var m44: CGFloat
static var identity: Matrix4x4 {
Matrix4x4(
m11: 1, m12: 0, m13: 0, m14: 0,
m21: 0, m22: 1, m23: 0, m24: 0,
m31: 0, m32: 0, m33: 1, m34: 0,
m41: 0, m42: 0, m43: 0, m44: 1)
}
}
enum AngularOrientation: Int {
case colinear
case clockwise
case counterclockwise
}
// To find orientation of ordered triplet (p, q, r).
// The function returns following values
// 0 --> p, q and r are colinear
// 1 --> Clockwise
// 2 --> Counterclockwise
func orientation(_ p: CGPoint, _ q: CGPoint, _ r: CGPoint) -> AngularOrientation {
let val = (q.y - p.y) * (r.x - q.x) -
(q.x - p.x) * (r.y - q.y);
if (val == 0) {
return .colinear
}
return val > 0 ? .clockwise : .counterclockwise
}
/// https://www.geeksforgeeks.org/convex-hull-set-1-jarviss-algorithm-or-wrapping/
/// For divide & conquer, see https://www.geeksforgeeks.org/convex-hull-using-divide-and-conquer-algorithm/
func convexHull(_ unordered: [CGPoint]) -> [CGPoint] {
if unordered.isEmpty {
return []
}
//differing from geeksforgeeks code: sort code so that first point is left bottommost
let points = unordered.sorted(by: { (p1: CGPoint, p2: CGPoint) -> Bool in p1.y < p2.y || (p1.y == p2.y && p1.x < p2.x) })
if points.count < 3 {
return points
}
var hull: [CGPoint] = []
let left = 0
var p = left
var q = 0
repeat
{
hull.append(points[p])
q = (p+1) % points.count
for i in 0 ..< points.count {
if orientation(points[p], points[i], points[q]) == .counterclockwise {
q = i;
}
}
p = q
} while (p != left)
return hull
}
/// Quadrilaterial defined using terminology of Eberly.
/// NOTE: points must be defined in the correct order! There's currently no convex hull or other method to enforce the correct order.
/// "The first convex quadrilateral has vertices p00, p10, p11 and p01, listed in counterclockwise order."
/// p11 (3rd)
/// p01 (4th)
/// p10 (2nd)
/// p00 (1st)
struct Quadrilateral {
var p00: CGPoint { point1}
var p10: CGPoint { point2 }
var p11: CGPoint { point3 }
var p01: CGPoint { point4 }
var point1: CGPoint
var point2: CGPoint
var point3: CGPoint
var point4: CGPoint
var points: [CGPoint] {
[point1, point2, point3, point4]
}
/// Ensure points are ordered counterclosewise, with point1 (p00) at bottom left.
/// This assumes CG coordinates, with the origin at bottom left, +x right, +y up
/// A suitable ordering function would be a traditional convex hull.
mutating func orderPoints(_ orderingFunction: ([CGPoint]) -> [CGPoint]) {
let ordered = orderingFunction(points)
point1 = ordered[0]
point2 = ordered[1]
point3 = ordered[2]
point4 = ordered[3]
}
}
/// Three points nominally defining a triangle, but possibly colinear.
/// Used as an argument to the function SAM.transform(t1:t2:)
struct Triangle {
var point1: CGPoint
var point2: CGPoint
var point3: CGPoint
var x1: CGFloat { point1.x }
var y1: CGFloat { point1.y }
var x2: CGFloat { point2.x }
var y2: CGFloat { point2.y }
var x3: CGFloat { point3.x }
var y3: CGFloat { point3.y }
/// Point1 as a 2D vector
var vector1: CGVector { CGVector(point1) }
/// Point2 as a 2D vector
var vector2: CGVector { CGVector(point2) }
/// Point3 as a 2D vector
var vector3: CGVector { CGVector(point3) }
/// Return a Triangle after applying an affine transform to self.
func applying(_ t: CGAffineTransform) -> Triangle {
Triangle(
point1: self.point1.applying(t),
point2: self.point2.applying(t),
point3: self.point3.applying(t)
)
}
init(point1: CGPoint, point2: CGPoint, point3: CGPoint) {
self.point1 = point1
self.point2 = point2
self.point3 = point3
}
init(x1: CGFloat, y1: CGFloat, x2: CGFloat, y2: CGFloat, x3: CGFloat, y3: CGFloat) {
point1 = CGPoint(x: x1, y: y1)
point2 = CGPoint(x: x2, y: y2)
point3 = CGPoint(x: x3, y: y3)
}
/// Returns a (Bool, CGFloat) tuple indicating whether the points in the Triangle are colinear, and the angle between vectors tested.
func colinear(degreesTolerance: CGFloat = 0.5) -> Bool {
let v1 = vector2 - vector1
let v2 = vector3 - vector2
let radians = CGVector.angleBetweenVectors(v1: v1, v2: v2)
if radians.isNaN {
return true
}
var degrees = radiansToDegrees(radians)
if degrees > 90 {
degrees = 180 - degrees
}
return degrees < degreesTolerance
}
/// | p1.x p2.x p3.x |
/// | p1.y p2.y p3.y |
/// | 1 1 1 |
func toMatrix() -> Matrix3x3 {
Matrix3x3(
m11: point1.x, m12: point2.x, m13: point3.x,
m21: point1.y, m22: point2.y, m23: point3.y,
m31: 1, m32: 1, m33: 1)
}
}
func affineTransform(from: Triangle, to: Triangle) -> Matrix3x3? {
// following example from https://stackoverflow.com/questions/18844000/transfer-coordinates-from-one-triangle-to-another-triangle
// M * A = B
// M = B * Inv(A)
let A = from.toMatrix()
guard let invA = A.inverted() else {
return nil
}
let B = to.toMatrix()
let M = B * invA
return M
}
func cgAffineTransform(from: Triangle, to: Triangle) -> CGAffineTransform? {
guard let M = affineTransform(from: from, to: to) else {
return nil
}
return M.toCGAffineTransform()
}
