1 /**************************************************************************
2 * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
4 * Author: The ALICE Off-line Project. *
5 * Contributors are mentioned in the code where appropriate. *
7 * Permission to use, copy, modify and distribute this software and its *
8 * documentation strictly for non-commercial purposes is hereby granted *
9 * without fee, provided that the above copyright notice appears in all *
10 * copies and that both the copyright notice and this permission notice *
11 * appear in the supporting documentation. The authors make no claims *
12 * about the suitability of this software for any purpose. It is *
13 * provided "as is" without express or implied warranty. *
14 **************************************************************************/
18 ///////////////////////////////////////////////////////////////////////////////
20 // Implementation of the external track parameterisation class. //
22 // This parameterisation is used to exchange tracks between the detectors. //
23 // A set of functions returning the position and the momentum of tracks //
24 // in the global coordinate system as well as the track impact parameters //
26 // Origin: I.Belikov, CERN, Jouri.Belikov@cern.ch //
27 ///////////////////////////////////////////////////////////////////////////////
31 #include <TMatrixDSym.h>
32 #include <TPolyMarker3D.h>
36 #include "AliExternalTrackParam.h"
37 #include "AliVVertex.h"
40 ClassImp(AliExternalTrackParam)
42 Double32_t AliExternalTrackParam::fgMostProbablePt=kMostProbablePt;
44 //_____________________________________________________________________________
45 AliExternalTrackParam::AliExternalTrackParam() :
51 // default constructor
53 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
54 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
57 //_____________________________________________________________________________
58 AliExternalTrackParam::AliExternalTrackParam(const AliExternalTrackParam &track):
66 for (Int_t i = 0; i < 5; i++) fP[i] = track.fP[i];
67 for (Int_t i = 0; i < 15; i++) fC[i] = track.fC[i];
71 //_____________________________________________________________________________
72 AliExternalTrackParam& AliExternalTrackParam::operator=(const AliExternalTrackParam &trkPar)
75 // assignment operator
79 AliVTrack::operator=(trkPar);
81 fAlpha = trkPar.fAlpha;
83 for (Int_t i = 0; i < 5; i++) fP[i] = trkPar.fP[i];
84 for (Int_t i = 0; i < 15; i++) fC[i] = trkPar.fC[i];
91 //_____________________________________________________________________________
92 AliExternalTrackParam::AliExternalTrackParam(Double_t x, Double_t alpha,
93 const Double_t param[5],
94 const Double_t covar[15]) :
100 // create external track parameters from given arguments
102 for (Int_t i = 0; i < 5; i++) fP[i] = param[i];
103 for (Int_t i = 0; i < 15; i++) fC[i] = covar[i];
107 //_____________________________________________________________________________
108 AliExternalTrackParam::AliExternalTrackParam(const AliVTrack *vTrack) :
114 // Constructor from virtual track,
115 // This is not a copy contructor !
118 if (vTrack->InheritsFrom("AliExternalTrackParam")) {
119 AliError("This is not a copy constructor. Use AliExternalTrackParam(const AliExternalTrackParam &) !");
120 AliWarning("Calling the default constructor...");
121 AliExternalTrackParam();
125 Double_t xyz[3],pxpypz[3],cv[21];
127 pxpypz[0]=vTrack->Px();
128 pxpypz[1]=vTrack->Py();
129 pxpypz[2]=vTrack->Pz();
130 vTrack->GetCovarianceXYZPxPyPz(cv);
131 Short_t sign = (Short_t)vTrack->Charge();
133 Set(xyz,pxpypz,cv,sign);
136 //_____________________________________________________________________________
137 AliExternalTrackParam::AliExternalTrackParam(Double_t xyz[3],Double_t pxpypz[3],
138 Double_t cv[21],Short_t sign) :
144 // constructor from the global parameters
147 Set(xyz,pxpypz,cv,sign);
150 //_____________________________________________________________________________
151 void AliExternalTrackParam::Set(Double_t xyz[3],Double_t pxpypz[3],
152 Double_t cv[21],Short_t sign)
155 // create external track parameters from the global parameters
156 // x,y,z,px,py,pz and their 6x6 covariance matrix
157 // A.Dainese 10.10.08
159 // Calculate alpha: the rotation angle of the corresponding local system.
161 // For global radial position inside the beam pipe, alpha is the
162 // azimuthal angle of the momentum projected on (x,y).
164 // For global radial position outside the ITS, alpha is the
165 // azimuthal angle of the centre of the TPC sector in which the point
168 Double_t radPos2 = xyz[0]*xyz[0]+xyz[1]*xyz[1];
169 Double_t radMax = 45.; // approximately ITS outer radius
170 if (radPos2 < radMax*radMax) { // inside the ITS
172 fAlpha = TMath::ATan2(pxpypz[1],pxpypz[0]);
173 } else { // outside the ITS
174 Float_t phiPos = TMath::Pi()+TMath::ATan2(-xyz[1], -xyz[0]);
176 TMath::DegToRad()*(20*((((Int_t)(phiPos*TMath::RadToDeg()))/20))+10);
179 // Get the vertex of origin and the momentum
180 TVector3 ver(xyz[0],xyz[1],xyz[2]);
181 TVector3 mom(pxpypz[0],pxpypz[1],pxpypz[2]);
183 // Rotate to the local coordinate system
184 ver.RotateZ(-fAlpha);
185 mom.RotateZ(-fAlpha);
187 // x of the reference plane
190 Double_t charge = (Double_t)sign;
194 fP[2] = TMath::Sin(mom.Phi());
195 fP[3] = mom.Pz()/mom.Pt();
196 fP[4] = TMath::Sign(1/mom.Pt(),charge);
198 // Covariance matrix (formulas to be simplified)
200 Double_t pt=1./TMath::Abs(fP[4]);
201 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
202 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
204 Double_t m00=-sn;// m10=cs;
205 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
206 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
207 Double_t m35=pt, m45=-pt*pt*fP[3];
213 Double_t cv34 = TMath::Sqrt(cv[3 ]*cv[3 ]+cv[4 ]*cv[4 ]);
214 Double_t a1=cv[13]-cv[9]*(m23*m44+m43*m24)/m23/m43;
215 Double_t a2=m23*m24-m23*(m23*m44+m43*m24)/m43;
216 Double_t a3=m43*m44-m43*(m23*m44+m43*m24)/m23;
217 Double_t a4=cv[14]-2.*cv[9]*m24*m44/m23/m43;
218 Double_t a5=m24*m24-2.*m24*m44*m23/m43;
219 Double_t a6=m44*m44-2.*m24*m44*m43/m23;
221 fC[0 ] = cv[0 ]+cv[2 ];
222 fC[1 ] = TMath::Sign(cv34,cv[3 ]/m00);
224 fC[3 ] = (cv[10]/m44-cv[6]/m43)/(m24/m44-m23/m43)/m00;
225 fC[10] = (cv[6]/m00-fC[3 ]*m23)/m43;
226 fC[6 ] = (cv[15]/m00-fC[10]*m45)/m35;
227 fC[4 ] = (cv[12]-cv[8]*m44/m43)/(m24-m23*m44/m43);
228 fC[11] = (cv[8]-fC[4]*m23)/m43;
229 fC[7 ] = cv[17]/m35-fC[11]*m45/m35;
230 fC[5 ] = TMath::Abs((a4-a6*a1/a3)/(a5-a6*a2/a3));
231 fC[14] = TMath::Abs(a1/a3-a2*fC[5]/a3);
232 fC[12] = (cv[9]-fC[5]*m23*m23-fC[14]*m43*m43)/m23/m43;
233 Double_t b1=cv[18]-fC[12]*m23*m45-fC[14]*m43*m45;
236 Double_t b4=cv[19]-fC[12]*m24*m45-fC[14]*m44*m45;
239 fC[8 ] = (b4-b6*b1/b3)/(b5-b6*b2/b3);
240 fC[13] = b1/b3-b2*fC[8]/b3;
241 fC[9 ] = TMath::Abs((cv[20]-fC[14]*(m45*m45)-fC[13]*2.*m35*m45)/(m35*m35));
248 //_____________________________________________________________________________
249 void AliExternalTrackParam::Reset() {
251 // Resets all the parameters to 0
254 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
255 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
258 //_____________________________________________________________________________
259 void AliExternalTrackParam::AddCovariance(const Double_t c[15]) {
261 // Add "something" to the track covarince matrix.
262 // May be needed to account for unknown mis-calibration/mis-alignment
265 fC[1] +=c[1]; fC[2] +=c[2];
266 fC[3] +=c[3]; fC[4] +=c[4]; fC[5] +=c[5];
267 fC[6] +=c[6]; fC[7] +=c[7]; fC[8] +=c[8]; fC[9] +=c[9];
268 fC[10]+=c[10]; fC[11]+=c[11]; fC[12]+=c[12]; fC[13]+=c[13]; fC[14]+=c[14];
273 Double_t AliExternalTrackParam::GetP() const {
274 //---------------------------------------------------------------------
275 // This function returns the track momentum
276 // Results for (nearly) straight tracks are meaningless !
277 //---------------------------------------------------------------------
278 if (TMath::Abs(fP[4])<=kAlmost0) return kVeryBig;
279 return TMath::Sqrt(1.+ fP[3]*fP[3])/TMath::Abs(fP[4]);
282 Double_t AliExternalTrackParam::Get1P() const {
283 //---------------------------------------------------------------------
284 // This function returns the 1/(track momentum)
285 //---------------------------------------------------------------------
286 return TMath::Abs(fP[4])/TMath::Sqrt(1.+ fP[3]*fP[3]);
289 //_______________________________________________________________________
290 Double_t AliExternalTrackParam::GetD(Double_t x,Double_t y,Double_t b) const {
291 //------------------------------------------------------------------
292 // This function calculates the transverse impact parameter
293 // with respect to a point with global coordinates (x,y)
294 // in the magnetic field "b" (kG)
295 //------------------------------------------------------------------
296 if (TMath::Abs(b) < kAlmost0Field) return GetLinearD(x,y);
297 Double_t rp4=GetC(b);
299 Double_t xt=fX, yt=fP[0];
301 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
302 Double_t a = x*cs + y*sn;
303 y = -x*sn + y*cs; x=a;
306 sn=rp4*xt - fP[2]; cs=rp4*yt + TMath::Sqrt((1.- fP[2])*(1.+fP[2]));
307 a=2*(xt*fP[2] - yt*TMath::Sqrt((1.-fP[2])*(1.+fP[2])))-rp4*(xt*xt + yt*yt);
308 return -a/(1 + TMath::Sqrt(sn*sn + cs*cs));
311 //_______________________________________________________________________
312 void AliExternalTrackParam::
313 GetDZ(Double_t x, Double_t y, Double_t z, Double_t b, Float_t dz[2]) const {
314 //------------------------------------------------------------------
315 // This function calculates the transverse and longitudinal impact parameters
316 // with respect to a point with global coordinates (x,y)
317 // in the magnetic field "b" (kG)
318 //------------------------------------------------------------------
319 Double_t f1 = fP[2], r1 = TMath::Sqrt((1.-f1)*(1.+f1));
320 Double_t xt=fX, yt=fP[0];
321 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
322 Double_t a = x*cs + y*sn;
323 y = -x*sn + y*cs; x=a;
326 Double_t rp4=GetC(b);
327 if ((TMath::Abs(b) < kAlmost0Field) || (TMath::Abs(rp4) < kAlmost0)) {
328 dz[0] = -(xt*f1 - yt*r1);
329 dz[1] = fP[1] + (dz[0]*f1 - xt)/r1*fP[3] - z;
333 sn=rp4*xt - f1; cs=rp4*yt + r1;
334 a=2*(xt*f1 - yt*r1)-rp4*(xt*xt + yt*yt);
335 Double_t rr=TMath::Sqrt(sn*sn + cs*cs);
337 Double_t f2 = -sn/rr, r2 = TMath::Sqrt((1.-f2)*(1.+f2));
338 dz[1] = fP[1] + fP[3]/rp4*TMath::ASin(f2*r1 - f1*r2) - z;
341 //_______________________________________________________________________
342 Double_t AliExternalTrackParam::GetLinearD(Double_t xv,Double_t yv) const {
343 //------------------------------------------------------------------
344 // This function calculates the transverse impact parameter
345 // with respect to a point with global coordinates (xv,yv)
346 // neglecting the track curvature.
347 //------------------------------------------------------------------
348 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
349 Double_t x= xv*cs + yv*sn;
350 Double_t y=-xv*sn + yv*cs;
352 Double_t d = (fX-x)*fP[2] - (fP[0]-y)*TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
357 Bool_t AliExternalTrackParam::CorrectForMeanMaterial
358 (Double_t xOverX0, Double_t xTimesRho, Double_t mass, Bool_t anglecorr,
359 Double_t (*Bethe)(Double_t)) {
360 //------------------------------------------------------------------
361 // This function corrects the track parameters for the crossed material.
362 // "xOverX0" - X/X0, the thickness in units of the radiation length.
363 // "xTimesRho" - is the product length*density (g/cm^2).
364 // "mass" - the mass of this particle (GeV/c^2).
365 //------------------------------------------------------------------
370 Double_t &fC22=fC[5];
371 Double_t &fC33=fC[9];
372 Double_t &fC43=fC[13];
373 Double_t &fC44=fC[14];
375 //Apply angle correction, if requested
377 Double_t angle=TMath::Sqrt((1.+ fP3*fP3)/((1-fP2)*(1.+fP2)));
384 Double_t beta2=p2/(p2 + mass*mass);
386 //Calculating the multiple scattering corrections******************
392 Double_t theta2=14.1*14.1/(beta2*p2*1e6)*TMath::Abs(xOverX0);
393 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
394 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
395 cC22 = theta2*((1.-fP2)*(1.+fP2))*(1. + fP3*fP3);
396 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
397 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
398 cC44 = theta2*fP3*fP4*fP3*fP4;
401 //Calculating the energy loss corrections************************
403 if ((xTimesRho != 0.) && (beta2 < 1.)) {
404 Double_t dE=Bethe(p/mass)*xTimesRho;
405 Double_t e=TMath::Sqrt(p2 + mass*mass);
406 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
408 if (TMath::Abs(fP4*cP4)>100.) return kFALSE; //Do not track below 10 MeV/c
411 // Approximate energy loss fluctuation (M.Ivanov)
412 const Double_t knst=0.07; // To be tuned.
413 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
414 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
418 //Applying the corrections*****************************
431 Bool_t AliExternalTrackParam::CorrectForMaterial
432 (Double_t d, Double_t x0, Double_t mass, Double_t (*Bethe)(Double_t)) {
433 //------------------------------------------------------------------
434 // Deprecated function !
435 // Better use CorrectForMeanMaterial instead of it.
437 // This function corrects the track parameters for the crossed material
438 // "d" - the thickness (fraction of the radiation length)
439 // "x0" - the radiation length (g/cm^2)
440 // "mass" - the mass of this particle (GeV/c^2)
441 //------------------------------------------------------------------
446 Double_t &fC22=fC[5];
447 Double_t &fC33=fC[9];
448 Double_t &fC43=fC[13];
449 Double_t &fC44=fC[14];
453 Double_t beta2=p2/(p2 + mass*mass);
454 d*=TMath::Sqrt((1.+ fP3*fP3)/((1.-fP2)*(1.+fP2)));
456 //Multiple scattering******************
462 Double_t theta2=14.1*14.1/(beta2*p2*1e6)*TMath::Abs(d);
463 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
464 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
465 cC22 = theta2*(1.-fP2)*(1.+fP2)*(1. + fP3*fP3);
466 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
467 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
468 cC44 = theta2*fP3*fP4*fP3*fP4;
471 //Energy losses************************
473 if (x0!=0. && beta2<1) {
475 Double_t dE=Bethe(p/mass)*d;
476 Double_t e=TMath::Sqrt(p2 + mass*mass);
477 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
480 // Approximate energy loss fluctuation (M.Ivanov)
481 const Double_t knst=0.07; // To be tuned.
482 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
483 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
498 Double_t AliExternalTrackParam::BetheBlochAleph(Double_t bg,
505 // This is the empirical ALEPH parameterization of the Bethe-Bloch formula.
506 // It is normalized to 1 at the minimum.
510 // The default values for the kp* parameters are for ALICE TPC.
511 // The returned value is in MIP units
514 Double_t beta = bg/TMath::Sqrt(1.+ bg*bg);
516 Double_t aa = TMath::Power(beta,kp4);
517 Double_t bb = TMath::Power(1./bg,kp5);
519 bb=TMath::Log(kp3+bb);
521 return (kp2-aa-bb)*kp1/aa;
524 Double_t AliExternalTrackParam::BetheBlochGeant(Double_t bg,
531 // This is the parameterization of the Bethe-Bloch formula inspired by Geant.
534 // kp0 - density [g/cm^3]
535 // kp1 - density effect first junction point
536 // kp2 - density effect second junction point
537 // kp3 - mean excitation energy [GeV]
540 // The default values for the kp* parameters are for silicon.
541 // The returned value is in [GeV/(g/cm^2)].
544 const Double_t mK = 0.307075e-3; // [GeV*cm^2/g]
545 const Double_t me = 0.511e-3; // [GeV/c^2]
546 const Double_t rho = kp0;
547 const Double_t x0 = kp1*2.303;
548 const Double_t x1 = kp2*2.303;
549 const Double_t mI = kp3;
550 const Double_t mZA = kp4;
551 const Double_t bg2 = bg*bg;
552 const Double_t maxT= 2*me*bg2; // neglecting the electron mass
556 const Double_t x=TMath::Log(bg);
557 const Double_t lhwI=TMath::Log(28.816*1e-9*TMath::Sqrt(rho*mZA)/mI);
561 const Double_t r=(x1-x)/(x1-x0);
562 d2 = lhwI + x - 0.5 + (0.5 - lhwI - x0)*r*r*r;
565 return mK*mZA*(1+bg2)/bg2*
566 (0.5*TMath::Log(2*me*bg2*maxT/(mI*mI)) - bg2/(1+bg2) - d2);
569 Double_t AliExternalTrackParam::BetheBlochSolid(Double_t bg) {
570 //------------------------------------------------------------------
571 // This is an approximation of the Bethe-Bloch formula,
572 // reasonable for solid materials.
573 // All the parameters are, in fact, for Si.
574 // The returned value is in [GeV/(g/cm^2)]
575 //------------------------------------------------------------------
577 return BetheBlochGeant(bg);
580 Double_t AliExternalTrackParam::BetheBlochGas(Double_t bg) {
581 //------------------------------------------------------------------
582 // This is an approximation of the Bethe-Bloch formula,
583 // reasonable for gas materials.
584 // All the parameters are, in fact, for Ne.
585 // The returned value is in [GeV/(g/cm^2)]
586 //------------------------------------------------------------------
588 const Double_t rho = 0.9e-3;
589 const Double_t x0 = 2.;
590 const Double_t x1 = 4.;
591 const Double_t mI = 140.e-9;
592 const Double_t mZA = 0.49555;
594 return BetheBlochGeant(bg,rho,x0,x1,mI,mZA);
597 Bool_t AliExternalTrackParam::Rotate(Double_t alpha) {
598 //------------------------------------------------------------------
599 // Transform this track to the local coord. system rotated
600 // by angle "alpha" (rad) with respect to the global coord. system.
601 //------------------------------------------------------------------
602 if (TMath::Abs(fP[2]) >= kAlmost1) {
603 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
607 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
608 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
612 Double_t &fC00=fC[0];
613 Double_t &fC10=fC[1];
614 Double_t &fC20=fC[3];
615 Double_t &fC21=fC[4];
616 Double_t &fC22=fC[5];
617 Double_t &fC30=fC[6];
618 Double_t &fC32=fC[8];
619 Double_t &fC40=fC[10];
620 Double_t &fC42=fC[12];
623 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
624 Double_t sf=fP2, cf=TMath::Sqrt((1.- fP2)*(1.+fP2)); // Improve precision
626 Double_t tmp=sf*ca - cf*sa;
627 if (TMath::Abs(tmp) >= kAlmost1) {
628 if (TMath::Abs(tmp) > 1.+ Double_t(FLT_EPSILON))
629 AliWarning(Form("Rotation failed ! %.10e",tmp));
638 if (TMath::Abs(cf)<kAlmost0) {
639 AliError(Form("Too small cosine value %f",cf));
643 Double_t rr=(ca+sf/cf*sa);
660 Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
661 //----------------------------------------------------------------
662 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
663 //----------------------------------------------------------------
665 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
667 Double_t crv=GetC(b);
668 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
670 Double_t f1=fP[2], f2=f1 + crv*dx;
671 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
672 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
674 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
677 &fC10=fC[1], &fC11=fC[2],
678 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
679 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
680 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
682 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
685 fP0 += dx*(f1+f2)/(r1+r2);
686 fP1 += dx*(r2 + f2*(f1+f2)/(r1+r2))*fP3; // Many thanks to P.Hristov !
691 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
692 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
693 Double_t f12= dx*fP3*f1/(r1*r1*r1);
694 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
696 Double_t f24= dx; f24*=cc;
699 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
700 Double_t b02=f24*fC40;
701 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
702 Double_t b12=f24*fC41;
703 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
704 Double_t b22=f24*fC42;
705 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
706 Double_t b42=f24*fC44;
707 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
708 Double_t b32=f24*fC43;
711 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
712 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
713 Double_t a22=f24*b42;
715 //F*C*Ft = C + (b + bt + a)
716 fC00 += b00 + b00 + a00;
717 fC10 += b10 + b01 + a01;
718 fC20 += b20 + b02 + a02;
721 fC11 += b11 + b11 + a11;
722 fC21 += b21 + b12 + a12;
725 fC22 += b22 + b22 + a22;
735 AliExternalTrackParam::Propagate(Double_t alpha, Double_t x, Double_t b) {
736 //------------------------------------------------------------------
737 // Transform this track to the local coord. system rotated
738 // by angle "alpha" (rad) with respect to the global coord. system,
739 // and propagate this track to the plane X=xk (cm) in the field "b" (kG)
740 //------------------------------------------------------------------
742 //Save the parameters
745 Double_t ps[5], cs[15];
746 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
747 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
750 if (PropagateTo(x,b)) return kTRUE;
752 //Restore the parameters, if the operation failed
755 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
756 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
760 Bool_t AliExternalTrackParam::PropagateBxByBz
761 (Double_t alpha, Double_t x, Double_t b[3]) {
762 //------------------------------------------------------------------
763 // Transform this track to the local coord. system rotated
764 // by angle "alpha" (rad) with respect to the global coord. system,
765 // and propagate this track to the plane X=xk (cm),
766 // taking into account all three components of the B field, "b[3]" (kG)
767 //------------------------------------------------------------------
769 //Save the parameters
772 Double_t ps[5], cs[15];
773 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
774 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
777 if (PropagateToBxByBz(x,b)) return kTRUE;
779 //Restore the parameters, if the operation failed
782 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
783 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
788 void AliExternalTrackParam::Propagate(Double_t len, Double_t x[3],
789 Double_t p[3], Double_t bz) const {
790 //+++++++++++++++++++++++++++++++++++++++++
791 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
792 // Extrapolate track along simple helix in magnetic field
793 // Arguments: len -distance alogn helix, [cm]
794 // bz - mag field, [kGaus]
795 // Returns: x and p contain extrapolated positon and momentum
796 // The momentum returned for straight-line tracks is meaningless !
797 //+++++++++++++++++++++++++++++++++++++++++
800 if (OneOverPt() < kAlmost0 || TMath::Abs(bz) < kAlmost0Field || GetC(bz) < kAlmost0){ //straight-line tracks
801 Double_t unit[3]; GetDirection(unit);
806 p[0]=unit[0]/kAlmost0;
807 p[1]=unit[1]/kAlmost0;
808 p[2]=unit[2]/kAlmost0;
812 Double_t a = -kB2C*bz*GetSign();
814 x[0] += p[0]*TMath::Sin(rho*len)/a - p[1]*(1-TMath::Cos(rho*len))/a;
815 x[1] += p[1]*TMath::Sin(rho*len)/a + p[0]*(1-TMath::Cos(rho*len))/a;
819 p[0] = p0 *TMath::Cos(rho*len) - p[1]*TMath::Sin(rho*len);
820 p[1] = p[1]*TMath::Cos(rho*len) + p0 *TMath::Sin(rho*len);
824 Bool_t AliExternalTrackParam::Intersect(Double_t pnt[3], Double_t norm[3],
826 //+++++++++++++++++++++++++++++++++++++++++
827 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
828 // Finds point of intersection (if exists) of the helix with the plane.
829 // Stores result in fX and fP.
830 // Arguments: planePoint,planeNorm - the plane defined by any plane's point
831 // and vector, normal to the plane
832 // Returns: kTrue if helix intersects the plane, kFALSE otherwise.
833 //+++++++++++++++++++++++++++++++++++++++++
834 Double_t x0[3]; GetXYZ(x0); //get track position in MARS
836 //estimates initial helix length up to plane
838 (pnt[0]-x0[0])*norm[0] + (pnt[1]-x0[1])*norm[1] + (pnt[2]-x0[2])*norm[2];
839 Double_t dist=99999,distPrev=dist;
841 while(TMath::Abs(dist)>0.00001){
842 //calculates helix at the distance s from x0 ALONG the helix
845 //distance between current helix position and plane
846 dist=(x[0]-pnt[0])*norm[0]+(x[1]-pnt[1])*norm[1]+(x[2]-pnt[2])*norm[2];
848 if(TMath::Abs(dist) >= TMath::Abs(distPrev)) {return kFALSE;}
852 //on exit pnt is intersection point,norm is track vector at that point,
854 for (Int_t i=0; i<3; i++) {pnt[i]=x[i]; norm[i]=p[i];}
859 AliExternalTrackParam::GetPredictedChi2(Double_t p[2],Double_t cov[3]) const {
860 //----------------------------------------------------------------
861 // Estimate the chi2 of the space point "p" with the cov. matrix "cov"
862 //----------------------------------------------------------------
863 Double_t sdd = fC[0] + cov[0];
864 Double_t sdz = fC[1] + cov[1];
865 Double_t szz = fC[2] + cov[2];
866 Double_t det = sdd*szz - sdz*sdz;
868 if (TMath::Abs(det) < kAlmost0) return kVeryBig;
870 Double_t d = fP[0] - p[0];
871 Double_t z = fP[1] - p[1];
873 return (d*szz*d - 2*d*sdz*z + z*sdd*z)/det;
876 Double_t AliExternalTrackParam::
877 GetPredictedChi2(Double_t p[3],Double_t covyz[3],Double_t covxyz[3]) const {
878 //----------------------------------------------------------------
879 // Estimate the chi2 of the 3D space point "p" and
880 // the full covariance matrix "covyz" and "covxyz"
882 // Cov(x,x) ... : covxyz[0]
883 // Cov(y,x) ... : covxyz[1] covyz[0]
884 // Cov(z,x) ... : covxyz[2] covyz[1] covyz[2]
885 //----------------------------------------------------------------
894 if (TMath::Abs(f) >= kAlmost1) return kVeryBig;
895 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
896 Double_t a=f/r, b=GetTgl()/r;
898 Double_t s2=333.*333.; //something reasonably big (cm^2)
901 v(0,0)= s2; v(0,1)= a*s2; v(0,2)= b*s2;;
902 v(1,0)=a*s2; v(1,1)=a*a*s2 + GetSigmaY2(); v(1,2)=a*b*s2 + GetSigmaZY();
903 v(2,0)=b*s2; v(2,1)=a*b*s2 + GetSigmaZY(); v(2,2)=b*b*s2 + GetSigmaZ2();
905 v(0,0)+=covxyz[0]; v(0,1)+=covxyz[1]; v(0,2)+=covxyz[2];
906 v(1,0)+=covxyz[1]; v(1,1)+=covyz[0]; v(1,2)+=covyz[1];
907 v(2,0)+=covxyz[2]; v(2,1)+=covyz[1]; v(2,2)+=covyz[2];
910 if (!v.IsValid()) return kVeryBig;
913 for (Int_t i = 0; i < 3; i++)
914 for (Int_t j = 0; j < 3; j++) chi2 += res[i]*res[j]*v(i,j);
919 Double_t AliExternalTrackParam::
920 GetPredictedChi2(const AliExternalTrackParam *t) const {
921 //----------------------------------------------------------------
922 // Estimate the chi2 (5 dof) of this track with respect to the track
923 // given by the argument.
924 // The two tracks must be in the same reference system
925 // and estimated at the same reference plane.
926 //----------------------------------------------------------------
928 if (TMath::Abs(1. - t->GetAlpha()/GetAlpha()) > FLT_EPSILON) {
929 AliError("The reference systems of the tracks differ !");
932 if (TMath::Abs(1. - t->GetX()/GetX()) > FLT_EPSILON) {
933 AliError("The reference of the tracks planes differ !");
939 c(1,0)=GetSigmaZY(); c(1,1)=GetSigmaZ2();
940 c(2,0)=GetSigmaSnpY(); c(2,1)=GetSigmaSnpZ(); c(2,2)=GetSigmaSnp2();
941 c(3,0)=GetSigmaTglY(); c(3,1)=GetSigmaTglZ(); c(3,2)=GetSigmaTglSnp(); c(3,3)=GetSigmaTgl2();
942 c(4,0)=GetSigma1PtY(); c(4,1)=GetSigma1PtZ(); c(4,2)=GetSigma1PtSnp(); c(4,3)=GetSigma1PtTgl(); c(4,4)=GetSigma1Pt2();
944 c(0,0)+=t->GetSigmaY2();
945 c(1,0)+=t->GetSigmaZY(); c(1,1)+=t->GetSigmaZ2();
946 c(2,0)+=t->GetSigmaSnpY();c(2,1)+=t->GetSigmaSnpZ();c(2,2)+=t->GetSigmaSnp2();
947 c(3,0)+=t->GetSigmaTglY();c(3,1)+=t->GetSigmaTglZ();c(3,2)+=t->GetSigmaTglSnp();c(3,3)+=t->GetSigmaTgl2();
948 c(4,0)+=t->GetSigma1PtY();c(4,1)+=t->GetSigma1PtZ();c(4,2)+=t->GetSigma1PtSnp();c(4,3)+=t->GetSigma1PtTgl();c(4,4)+=t->GetSigma1Pt2();
950 c(0,2)=c(2,0); c(1,2)=c(2,1);
951 c(0,3)=c(3,0); c(1,3)=c(3,1); c(2,3)=c(3,2);
952 c(0,4)=c(4,0); c(1,4)=c(4,1); c(2,4)=c(4,2); c(3,4)=c(4,3);
955 if (!c.IsValid()) return kVeryBig;
961 GetSnp() - t->GetSnp(),
962 GetTgl() - t->GetTgl(),
963 GetSigned1Pt() - t->GetSigned1Pt()
967 for (Int_t i = 0; i < 5; i++)
968 for (Int_t j = 0; j < 5; j++) chi2 += res[i]*res[j]*c(i,j);
973 Bool_t AliExternalTrackParam::
974 PropagateTo(Double_t p[3],Double_t covyz[3],Double_t covxyz[3],Double_t bz) {
975 //----------------------------------------------------------------
976 // Propagate this track to the plane
977 // the 3D space point "p" (with the covariance matrix "covyz" and "covxyz")
979 // The magnetic field is "bz" (kG)
981 // The track curvature and the change of the covariance matrix
982 // of the track parameters are negleted !
983 // (So the "step" should be small compared with 1/curvature)
984 //----------------------------------------------------------------
987 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
988 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
989 Double_t a=f/r, b=GetTgl()/r;
991 Double_t s2=333.*333.; //something reasonably big (cm^2)
994 tV(0,0)= s2; tV(0,1)= a*s2; tV(0,2)= b*s2;
995 tV(1,0)=a*s2; tV(1,1)=a*a*s2; tV(1,2)=a*b*s2;
996 tV(2,0)=b*s2; tV(2,1)=a*b*s2; tV(2,2)=b*b*s2;
999 pV(0,0)=covxyz[0]; pV(0,1)=covxyz[1]; pV(0,2)=covxyz[2];
1000 pV(1,0)=covxyz[1]; pV(1,1)=covyz[0]; pV(1,2)=covyz[1];
1001 pV(2,0)=covxyz[2]; pV(2,1)=covyz[1]; pV(2,2)=covyz[2];
1003 TMatrixDSym tpV(tV);
1006 if (!tpV.IsValid()) return kFALSE;
1008 TMatrixDSym pW(3),tW(3);
1009 for (Int_t i=0; i<3; i++)
1010 for (Int_t j=0; j<3; j++) {
1012 for (Int_t k=0; k<3; k++) {
1013 pW(i,j) += tV(i,k)*tpV(k,j);
1014 tW(i,j) += pV(i,k)*tpV(k,j);
1018 Double_t t[3] = {GetX(), GetY(), GetZ()};
1021 for (Int_t i=0; i<3; i++) x += (tW(0,i)*t[i] + pW(0,i)*p[i]);
1022 Double_t crv=GetC(bz);
1023 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1025 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1029 for (Int_t i=0; i<3; i++) fP[0] += (tW(1,i)*t[i] + pW(1,i)*p[i]);
1031 for (Int_t i=0; i<3; i++) fP[1] += (tW(2,i)*t[i] + pW(2,i)*p[i]);
1036 Double_t *AliExternalTrackParam::GetResiduals(
1037 Double_t *p,Double_t *cov,Bool_t updated) const {
1038 //------------------------------------------------------------------
1039 // Returns the track residuals with the space point "p" having
1040 // the covariance matrix "cov".
1041 // If "updated" is kTRUE, the track parameters expected to be updated,
1042 // otherwise they must be predicted.
1043 //------------------------------------------------------------------
1044 static Double_t res[2];
1046 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1048 r00-=fC[0]; r01-=fC[1]; r11-=fC[2];
1050 r00+=fC[0]; r01+=fC[1]; r11+=fC[2];
1052 Double_t det=r00*r11 - r01*r01;
1054 if (TMath::Abs(det) < kAlmost0) return 0;
1056 Double_t tmp=r00; r00=r11/det; r11=tmp/det;
1058 if (r00 < 0.) return 0;
1059 if (r11 < 0.) return 0;
1061 Double_t dy = fP[0] - p[0];
1062 Double_t dz = fP[1] - p[1];
1064 res[0]=dy*TMath::Sqrt(r00);
1065 res[1]=dz*TMath::Sqrt(r11);
1070 Bool_t AliExternalTrackParam::Update(Double_t p[2], Double_t cov[3]) {
1071 //------------------------------------------------------------------
1072 // Update the track parameters with the space point "p" having
1073 // the covariance matrix "cov"
1074 //------------------------------------------------------------------
1075 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
1078 &fC10=fC[1], &fC11=fC[2],
1079 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1080 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1081 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1083 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1084 r00+=fC00; r01+=fC10; r11+=fC11;
1085 Double_t det=r00*r11 - r01*r01;
1087 if (TMath::Abs(det) < kAlmost0) return kFALSE;
1090 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
1092 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
1093 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
1094 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
1095 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
1096 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
1098 Double_t dy=p[0] - fP0, dz=p[1] - fP1;
1099 Double_t sf=fP2 + k20*dy + k21*dz;
1100 if (TMath::Abs(sf) > kAlmost1) return kFALSE;
1102 fP0 += k00*dy + k01*dz;
1103 fP1 += k10*dy + k11*dz;
1105 fP3 += k30*dy + k31*dz;
1106 fP4 += k40*dy + k41*dz;
1108 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
1109 Double_t c12=fC21, c13=fC31, c14=fC41;
1111 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
1112 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
1113 fC40-=k00*c04+k01*c14;
1115 fC11-=k10*c01+k11*fC11;
1116 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
1117 fC41-=k10*c04+k11*c14;
1119 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
1120 fC42-=k20*c04+k21*c14;
1122 fC33-=k30*c03+k31*c13;
1123 fC43-=k30*c04+k31*c14;
1125 fC44-=k40*c04+k41*c14;
1133 AliExternalTrackParam::GetHelixParameters(Double_t hlx[6], Double_t b) const {
1134 //--------------------------------------------------------------------
1135 // External track parameters -> helix parameters
1136 // "b" - magnetic field (kG)
1137 //--------------------------------------------------------------------
1138 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1140 hlx[0]=fP[0]; hlx[1]=fP[1]; hlx[2]=fP[2]; hlx[3]=fP[3];
1142 hlx[5]=fX*cs - hlx[0]*sn; // x0
1143 hlx[0]=fX*sn + hlx[0]*cs; // y0
1145 hlx[2]=TMath::ASin(hlx[2]) + fAlpha; // phi0
1147 hlx[4]=GetC(b); // C
1151 static void Evaluate(const Double_t *h, Double_t t,
1152 Double_t r[3], //radius vector
1153 Double_t g[3], //first defivatives
1154 Double_t gg[3]) //second derivatives
1156 //--------------------------------------------------------------------
1157 // Calculate position of a point on a track and some derivatives
1158 //--------------------------------------------------------------------
1159 Double_t phase=h[4]*t+h[2];
1160 Double_t sn=TMath::Sin(phase), cs=TMath::Cos(phase);
1164 if (TMath::Abs(h[4])>kAlmost0) {
1165 r[0] += (sn - h[6])/h[4];
1166 r[1] -= (cs - h[7])/h[4];
1168 r[2] = h[1] + h[3]*t;
1170 g[0] = cs; g[1]=sn; g[2]=h[3];
1172 gg[0]=-h[4]*sn; gg[1]=h[4]*cs; gg[2]=0.;
1175 Double_t AliExternalTrackParam::GetDCA(const AliExternalTrackParam *p,
1176 Double_t b, Double_t &xthis, Double_t &xp) const {
1177 //------------------------------------------------------------
1178 // Returns the (weighed !) distance of closest approach between
1179 // this track and the track "p".
1180 // Other returned values:
1181 // xthis, xt - coordinates of tracks' reference planes at the DCA
1182 //-----------------------------------------------------------
1183 Double_t dy2=GetSigmaY2() + p->GetSigmaY2();
1184 Double_t dz2=GetSigmaZ2() + p->GetSigmaZ2();
1187 Double_t p1[8]; GetHelixParameters(p1,b);
1188 p1[6]=TMath::Sin(p1[2]); p1[7]=TMath::Cos(p1[2]);
1189 Double_t p2[8]; p->GetHelixParameters(p2,b);
1190 p2[6]=TMath::Sin(p2[2]); p2[7]=TMath::Cos(p2[2]);
1193 Double_t r1[3],g1[3],gg1[3]; Double_t t1=0.;
1194 Evaluate(p1,t1,r1,g1,gg1);
1195 Double_t r2[3],g2[3],gg2[3]; Double_t t2=0.;
1196 Evaluate(p2,t2,r2,g2,gg2);
1198 Double_t dx=r2[0]-r1[0], dy=r2[1]-r1[1], dz=r2[2]-r1[2];
1199 Double_t dm=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1203 Double_t gt1=-(dx*g1[0]/dx2 + dy*g1[1]/dy2 + dz*g1[2]/dz2);
1204 Double_t gt2=+(dx*g2[0]/dx2 + dy*g2[1]/dy2 + dz*g2[2]/dz2);
1205 Double_t h11=(g1[0]*g1[0] - dx*gg1[0])/dx2 +
1206 (g1[1]*g1[1] - dy*gg1[1])/dy2 +
1207 (g1[2]*g1[2] - dz*gg1[2])/dz2;
1208 Double_t h22=(g2[0]*g2[0] + dx*gg2[0])/dx2 +
1209 (g2[1]*g2[1] + dy*gg2[1])/dy2 +
1210 (g2[2]*g2[2] + dz*gg2[2])/dz2;
1211 Double_t h12=-(g1[0]*g2[0]/dx2 + g1[1]*g2[1]/dy2 + g1[2]*g2[2]/dz2);
1213 Double_t det=h11*h22-h12*h12;
1216 if (TMath::Abs(det)<1.e-33) {
1217 //(quasi)singular Hessian
1220 dt1=-(gt1*h22 - gt2*h12)/det;
1221 dt2=-(h11*gt2 - h12*gt1)/det;
1224 if ((dt1*gt1+dt2*gt2)>0) {dt1=-dt1; dt2=-dt2;}
1226 //check delta(phase1) ?
1227 //check delta(phase2) ?
1229 if (TMath::Abs(dt1)/(TMath::Abs(t1)+1.e-3) < 1.e-4)
1230 if (TMath::Abs(dt2)/(TMath::Abs(t2)+1.e-3) < 1.e-4) {
1231 if ((gt1*gt1+gt2*gt2) > 1.e-4/dy2/dy2)
1232 AliDebug(1," stopped at not a stationary point !");
1233 Double_t lmb=h11+h22; lmb=lmb-TMath::Sqrt(lmb*lmb-4*det);
1235 AliDebug(1," stopped at not a minimum !");
1240 for (Int_t div=1 ; ; div*=2) {
1241 Evaluate(p1,t1+dt1,r1,g1,gg1);
1242 Evaluate(p2,t2+dt2,r2,g2,gg2);
1243 dx=r2[0]-r1[0]; dy=r2[1]-r1[1]; dz=r2[2]-r1[2];
1244 dd=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1248 AliDebug(1," overshoot !"); break;
1258 if (max<=0) AliDebug(1," too many iterations !");
1260 Double_t cs=TMath::Cos(GetAlpha());
1261 Double_t sn=TMath::Sin(GetAlpha());
1262 xthis=r1[0]*cs + r1[1]*sn;
1264 cs=TMath::Cos(p->GetAlpha());
1265 sn=TMath::Sin(p->GetAlpha());
1266 xp=r2[0]*cs + r2[1]*sn;
1268 return TMath::Sqrt(dm*TMath::Sqrt(dy2*dz2));
1271 Double_t AliExternalTrackParam::
1272 PropagateToDCA(AliExternalTrackParam *p, Double_t b) {
1273 //--------------------------------------------------------------
1274 // Propagates this track and the argument track to the position of the
1275 // distance of closest approach.
1276 // Returns the (weighed !) distance of closest approach.
1277 //--------------------------------------------------------------
1279 Double_t dca=GetDCA(p,b,xthis,xp);
1281 if (!PropagateTo(xthis,b)) {
1282 //AliWarning(" propagation failed !");
1286 if (!p->PropagateTo(xp,b)) {
1287 //AliWarning(" propagation failed !";
1295 Bool_t AliExternalTrackParam::PropagateToDCA(const AliVVertex *vtx,
1296 Double_t b, Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1298 // Propagate this track to the DCA to vertex "vtx",
1299 // if the (rough) transverse impact parameter is not bigger then "maxd".
1300 // Magnetic field is "b" (kG).
1302 // a) The track gets extapolated to the DCA to the vertex.
1303 // b) The impact parameters and their covariance matrix are calculated.
1305 // In the case of success, the returned value is kTRUE
1306 // (otherwise, it's kFALSE)
1308 Double_t alpha=GetAlpha();
1309 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1310 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1311 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1312 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1315 //Estimate the impact parameter neglecting the track curvature
1316 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1317 if (d > maxd) return kFALSE;
1319 //Propagate to the DCA
1320 Double_t crv=GetC(b);
1321 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1323 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1324 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1325 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1329 yv=-xv*sn + yv*cs; xv=x;
1331 if (!Propagate(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1333 if (dz==0) return kTRUE;
1334 dz[0] = GetParameter()[0] - yv;
1335 dz[1] = GetParameter()[1] - zv;
1337 if (covar==0) return kTRUE;
1338 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1340 //***** Improvements by A.Dainese
1341 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1342 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1343 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1344 covar[1] = GetCovariance()[1]; // between (x,y) and z
1345 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1351 Bool_t AliExternalTrackParam::PropagateToDCABxByBz(const AliVVertex *vtx,
1352 Double_t b[3], Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1354 // Propagate this track to the DCA to vertex "vtx",
1355 // if the (rough) transverse impact parameter is not bigger then "maxd".
1357 // This function takes into account all three components of the magnetic
1358 // field given by the b[3] arument (kG)
1360 // a) The track gets extapolated to the DCA to the vertex.
1361 // b) The impact parameters and their covariance matrix are calculated.
1363 // In the case of success, the returned value is kTRUE
1364 // (otherwise, it's kFALSE)
1366 Double_t alpha=GetAlpha();
1367 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1368 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1369 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1370 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1373 //Estimate the impact parameter neglecting the track curvature
1374 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1375 if (d > maxd) return kFALSE;
1377 //Propagate to the DCA
1378 Double_t crv=GetC(b[2]);
1379 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1381 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1382 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1383 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1387 yv=-xv*sn + yv*cs; xv=x;
1389 if (!PropagateBxByBz(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1391 if (dz==0) return kTRUE;
1392 dz[0] = GetParameter()[0] - yv;
1393 dz[1] = GetParameter()[1] - zv;
1395 if (covar==0) return kTRUE;
1396 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1398 //***** Improvements by A.Dainese
1399 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1400 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1401 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1402 covar[1] = GetCovariance()[1]; // between (x,y) and z
1403 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1409 void AliExternalTrackParam::GetDirection(Double_t d[3]) const {
1410 //----------------------------------------------------------------
1411 // This function returns a unit vector along the track direction
1412 // in the global coordinate system.
1413 //----------------------------------------------------------------
1414 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1416 Double_t csp =TMath::Sqrt((1.-snp)*(1.+snp));
1417 Double_t norm=TMath::Sqrt(1.+ fP[3]*fP[3]);
1418 d[0]=(csp*cs - snp*sn)/norm;
1419 d[1]=(snp*cs + csp*sn)/norm;
1423 Bool_t AliExternalTrackParam::GetPxPyPz(Double_t p[3]) const {
1424 //---------------------------------------------------------------------
1425 // This function returns the global track momentum components
1426 // Results for (nearly) straight tracks are meaningless !
1427 //---------------------------------------------------------------------
1428 p[0]=fP[4]; p[1]=fP[2]; p[2]=fP[3];
1429 return Local2GlobalMomentum(p,fAlpha);
1432 Double_t AliExternalTrackParam::Px() const {
1433 //---------------------------------------------------------------------
1434 // Returns x-component of momentum
1435 // Result for (nearly) straight tracks is meaningless !
1436 //---------------------------------------------------------------------
1438 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1444 Double_t AliExternalTrackParam::Py() const {
1445 //---------------------------------------------------------------------
1446 // Returns y-component of momentum
1447 // Result for (nearly) straight tracks is meaningless !
1448 //---------------------------------------------------------------------
1450 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1456 Double_t AliExternalTrackParam::Xv() const {
1457 //---------------------------------------------------------------------
1458 // Returns x-component of first track point
1459 //---------------------------------------------------------------------
1461 Double_t r[3]={0.,0.,0.};
1467 Double_t AliExternalTrackParam::Yv() const {
1468 //---------------------------------------------------------------------
1469 // Returns y-component of first track point
1470 //---------------------------------------------------------------------
1472 Double_t r[3]={0.,0.,0.};
1478 Double_t AliExternalTrackParam::Theta() const {
1479 // return theta angle of momentum
1481 return 0.5*TMath::Pi() - TMath::ATan(fP[3]);
1484 Double_t AliExternalTrackParam::Phi() const {
1485 //---------------------------------------------------------------------
1486 // Returns the azimuthal angle of momentum
1488 //---------------------------------------------------------------------
1490 Double_t phi=TMath::ASin(fP[2]) + fAlpha;
1491 if (phi<0.) phi+=2.*TMath::Pi();
1492 else if (phi>=2.*TMath::Pi()) phi-=2.*TMath::Pi();
1497 Double_t AliExternalTrackParam::M() const {
1498 // return particle mass
1500 // No mass information available so far.
1501 // Redifine in derived class!
1506 Double_t AliExternalTrackParam::E() const {
1507 // return particle energy
1509 // No PID information available so far.
1510 // Redifine in derived class!
1515 Double_t AliExternalTrackParam::Eta() const {
1516 // return pseudorapidity
1518 return -TMath::Log(TMath::Tan(0.5 * Theta()));
1521 Double_t AliExternalTrackParam::Y() const {
1524 // No PID information available so far.
1525 // Redifine in derived class!
1530 Bool_t AliExternalTrackParam::GetXYZ(Double_t *r) const {
1531 //---------------------------------------------------------------------
1532 // This function returns the global track position
1533 //---------------------------------------------------------------------
1534 r[0]=fX; r[1]=fP[0]; r[2]=fP[1];
1535 return Local2GlobalPosition(r,fAlpha);
1538 Bool_t AliExternalTrackParam::GetCovarianceXYZPxPyPz(Double_t cv[21]) const {
1539 //---------------------------------------------------------------------
1540 // This function returns the global covariance matrix of the track params
1542 // Cov(x,x) ... : cv[0]
1543 // Cov(y,x) ... : cv[1] cv[2]
1544 // Cov(z,x) ... : cv[3] cv[4] cv[5]
1545 // Cov(px,x)... : cv[6] cv[7] cv[8] cv[9]
1546 // Cov(py,x)... : cv[10] cv[11] cv[12] cv[13] cv[14]
1547 // Cov(pz,x)... : cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
1549 // Results for (nearly) straight tracks are meaningless !
1550 //---------------------------------------------------------------------
1551 if (TMath::Abs(fP[4])<=kAlmost0) {
1552 for (Int_t i=0; i<21; i++) cv[i]=0.;
1555 if (TMath::Abs(fP[2]) > kAlmost1) {
1556 for (Int_t i=0; i<21; i++) cv[i]=0.;
1559 Double_t pt=1./TMath::Abs(fP[4]);
1560 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1561 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
1563 Double_t m00=-sn, m10=cs;
1564 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
1565 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
1566 Double_t m35=pt, m45=-pt*pt*fP[3];
1572 cv[0 ] = fC[0]*m00*m00;
1573 cv[1 ] = fC[0]*m00*m10;
1574 cv[2 ] = fC[0]*m10*m10;
1578 cv[6 ] = m00*(fC[3]*m23 + fC[10]*m43);
1579 cv[7 ] = m10*(fC[3]*m23 + fC[10]*m43);
1580 cv[8 ] = fC[4]*m23 + fC[11]*m43;
1581 cv[9 ] = m23*(fC[5]*m23 + fC[12]*m43) + m43*(fC[12]*m23 + fC[14]*m43);
1582 cv[10] = m00*(fC[3]*m24 + fC[10]*m44);
1583 cv[11] = m10*(fC[3]*m24 + fC[10]*m44);
1584 cv[12] = fC[4]*m24 + fC[11]*m44;
1585 cv[13] = m23*(fC[5]*m24 + fC[12]*m44) + m43*(fC[12]*m24 + fC[14]*m44);
1586 cv[14] = m24*(fC[5]*m24 + fC[12]*m44) + m44*(fC[12]*m24 + fC[14]*m44);
1587 cv[15] = m00*(fC[6]*m35 + fC[10]*m45);
1588 cv[16] = m10*(fC[6]*m35 + fC[10]*m45);
1589 cv[17] = fC[7]*m35 + fC[11]*m45;
1590 cv[18] = m23*(fC[8]*m35 + fC[12]*m45) + m43*(fC[13]*m35 + fC[14]*m45);
1591 cv[19] = m24*(fC[8]*m35 + fC[12]*m45) + m44*(fC[13]*m35 + fC[14]*m45);
1592 cv[20] = m35*(fC[9]*m35 + fC[13]*m45) + m45*(fC[13]*m35 + fC[14]*m45);
1599 AliExternalTrackParam::GetPxPyPzAt(Double_t x, Double_t b, Double_t *p) const {
1600 //---------------------------------------------------------------------
1601 // This function returns the global track momentum extrapolated to
1602 // the radial position "x" (cm) in the magnetic field "b" (kG)
1603 //---------------------------------------------------------------------
1605 p[1]=fP[2]+(x-fX)*GetC(b);
1607 return Local2GlobalMomentum(p,fAlpha);
1611 AliExternalTrackParam::GetYAt(Double_t x, Double_t b, Double_t &y) const {
1612 //---------------------------------------------------------------------
1613 // This function returns the local Y-coordinate of the intersection
1614 // point between this track and the reference plane "x" (cm).
1615 // Magnetic field "b" (kG)
1616 //---------------------------------------------------------------------
1618 if(TMath::Abs(dx)<=kAlmost0) {y=fP[0]; return kTRUE;}
1620 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1622 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1623 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1625 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1626 y = fP[0] + dx*(f1+f2)/(r1+r2);
1631 AliExternalTrackParam::GetZAt(Double_t x, Double_t b, Double_t &z) const {
1632 //---------------------------------------------------------------------
1633 // This function returns the local Z-coordinate of the intersection
1634 // point between this track and the reference plane "x" (cm).
1635 // Magnetic field "b" (kG)
1636 //---------------------------------------------------------------------
1638 if(TMath::Abs(dx)<=kAlmost0) {z=fP[1]; return kTRUE;}
1640 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1642 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1643 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1645 Double_t r1=sqrt((1.-f1)*(1.+f1)), r2=sqrt((1.-f2)*(1.+f2));
1646 z = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3]; // Many thanks to P.Hristov !
1651 AliExternalTrackParam::GetXYZAt(Double_t x, Double_t b, Double_t *r) const {
1652 //---------------------------------------------------------------------
1653 // This function returns the global track position extrapolated to
1654 // the radial position "x" (cm) in the magnetic field "b" (kG)
1655 //---------------------------------------------------------------------
1657 if(TMath::Abs(dx)<=kAlmost0) return GetXYZ(r);
1659 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1661 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1662 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1664 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1666 r[1] = fP[0] + dx*(f1+f2)/(r1+r2);
1667 r[2] = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3];//Thanks to Andrea & Peter
1669 return Local2GlobalPosition(r,fAlpha);
1672 //_____________________________________________________________________________
1673 void AliExternalTrackParam::Print(Option_t* /*option*/) const
1675 // print the parameters and the covariance matrix
1677 printf("AliExternalTrackParam: x = %-12g alpha = %-12g\n", fX, fAlpha);
1678 printf(" parameters: %12g %12g %12g %12g %12g\n",
1679 fP[0], fP[1], fP[2], fP[3], fP[4]);
1680 printf(" covariance: %12g\n", fC[0]);
1681 printf(" %12g %12g\n", fC[1], fC[2]);
1682 printf(" %12g %12g %12g\n", fC[3], fC[4], fC[5]);
1683 printf(" %12g %12g %12g %12g\n",
1684 fC[6], fC[7], fC[8], fC[9]);
1685 printf(" %12g %12g %12g %12g %12g\n",
1686 fC[10], fC[11], fC[12], fC[13], fC[14]);
1689 Double_t AliExternalTrackParam::GetSnpAt(Double_t x,Double_t b) const {
1691 // Get sinus at given x
1693 Double_t crv=GetC(b);
1694 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1696 Double_t res = fP[2]+dx*crv;
1700 Bool_t AliExternalTrackParam::GetDistance(AliExternalTrackParam *param2, Double_t x, Double_t dist[3], Double_t bz){
1701 //------------------------------------------------------------------------
1702 // Get the distance between two tracks at the local position x
1703 // working in the local frame of this track.
1704 // Origin : Marian.Ivanov@cern.ch
1705 //-----------------------------------------------------------------------
1709 if (!GetYAt(x,bz,xyz[1])) return kFALSE;
1710 if (!GetZAt(x,bz,xyz[2])) return kFALSE;
1713 if (TMath::Abs(GetAlpha()-param2->GetAlpha())<kAlmost0){
1715 if (!param2->GetYAt(x,bz,xyz2[1])) return kFALSE;
1716 if (!param2->GetZAt(x,bz,xyz2[2])) return kFALSE;
1720 Double_t dfi = param2->GetAlpha()-GetAlpha();
1721 Double_t ca = TMath::Cos(dfi), sa = TMath::Sin(dfi);
1722 xyz2[0] = xyz[0]*ca+xyz[1]*sa;
1723 xyz2[1] = -xyz[0]*sa+xyz[1]*ca;
1726 if (!param2->GetYAt(xyz2[0],bz,xyz1[1])) return kFALSE;
1727 if (!param2->GetZAt(xyz2[0],bz,xyz1[2])) return kFALSE;
1729 xyz2[0] = xyz1[0]*ca-xyz1[1]*sa;
1730 xyz2[1] = +xyz1[0]*sa+xyz1[1]*ca;
1733 dist[0] = xyz[0]-xyz2[0];
1734 dist[1] = xyz[1]-xyz2[1];
1735 dist[2] = xyz[2]-xyz2[2];
1742 // Draw functionality.
1743 // Origin: Marian Ivanov, Marian.Ivanov@cern.ch
1746 void AliExternalTrackParam::DrawTrack(Float_t magf, Float_t minR, Float_t maxR, Float_t stepR){
1750 if (minR>maxR) return ;
1751 if (stepR<=0) return ;
1752 Int_t npoints = TMath::Nint((maxR-minR)/stepR)+1;
1753 if (npoints<1) return;
1754 TPolyMarker3D *polymarker = new TPolyMarker3D(npoints);
1755 FillPolymarker(polymarker, magf,minR,maxR,stepR);
1760 void AliExternalTrackParam::FillPolymarker(TPolyMarker3D *pol, Float_t magF, Float_t minR, Float_t maxR, Float_t stepR){
1762 // Fill points in the polymarker
1765 for (Double_t r=minR; r<maxR; r+=stepR){
1767 GetXYZAt(r,magF,point);
1768 pol->SetPoint(counter,point[0],point[1], point[2]);
1769 printf("xyz\t%f\t%f\t%f\n",point[0], point[1],point[2]);
1774 Int_t AliExternalTrackParam::GetIndex(Int_t i, Int_t j) const {
1776 Int_t min = TMath::Min(i,j);
1777 Int_t max = TMath::Max(i,j);
1779 return min+(max+1)*max/2;
1783 void AliExternalTrackParam::g3helx3(Double_t qfield,
1786 /******************************************************************
1788 * GEANT3 tracking routine in a constant field oriented *
1790 * Tracking is performed with a conventional *
1791 * helix step method *
1793 * Authors R.Brun, M.Hansroul ********* *
1794 * Rewritten V.Perevoztchikov *
1796 * Rewritten in C++ by I.Belikov *
1798 * qfield (kG) - particle charge times magnetic field *
1799 * step (cm) - step length along the helix *
1800 * vect[7](cm,GeV/c) - input/output x, y, z, px/p, py/p ,pz/p, p *
1802 ******************************************************************/
1803 const Int_t ix=0, iy=1, iz=2, ipx=3, ipy=4, ipz=5, ipp=6;
1804 const Double_t kOvSqSix=TMath::Sqrt(1./6.);
1806 Double_t cosx=vect[ipx], cosy=vect[ipy], cosz=vect[ipz];
1808 Double_t rho = qfield*kB2C/vect[ipp];
1809 Double_t tet = rho*step;
1811 Double_t tsint, sintt, sint, cos1t;
1812 if (TMath::Abs(tet) > 0.15) {
1813 sint = TMath::Sin(tet);
1815 tsint = (tet - sint)/tet;
1816 Double_t t=TMath::Sin(0.5*tet);
1820 sintt = (1.-tet*kOvSqSix)*(1.+tet*kOvSqSix); // 1.- tsint;
1825 Double_t f1 = step*sintt;
1826 Double_t f2 = step*cos1t;
1827 Double_t f3 = step*tsint*cosz;
1828 Double_t f4 = -tet*cos1t;
1831 vect[ix] += f1*cosx - f2*cosy;
1832 vect[iy] += f1*cosy + f2*cosx;
1833 vect[iz] += f1*cosz + f3;
1835 vect[ipx] += f4*cosx - f5*cosy;
1836 vect[ipy] += f4*cosy + f5*cosx;
1840 Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]) {
1841 //----------------------------------------------------------------
1842 // Extrapolate this track to the plane X=xk in the field b[].
1844 // X [cm] is in the "tracking coordinate system" of this track.
1845 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
1846 //----------------------------------------------------------------
1849 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
1851 Double_t crv=GetC(b[2]);
1852 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1854 Double_t f1=fP[2], f2=f1 + crv*dx;
1855 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1856 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1859 // Estimate the covariance matrix
1860 Double_t &fP3=fP[3], &fP4=fP[4];
1863 &fC10=fC[1], &fC11=fC[2],
1864 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1865 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1866 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1868 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1871 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
1872 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
1873 Double_t f12= dx*fP3*f1/(r1*r1*r1);
1874 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
1875 Double_t f13= dx/r1;
1876 Double_t f24= dx; f24*=cc;
1879 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
1880 Double_t b02=f24*fC40;
1881 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
1882 Double_t b12=f24*fC41;
1883 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
1884 Double_t b22=f24*fC42;
1885 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
1886 Double_t b42=f24*fC44;
1887 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
1888 Double_t b32=f24*fC43;
1891 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
1892 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
1893 Double_t a22=f24*b42;
1895 //F*C*Ft = C + (b + bt + a)
1896 fC00 += b00 + b00 + a00;
1897 fC10 += b10 + b01 + a01;
1898 fC20 += b20 + b02 + a02;
1901 fC11 += b11 + b11 + a11;
1902 fC21 += b21 + b12 + a12;
1905 fC22 += b22 + b22 + a22;
1911 // Appoximate step length
1912 Double_t step=dx*TMath::Abs(r2 + f2*(f1+f2)/(r1+r2));
1913 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
1916 // Get the track's (x,y,z) and (px,py,pz) in the Global System
1917 Double_t r[3]; GetXYZ(r);
1918 Double_t p[3]; GetPxPyPz(p);
1925 // Rotate to the system where Bx=By=0.
1926 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
1927 Double_t cosphi=1., sinphi=0.;
1928 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
1929 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
1930 Double_t costet=1., sintet=0.;
1931 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
1934 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
1935 vect[1] = -sinphi*r[0] + cosphi*r[1];
1936 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
1938 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
1939 vect[4] = -sinphi*p[0] + cosphi*p[1];
1940 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
1945 // Do the helix step
1946 g3helx3(GetSign()*bb,step,vect);
1949 // Rotate back to the Global System
1950 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
1951 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
1952 r[2] = -sintet*vect[0] + costet*vect[2];
1954 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
1955 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
1956 p[2] = -sintet*vect[3] + costet*vect[5];
1959 // Rotate back to the Tracking System
1960 Double_t cosalp = TMath::Cos(fAlpha);
1961 Double_t sinalp =-TMath::Sin(fAlpha);
1964 t = cosalp*r[0] - sinalp*r[1];
1965 r[1] = sinalp*r[0] + cosalp*r[1];
1968 t = cosalp*p[0] - sinalp*p[1];
1969 p[1] = sinalp*p[0] + cosalp*p[1];
1973 // Do the final correcting step to the target plane (linear approximation)
1974 Double_t x=r[0], y=r[1], z=r[2];
1975 if (TMath::Abs(dx) > kAlmost0) {
1976 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
1984 // Calculate the track parameters
1985 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
1991 fP[4] = GetSign()/(t*pp);
1996 Bool_t AliExternalTrackParam::Translate(Double_t *vTrasl,Double_t *covV){
1998 //Translation: in the event mixing, the tracks can be shifted
1999 //of the difference among primary vertices (vTrasl) and
2000 //the covariance matrix is changed accordingly
2001 //(covV = covariance of the primary vertex).
2002 //Origin: "Romita, Rossella" <R.Romita@gsi.de>
2004 TVector3 translation;
2005 // vTrasl coordinates in the local system
2006 translation.SetXYZ(vTrasl[0],vTrasl[1],vTrasl[2]);
2007 translation.RotateZ(-fAlpha);
2008 translation.GetXYZ(vTrasl);
2010 //compute the new x,y,z of the track
2011 Double_t newX=fX-vTrasl[0];
2012 Double_t newY=fP[0]-vTrasl[1];
2013 Double_t newZ=fP[1]-vTrasl[2];
2015 //define the new parameters
2016 Double_t newParam[5];
2023 // recompute the covariance matrix:
2024 // 1. covV in the local system
2025 Double_t cosRot=TMath::Cos(fAlpha), sinRot=TMath::Sin(fAlpha);
2046 if(uUi.Determinant() <= 0.) {return kFALSE;}
2047 TMatrixD uUiQi(uUi,TMatrixD::kMult,qQi);
2048 TMatrixD m(qQi,TMatrixD::kTransposeMult,uUiQi);
2050 //2. compute the new covariance matrix of the track
2051 Double_t sigmaXX=m(0,0);
2052 Double_t sigmaXZ=m(2,0);
2053 Double_t sigmaXY=m(1,0);
2054 Double_t sigmaYY=GetSigmaY2()+m(1,1);
2055 Double_t sigmaYZ=fC[1]+m(1,2);
2056 Double_t sigmaZZ=fC[2]+m(2,2);
2057 Double_t covarianceYY=sigmaYY + (-1.)*((sigmaXY*sigmaXY)/sigmaXX);
2058 Double_t covarianceYZ=sigmaYZ-(sigmaXZ*sigmaXY/sigmaXX);
2059 Double_t covarianceZZ=sigmaZZ-((sigmaXZ*sigmaXZ)/sigmaXX);
2061 Double_t newCov[15];
2062 newCov[0]=covarianceYY;
2063 newCov[1]=covarianceYZ;
2064 newCov[2]=covarianceZZ;
2065 for(Int_t i=3;i<15;i++){
2069 // set the new parameters
2071 Set(newX,fAlpha,newParam,newCov);
2076 void AliExternalTrackParam::CheckCovariance() {
2078 // This function forces the diagonal elements of the covariance matrix to be positive.
2079 // In case the diagonal element is bigger than the maximal allowed value, it is set to
2080 // the limit and the off-diagonal elements that correspond to it are set to zero.
2082 fC[0] = TMath::Abs(fC[0]);
2090 fC[2] = TMath::Abs(fC[2]);
2098 fC[5] = TMath::Abs(fC[5]);
2106 fC[9] = TMath::Abs(fC[9]);
2114 fC[14] = TMath::Abs(fC[14]);
2115 if (fC[14]>kC14max) {
2123 // The part below is used for tests and normally is commented out
2124 // TMatrixDSym m(5);
2128 // m(1,0)=fC[1]; m(1,1)=fC[2];
2129 // m(2,0)=fC[3]; m(2,1)=fC[4]; m(2,2)=fC[5];
2130 // m(3,0)=fC[6]; m(3,1)=fC[7]; m(3,2)=fC[8]; m(3,3)=fC[9];
2131 // m(4,0)=fC[10]; m(4,1)=fC[11]; m(4,2)=fC[12]; m(4,3)=fC[13]; m(4,4)=fC[14];
2134 // m(0,2)=m(2,0); m(1,2)=m(2,1);
2135 // m(0,3)=m(3,0); m(1,3)=m(3,1); m(2,3)=m(3,2);
2136 // m(0,4)=m(4,0); m(1,4)=m(4,1); m(2,4)=m(4,2); m(3,4)=m(4,3);
2137 // m.EigenVectors(eig);
2139 // // assert(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0);
2140 // if (!(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0)) {
2141 // AliWarning("Negative eigenvalues of the covariance matrix!");