1)Corrected the step calculation from chord to arc for propagation over |x*curv|...

2)the sin(x)->x approximation if g3helix3 will be used only for |x|<0.03 instead of 0.15

diff --git a/STEER/AliExternalTrackParam.cxx b/STEER/AliExternalTrackParam.cxx
index 2d2ad64ac9bb558b2c3cda3b1633938962cb4cb6..295830b47930627c00ffdc667aa60a650707ab91 100644 (file)
--- a/STEER/AliExternalTrackParam.cxx
+++ b/STEER/AliExternalTrackParam.cxx
@@ -672,7 +672,8 @@ Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
   Double_t crv=GetC(b);
   if (TMath::Abs(b) < kAlmost0Field) crv=0.;
 
-  Double_t f1=fP[2], f2=f1 + crv*dx;
+  Double_t x2r = crv*dx;
+  Double_t f1=fP[2], f2=f1 + x2r;
   if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
   if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
 
@@ -687,9 +688,23 @@ Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
   Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
 
   fX=xk;
-  fP0 += dx*(f1+f2)/(r1+r2);
-  fP1 += dx*(r2 + f2*(f1+f2)/(r1+r2))*fP3;  // Many thanks to P.Hristov !
-  fP2 += dx*crv;
+  double dy2dx = (f1+f2)/(r1+r2);
+  fP0 += dx*dy2dx;
+  if (TMath::Abs(x2r)<0.05) {
+    fP1 += dx*(r2 + f2*dy2dx)*fP3;  // Many thanks to P.Hristov !
+    fP2 += x2r;
+  }
+  else { 
+    // for small dx/R the linear apporximation of the arc by the segment is OK,
+    // but at large dx/R the error is very large and leads to incorrect Z propagation
+    // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
+    // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
+    // Similarly, the rotation angle in linear in dx only for dx<<R
+    double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx);   // distance from old position to new one
+    double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
+    fP1 += rot/crv*fP3;
+    fP2  = TMath::Sin(rot + TMath::ASin(fP2));
+  }
 
   //f = F - 1
    
@@ -1814,7 +1829,7 @@ void AliExternalTrackParam::g3helx3(Double_t qfield,
   Double_t tet = rho*step;
 
   Double_t tsint, sintt, sint, cos1t; 
-  if (TMath::Abs(tet) > 0.15) {
+  if (TMath::Abs(tet) > 0.03) {
      sint  = TMath::Sin(tet);
      sintt = sint/tet;
      tsint = (tet - sint)/tet;
@@ -1857,7 +1872,8 @@ Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]
   Double_t crv=GetC(b[2]);
   if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
 
-  Double_t f1=fP[2], f2=f1 + crv*dx;
+  Double_t x2r = crv*dx;
+  Double_t f1=fP[2], f2=f1 + x2r;
   if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
   if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
 
@@ -1915,10 +1931,11 @@ Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]
   CheckCovariance();
   
   // Appoximate step length
-  Double_t step=dx*TMath::Abs(r2 + f2*(f1+f2)/(r1+r2));
+  double dy2dx = (f1+f2)/(r1+r2);
+  Double_t step = (TMath::Abs(x2r)<0.05) ? dx*TMath::Abs(r2 + f2*dy2dx)  // chord
+    : 2.*TMath::ASin(0.5*dx*TMath::Sqrt(1.+dy2dx*dy2dx)*crv)/crv;        // arc
   step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
 
-
   // Get the track's (x,y,z) and (px,py,pz) in the Global System
   Double_t r[3]; GetXYZ(r);
   Double_t p[3]; GetPxPyPz(p);