Quaternion from transition matrix

Hi,

i need to calculate the quaternion from a transition matrix. What i name a transition matrix is the matrix that allow to express the coordinates of a vector in a first referential R1 (x,y,z) with the combinaison of vectors defining a second referential R2 => v2 = M . v1

I know coordinates of 3 points in R1 and the same points in R2, so i can calculate the matrix and it works fine : i can convert coordinates of any point from R1 to R2 and vice versa.

However, i need to convert this matrix in a transform, so in a translation and a rotation (the quaternion i'm searching).

The functions to convert one matrix4x4 to quaternion doesn't work because i think them works for rotation matrix, but here it's a transition matrix.

May be there is another solution from known coordinates of 3 points in two referential to calculate translation and rotation ?

I tryed with 2 successives rotations and even 3 successives rotations, but it's not exact each times : from the 3 points => calculate quaternion q1 to transform v1 to v1', calculate quaternion to transform q1.v2 to v2', and combine the 2 quaternions... and even tryed to do the same with v1,v2,v3...

Any idea please ?

here it's my code to calculate transition matrix : it's 3 équations systems with each 3 unkown variables, so i use matrix to solve each system in 3 times, result is matrix MatR1R2 (or MatR2R1) : (abc) (def) (ghi)

 // résolution du systèmes de 3 x 3 équations à 3 inconnues pour connaitre les coeff de la matrice de passage
 // matrix de passage : |abc0|
 //                                     |def0|
 //                                     |ghi0|
 //                                     |0001|
 
 Matrix4x4 m;
 Vector4 eq1,eq2,eq3;
 Vector4 abc, def, ghi;
 
 // les 3 coeff a,b,c
 eq1 = new Vector4(x2, y2, z2, x5);
 eq2 = new Vector4(x1, y1, z1, x4);
 eq3 = new Vector4(x, y, z, x3);
 m = assignMatrix(eq1, eq2, eq3);
 //Debug.Log("m pour abc =" + m.ToString("F3"));
 abc = solveSystem3(m);
 
 // les 3 coeff d,e,f
 eq1 = new Vector4(x2, y2, z2, y5);
 eq2 = new Vector4(x1, y1, z1, y4);
 eq3 = new Vector4(x, y, z, y3);
 m = assignMatrix(eq1, eq2, eq3);
 //Debug.Log("m pour def =" + m.ToString("F3"));
 def = solveSystem3(m);
 
 // les 3 coeff g,h,i
 eq1 = new Vector4(x2, y2, z2, z5);
 eq2 = new Vector4(x1, y1, z1, z4);
 eq3 = new Vector4(x, y, z, z3);
 m = assignMatrix(eq1, eq2, eq3);
 //Debug.Log("m pour ghi =" + m.ToString("F3"));
 ghi = solveSystem3(m);
 
 //Calcul matrice de passage du nouveau référentiel à l'ancien
 MatR2R1 = assignMatrix(abc, def, ghi);
 
 //Calcul matrice de passage de l'ancien référentiel au nouveau
 MatR1R2 = MatR2R1.inverse;
 
 Debug.Log(MatR1R2.ToString("F3"));
 
 // TEST
 vDelta1 = new Vector4(x2,y2,z2,0.0f);
 vDelta2 = new Vector4(x5, y5, z5, 0.0f);
 Debug.Log("vDelta1 : x=" + vDelta1.x.ToString("F3") + " y=" + vDelta1.y.ToString("F3") + " z=" + vDelta1.z.ToString("F3"));
 Debug.Log("vDelta2 : x=" + vDelta2.x.ToString("F3") + " y=" + vDelta2.y.ToString("F3") + " z=" + vDelta2.z.ToString("F3"));
 vDelta2 = MatR1R2 * vDelta2;
 Debug.Log("MatR1R2 * vDelta2 : x=" + vDelta2.x.ToString("F3") + " y=" + vDelta2.y.ToString("F3") + " z=" + vDelta2.z.ToString("F3"));
 
 The function to solve one system
     Vector4 solveSystem3 (Matrix4x4 system3)
     {
         Vector4 result = Vector3.zero;
         Matrix4x4 m = system3;
         Matrix4x4 m1 = system3;
         Matrix4x4 m2 = system3;
         Matrix4x4 m3 = system3;
         float d = 0.0f;
         float d1 = 0.0f;
         float d2 = 0.0f;
         float d3 = 0.0f;
 
         // on fixe la dernière colone et dernière ligne de chaque matrice
         m.SetColumn(3, new Vector4(0, 0, 0, 1));
         m.SetRow(3, new Vector4(0, 0, 0, 1));
         m1.SetColumn(3, new Vector4(0, 0, 0, 1));
         m1.SetRow(3, new Vector4(0, 0, 0, 1));
         m2.SetColumn(3, new Vector4(0, 0, 0, 1));
         m2.SetRow(3, new Vector4(0, 0, 0, 1));
         m3.SetColumn(3, new Vector4(0, 0, 0, 1));
         m3.SetRow(3, new Vector4(0, 0, 0, 1));
 
         // modifs de colone 0 pour la matrice m1
         m1[0, 0] = system3[0, 3];
         m1[1, 0] = system3[1, 3];
         m1[2, 0] = system3[2, 3];
 
         // modifs de colone 1 pour la matrice m2
         m2[0, 1] = system3[0, 3];
         m2[1, 1] = system3[1, 3];
         m2[2, 1] = system3[2, 3];
 
         // modifs de colone 2 pour la matrice m3
         m3[0, 2] = system3[0, 3];
         m3[1, 2] = system3[1, 3];
         m3[2, 2] = system3[2, 3];
 
         // calcul des déterminants
         d = m.determinant;
         d1 = m1.determinant;
         d2 = m2.determinant;
         d3 = m3.determinant;
 
         // calcul des coefficients résultats a,b,c
         result = new Vector4(d1 / d, d2 / d, d3 / d,0.0f);
 
         return result;
     }