extracting rotation, scale values from 2d transformation matrix

Question

How can I extract rotation and scale values from a 2D transformation matrix?

matrix = [1, 0, 0, 1, 0, 0]

matrix.rotate(45 / 180 * PI)
matrix.scale(3, 4)
matrix.translate(50, 100)
matrix.rotate(30 / 180 * PI)
matrix.scale(-2, 4)

Now my matrix have values [a, b, c, d, tx, ty]. Lets forget about the processes above and imagine that we have only the values a, b, c, d, tx, and ty. How can I find final rotation and scale values?

Answer 1

Essentially you need to solve the following

$$\left[\begin{array}{ccc} \mathrm{a} & \mathrm{b} & \mathrm{tx}\\ \mathrm{c} & \mathrm{d} & \mathrm{ty}\end{array}\right]=\left[\begin{array}{ccc} s_{x}\cos\psi & -s_{x}\sin\psi & x_{c}\\ s_{y}\sin\psi & s_{y}\cos\psi & y_{c}\end{array}\right]$$

where $s_x$, $s_y$ are the scalings, $x_c$, $y_c$ is the translation and $\psi$ is the rotation angle. The results I get are:

$$x_{c}=\mathrm{tx}$$ $$y_{c}=\mathrm{ty}$$ $$s_{x}=\mathrm{sign(a)\,}\sqrt{\mathrm{a}^{2}+\mathrm{b}^{2}}$$ $$s_{y}=\mathrm{sign(d)\,}\sqrt{\mathrm{c}^{2}+\mathrm{d}^{2}}$$ $$\tan\psi=-\frac{\mathrm{b}}{\mathrm{a}}=\frac{\mathrm{c}}{\mathrm{d}}$$

So the angle is either $\psi = {\rm atan2}(-b,a)$ or $\psi = {\rm atan2}(c,d)$

Answer 2

The term for this is matrix decomposition. Here is a solution that includes skew as described by Frédéric Wang.

It operates on a 2d matrix defined as such:

$$\left[\begin{array}{ccc} \mathrm{a} & \mathrm{c} & \mathrm{tx}\\ \mathrm{b} & \mathrm{d} & \mathrm{ty}\end{array}\right]$$

function decompose_2d_matrix(mat) {
  var a = mat[0];
  var b = mat[1];
  var c = mat[2];
  var d = mat[3];
  var e = mat[4];
  var f = mat[5];

  var delta = a * d - b * c;

  let result = {
    translation: [e, f],
    rotation: 0,
    scale: [0, 0],
    skew: [0, 0],
  };

  // Apply the QR-like decomposition.
  if (a != 0 || b != 0) {
    var r = Math.sqrt(a * a + b * b);
    result.rotation = b > 0 ? Math.acos(a / r) : -Math.acos(a / r);
    result.scale = [r, delta / r];
    result.skew = [Math.atan((a * c + b * d) / (r * r)), 0];
  } else if (c != 0 || d != 0) {
    var s = Math.sqrt(c * c + d * d);
    result.rotation =
      Math.PI / 2 - (d > 0 ? Math.acos(-c / s) : -Math.acos(c / s));
    result.scale = [delta / s, s];
    result.skew = [0, Math.atan((a * c + b * d) / (s * s))];
  } else {
    // a = b = c = d = 0
  }

  return result;
}

Answer 3

Scale and Rotation Extraction for Action Script 3

package nid.utils 
{
    import flash.geom.Matrix;
    import flash.geom.Point;
    import nid.geom.DMatrix;
    /**
     * ...
     * @author Nidin P Vinayakan
     */
    public class MatrixConvertor 
    {
        public static const degree:Number = 180 / Math.PI;
        public static const radian:Number = Math.PI / 180;

        public function MatrixConvertor()
        {

        }
        public static function convert(mat:Matrix):DMatrix 
        {
            var dmat:DMatrix = new DMatrix(mat.a, mat.b, mat.c, mat.d, mat.tx, mat.ty);
            var rad:Number;
            var deg:Number;
            var sign:Number;
            /**
             * scaleX = √(a^2+c^2)
             * scaleY = √(b^2+d^2)
             * rotation = tan^-1(c/d) = tan^-1(-b/a) it will not work sometimes 
             * rotation = a / scaleX  = d / scaleY
             */
            with (dmat)
            {
                scaleX = Math.sqrt((a * a) + (c * c));
                scaleY = Math.sqrt((b * b) + (d * d));

                sign = Math.atan(-c / a);
                rad  = Math.acos(a / scaleX);
                deg  = rad * degree;

                if (deg > 90 && sign > 0)
                {
                    rotation = (360 - deg) * radian;
                }
                else if (deg < 90 && sign < 0)
                {
                    rotation = (360 - deg) * radian;
                }
                else
                {
                    rotation = rad;
                }
                rotationInDegree = rotation * degree;
            }
            return dmat;
        }
    }

}

/**
* DMatrix Class
*/
package nid.geom 
{
    import flash.geom.Matrix;
    /**
     * ...
     * @author Nidin P Vinayakan
     */
    public class DMatrix extends Matrix
    {
        public var rotation:Number=0;
        public var rotationInDegree:Number=0;
        public var scaleX:Number=1;
        public var scaleY:Number=1;

        public function DMatrix(a:Number=1, b:Number=0, c:Number=0, d:Number=1, tx:Number=0, ty:Number=0)
        {
            super(a, b, c, d, tx, ty);
        }

    }

}