+echo
+
+# determine trafo of cartesian coordinates to this one
+normx=`echo $x1 $x2 $x3 | awk '{ print sqrt($1*$1+$2*$2+$3*$3) }'`
+normy=`echo $y1 $y2 $y3 | awk '{ print sqrt($1*$1+$2*$2+$3*$3) }'`
+normz=`echo $z1 $z2 $z3 | awk '{ print sqrt($1*$1+$2*$2+$3*$3) }'`
+echo $normx $normz $normy
+t11=`echo $x1 $normx | awk '{ print $1/$2 }'`
+t12=`echo $x2 $normx | awk '{ print $1/$2 }'`
+t13=`echo $x3 $normx | awk '{ print $1/$2 }'`
+t21=`echo $y1 $normy | awk '{ print $1/$2 }'`
+t22=`echo $y2 $normy | awk '{ print $1/$2 }'`
+t23=`echo $y3 $normy | awk '{ print $1/$2 }'`
+t31=`echo $z1 $normz | awk '{ print $1/$2 }'`
+t32=`echo $z2 $normz | awk '{ print $1/$2 }'`
+t33=`echo $z3 $normz | awk '{ print $1/$2 }'`
+echo " Matrix from cartesian to used coordinates:"
+echo " | $t11 $t21 $t31 |"
+echo " | $t12 $t22 $t32 |"
+echo " | $t13 $t23 $t33 |"
+echo
+i11=$t11
+i12=$t21
+i13=$t31
+i21=$t12
+i22=$t22
+i23=$t32
+i31=$t13
+i32=$t23
+i33=$t33
+echo " Inverse matrix (i=t^T):"
+echo " | $i11 $i21 $i31 |"
+echo " | $i12 $i22 $i32 |"
+echo " | $i13 $i23 $i33 |"
+echo
+# i * basis
+X1=`echo $i11 $i21 $i31 $x1 $x2 $x3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+X2=`echo $i12 $i22 $i32 $x1 $x2 $x3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+X3=`echo $i13 $i23 $i33 $x1 $x2 $x3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+
+Y1=`echo $i11 $i21 $i31 $y1 $y2 $y3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+Y2=`echo $i12 $i22 $i32 $y1 $y2 $y3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+Y3=`echo $i13 $i23 $i33 $y1 $y2 $y3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+
+Z1=`echo $i11 $i21 $i31 $z1 $z2 $z3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+Z2=`echo $i12 $i22 $i32 $z1 $z2 $z3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+Z3=`echo $i13 $i23 $i33 $z1 $z2 $z3 | awk '{ print $1*$4+$2*$5+$3*$6 }'`
+echo " Basis (in rotation system):"
+echo " $X1 $Y1 $Z1"
+echo " x = $X2 y = $Y2 z = $Z2"
+echo " $X3 $Y3 $Z3"
+echo