extracting rotation, scale values from 2d transformation matrix

Question

how can i extract rotation, scale values from 2d transformation matrix? i mean a have a 2d transformation

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, ty

how can i find total rotation and scale values via a, b, c, d, tx, ty

sorry for my english

Thanks your advance

Answer 1

Esentially you need to solve the following

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

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:

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

Answer 2

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 

Answer 3

Is there a way to get the sign of the calculated angle? Right now the calculation gives me an angle of 90° no matter wether I rotate clockwise or counterclockwise.