Decompose a 2D transform matrix into [rotate scale rotate translate]

decompose-matrix.js

function decomposeMatrix(m) {
	var t,r,s,k,E,F,G,H,Q,R,sx,sy,a1,a2,theta,phi,sqrt=Math.sqrt,atan2=Math.atan2;
	// http://math.stackexchange.com/questions/861674/decompose-a-2d-arbitrary-transform-into-only-scaling-and-rotation
	// 
	// It works wonderfully! Thanks. 
	// The input matrix is transposed though, 
	// so let me spell the solution out. 
	

	E=(m[0]+m[3])/2
	F=(m[0]-m[3])/2
	G=(m[2]+m[1])/2
	H=(m[2]-m[1])/2

	Q=sqrt(E*E+H*H);
	R=sqrt(F*F+G*G);
	sx=Q+R; 
	sy=Q-R; 
	a1=atan2(G,F); 
	a2=atan2(H,E); 
	theta=(a2-a1)/2; 
	phi=(a2+a1)/2;

	// The requested parameters are then theta, 
	// sx, sy, phi,
	//  i.e. rotate by theta, 
	k=-theta*180/Math.PI;
	//  scale by sx,sy, 
	s=[sx,sy];
	//  rotate by phi. 
	r=-phi*180/Math.PI;
	//No division by zero or sqrt(negative) hazard. Excellent. 
	t=[m[4],m[5]];
	return {translate:t,rotate:r,scale:s,skew:k};
}

decomposed_toString = function(tr) {
  return [ 
		"translate(" + tr.translate.join(",") + ")",
		"rotate(" + tr.skew + ")", 
		"scale(" + tr.scale.join(",") + ")",
		"rotate(" + tr.rotate + ")",
  ].join(" ");
};

// http://frederic-wang.fr/decomposition-of-2d-transform-matrices.html


       var TransformName = null;
       var CSSdecomposition, SVGdecomposition, MatrixDecomposition;

       function initTransformName()
       {
          // Initialize TransformName with the appropriate CSS property name.
          if (TransformName) return true;
          var test = document.getElementById("test");

          var nameList = ["transform", "-moz-transform", "-webkit-transform",
                          "-o-transform"];
          for (var i in nameList) {
            TransformName = nameList[i];
            if (getComputedStyle(test)[TransformName] === "none") return true;
          }

          return false;
       }

       function radToDeg(a)
       {
         // Radian to degree.
         return 180 * a / Math.PI;
       }

       function mn(x)
       {
         // MathML Number.
         if (x < 0)
           return "<mrow><mo>&#x2212;</mo><mn>"+Math.abs(x)+"</mn></mrow>";
         return "<mn>"+x+"</mn>";
       }

       function trig(func, arg)
       {
         // MathML trigonometric function.
         return "<mrow><mi>"+func+"</mi><mo>&#x2061;</mo><mrow><mo>(</mo>"+
                mn(arg / Math.PI)+"<mi>&#x3c0;</mi><mo>)</mo></mrow></mrow>";
       }

       function matrix(a, b, c, d, e, f)
       {
         // MathML 2D matrix.
         return "<mrow><mo>(</mo><mtable><mtr><mtd>"+a+"</mtd><mtd>"+c+"</mtd><mtd>"+e+"</mtd></mtr><mtr><mtd>"+b+"</mtd><mtd>"+d+"</mtd><mtd>"+f+"</mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr></mtable><mo>)</mo></mrow>"
       }

       function newTranslate(tx, ty)
       {
         // Add a translate.
         if (tx == 0 && ty == 0) return;
         MatrixDecomposition +=
           matrix(mn(1), mn(0), mn(0), mn(1), mn(tx), mn(ty));
         if (ty == 0) {
           SVGdecomposition += "translate(" + tx + ") ";
           CSSdecomposition += "translate(" + tx + "px) ";
         } else {
           SVGdecomposition += "translate(" + tx + ", " + ty + ") ";
           CSSdecomposition += "translate(" + tx + "px, " + ty + "px) ";
         }
       }

       function newScale(sx, sy)
       {
         // Add a scale.
         if (sx == 1 && sy == 1) return;
         MatrixDecomposition +=
           matrix(mn(sx), mn(0),
                  mn(0), mn(sy), mn(0), mn(0));
         var s = "scale(" + sx + (sx == sy ? "" : "," + sy) + ") ";
         SVGdecomposition += s;
         CSSdecomposition += s;
       }

       function newRotate(a)
       {
         // Add a rotation.
         if (a == 0) return "";
         MatrixDecomposition +=
             matrix(trig("cos", a), trig("sin", a),
                    trig("sin", -a), trig("cos", a), mn(0), mn(0));
         a = radToDeg(a);
         SVGdecomposition += "rotate(" + a + ") ";
         CSSdecomposition += "rotate(" + a + "deg) ";
       }

       function newSkewX(a)
       {
         // Add a skewX.
         if (a == 0) return;
         MatrixDecomposition +=
           matrix(mn(1), mn(0),
                  trig("tan", a), mn(1), mn(0), mn(0), mn(0));
         a = radToDeg(a);
         SVGdecomposition += "skewX(" + a + ") ";
         CSSdecomposition += "skewX(" + a + "deg) ";
       }

       function newSkewY(a)
       {
         // Add a skewY.
         if (a == 0) return;
         MatrixDecomposition +=
           matrix(mn(1), trig("tan", a),
                  mn(0), mn(1), mn(0), mn(0), mn(0));
         a = radToDeg(a);
         SVGdecomposition += "skewY(" + a + ") ";
         CSSdecomposition += "skewY(" + a + "deg) ";
       }
 
       function decompose()
       {
         // Verify if a CSS transform is available.
         if (!initTransformName())
           throw "Your browser does not support CSS transforms."

         // Apply the transform specified by the user.
         var cssRect1 = document.getElementById("cssRect1");
         var CSS2Dtransform = document.getElementById("CSS2Dtransform").value;
         cssRect1.style[TransformName] = "none";
         cssRect1.style[TransformName] = CSS2Dtransform;
         
         // Get the matrix computed by the rendering engine.
         var CSS2Dmatrix = getComputedStyle(cssRect1)[TransformName];
         var regexp = /matrix\((.*),(.*),(.*),(.*),(.*),(.*)\)/;
         var match = regexp.exec(CSS2Dmatrix);
         if (match === null)
            throw "Syntax Error. Please enter a valid CSS 2D transform."
         var a = parseFloat(match[1]);
         var b = parseFloat(match[2]);
         var c = parseFloat(match[3]);
         var d = parseFloat(match[4]);
         var e = parseFloat(match[5]);
         var f = parseFloat(match[6]);
         document.getElementById("CSS2Dmatrix").innerHTML = CSS2Dmatrix;
         document.getElementById("CSS2DmatrixMathML").innerHTML =
           "<math display='block'>" + 
           matrix(mn(a), mn(b), mn(c), mn(d), mn(e), mn(f)) + "</math>"

         // Apply the decomposition algorithm.
         CSSdecomposition = ""; SVGdecomposition = ""; MatrixDecomposition = "";
         newTranslate(e, f);

         var Delta = a * d - b * c;

         if (document.getElementById("decompo").value == "QR-like") {
           // Apply the QR-like decomposition.
           if (a != 0 || b != 0) {
             var r = Math.sqrt(a*a+b*b);
             newRotate(b > 0 ? Math.acos(a/r) : -Math.acos(a/r));
             newScale(r, Delta/r);
             newSkewX(Math.atan((a*c+b*d)/(r*r)));
           } else if (c != 0 || d != 0) {
             var s = Math.sqrt(c*c+d*d);
             newRotate(Math.PI/2 - (d > 0 ? Math.acos(-c/s) : -Math.acos(c/s)));
             newScale(Delta/s, s);
             newSkewY(Math.atan((a*c+b*d)/(s*s)));
           } else { // a = b = c = d = 0
             newScale(0, 0);
           }
         } else {
           // Apply the LU-like decomposition.
           if (a != 0) {
             newSkewY(Math.atan(b/a));
             newScale(a, Delta/a);
             newSkewX(Math.atan(c/a));
           } else if (b != 0) {
             newRotate(Math.PI / 2);
             newScale(b, Delta/b);
             newSkewX(Math.atan(d/b));
           } else { // a = b = 0
             newScale(c, d);
             newSkewX(Math.PI/4);
             newScale(0, 1);
           }
         }

         // Display something if the transform is the identity.
         if (MatrixDecomposition === "") {
           MatrixDecomposition =
             matrix(mn(1), mn(0), mn(0), mn(1), mn(0), mn(0));
           CSSdecomposition = SVGdecomposition = "scale(1)";
         }

         // Display the result.
         document.getElementById("SVGdecomposition").innerHTML =
           SVGdecomposition;
         document.getElementById("CSSdecomposition").innerHTML =
           CSSdecomposition;
         document.getElementById("MatrixDecomposition").innerHTML =
           "<math display='block'>"+MatrixDecomposition+"</math>"

         // Apply the (decomposed) transformation to the SVG and CSS elements.
         document.getElementById("svgRect").
           setAttribute("transform", SVGdecomposition);
         cssRect2.style[TransformName] = CSSdecomposition;
       }

       function run()
       {
         var error = document.getElementById("error");
         try {
           error.innerHTML = "";
           decompose();
         } catch (e) {
           error.innerHTML = e;
         }
       }

       function update(v)
       {
          if (v === "") return;
          document.getElementById("CSS2Dtransform").value = v;
          run();
       }

@breton Thank you for answering, really appreciate it. I can confirm that it works great (I tried the first solution). What a fun mathematical tool :)