math - extract rotation, scale values from 2d transformation matrix

Question

how can i extract rotation, scale and translation 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

EDIT

I think it should be an answer somewhere...

i just tried in Flash Builder (AS3) like this

   var m:Matrix = new Matrix;
   m.rotate(.25 * Math.PI);
   m.scale(4, 5);
   m.translate(100, 50);
   m.rotate(.33 * Math.PI);
   m.scale(-3, 2.5);

   var shape:Shape = new Shape;
   shape.transform.matrix = m;

   trace(shape.x, shape.y, shape.scaleX, shape.scaleY, shape.rotation);

and the output is:

x = -23.6 
y = 278.8 
scaleX = 11.627334873920528 
scaleY = -13.54222263865791 
rotation = 65.56274134518259 (in degrees)

Answer 1

Not all values of a,b,c,d,tx,ty will yield a valid rotation sequence. I assume the above values are part of a 3x3 homogeneous rotation matrix in 2D

    | a  b  tx |
A = | c  d  ty |
    | 0  0  1  |

which transforms the coordinates [x, y, 1] into:

[x', y', 1] = A * |x|
                  |y|
                  |z|

• Thus set the traslation into [dx, dy]=[tx, ty]
• The scale is sx = sqrt(a2 + c2) and sy = sqrt(b2 + d2)
• The rotation angle is t = atan(c/d) or t = atan(-b/a) as also they should be the same.

Otherwise you don't have a valid rotation matrix.

---

The above transformation is expanded to:

x' = tx + sx (x Cos θ - y Sin θ)
y' = ty + sy (x Sin θ + y Cos θ)

when the order is rotation, followed by scale and then translation.