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.

Answer 2

I ran into this problem today and found the easiest solution to transform a point using the matrix. This way, you can extract the translation first, then rotation and scaling.

This only works if x and y are always scaled the same (uniform scaling).

Given your matrix m which has undergone a series of transforms,

var translate:Point; var rotate:Number; var scale:Number;  // extract translation var p:Point = new Point(); translate = m.transformPoint(p); m.translate( -translate.x, -translate.y);  // extract (uniform) scale p.x = 1.0; p.y = 0.0; p = m.transformPoint(p); scale = p.length;  // and rotation rotate = Math.atan2(p.y, p.x); 

There you go!