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 *
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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;
43 Bool_t AliExternalTrackParam::fgUseLogTermMS = kFALSE;;
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 void AliExternalTrackParam::CopyFromVTrack(const AliVTrack *vTrack)
111 // Recreate TrackParams from VTrack
112 // This is not a copy contructor !
115 AliError("Source VTrack is NULL");
119 AliError("Copy of itself is requested");
123 if (vTrack->InheritsFrom(AliExternalTrackParam::Class())) {
124 AliDebug(1,"Source itself is AliExternalTrackParam, using assignment operator");
125 *this = *(AliExternalTrackParam*)vTrack;
129 AliVTrack::operator=( *vTrack );
131 Double_t xyz[3],pxpypz[3],cv[21];
133 pxpypz[0]=vTrack->Px();
134 pxpypz[1]=vTrack->Py();
135 pxpypz[2]=vTrack->Pz();
136 vTrack->GetCovarianceXYZPxPyPz(cv);
137 Short_t sign = (Short_t)vTrack->Charge();
138 Set(xyz,pxpypz,cv,sign);
141 //_____________________________________________________________________________
142 AliExternalTrackParam::AliExternalTrackParam(const AliVTrack *vTrack) :
148 // Constructor from virtual track,
149 // This is not a copy contructor !
152 if (vTrack->InheritsFrom("AliExternalTrackParam")) {
153 AliError("This is not a copy constructor. Use AliExternalTrackParam(const AliExternalTrackParam &) !");
154 AliWarning("Calling the default constructor...");
155 AliExternalTrackParam();
159 Double_t xyz[3],pxpypz[3],cv[21];
161 pxpypz[0]=vTrack->Px();
162 pxpypz[1]=vTrack->Py();
163 pxpypz[2]=vTrack->Pz();
164 vTrack->GetCovarianceXYZPxPyPz(cv);
165 Short_t sign = (Short_t)vTrack->Charge();
167 Set(xyz,pxpypz,cv,sign);
170 //_____________________________________________________________________________
171 AliExternalTrackParam::AliExternalTrackParam(Double_t xyz[3],Double_t pxpypz[3],
172 Double_t cv[21],Short_t sign) :
178 // constructor from the global parameters
181 Set(xyz,pxpypz,cv,sign);
184 //_____________________________________________________________________________
185 void AliExternalTrackParam::Set(Double_t xyz[3],Double_t pxpypz[3],
186 Double_t cv[21],Short_t sign)
189 // create external track parameters from the global parameters
190 // x,y,z,px,py,pz and their 6x6 covariance matrix
191 // A.Dainese 10.10.08
193 // Calculate alpha: the rotation angle of the corresponding local system.
195 // For global radial position inside the beam pipe, alpha is the
196 // azimuthal angle of the momentum projected on (x,y).
198 // For global radial position outside the ITS, alpha is the
199 // azimuthal angle of the centre of the TPC sector in which the point
202 const double kSafe = 1e-5;
203 Double_t radPos2 = xyz[0]*xyz[0]+xyz[1]*xyz[1];
204 Double_t radMax = 45.; // approximately ITS outer radius
205 if (radPos2 < radMax*radMax) { // inside the ITS
206 fAlpha = TMath::ATan2(pxpypz[1],pxpypz[0]);
207 } else { // outside the ITS
208 Float_t phiPos = TMath::Pi()+TMath::ATan2(-xyz[1], -xyz[0]);
210 TMath::DegToRad()*(20*((((Int_t)(phiPos*TMath::RadToDeg()))/20))+10);
213 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
214 // protection: avoid alpha being too close to 0 or +-pi/2
215 if (TMath::Abs(sn)<kSafe) {
216 if (fAlpha>0) fAlpha += fAlpha< TMath::Pi()/2. ? kSafe : -kSafe;
217 else fAlpha += fAlpha>-TMath::Pi()/2. ? -kSafe : kSafe;
218 cs=TMath::Cos(fAlpha);
219 sn=TMath::Sin(fAlpha);
221 else if (TMath::Abs(cs)<kSafe) {
222 if (fAlpha>0) fAlpha += fAlpha> TMath::Pi()/2. ? kSafe : -kSafe;
223 else fAlpha += fAlpha>-TMath::Pi()/2. ? kSafe : -kSafe;
224 cs=TMath::Cos(fAlpha);
225 sn=TMath::Sin(fAlpha);
227 // Get the vertex of origin and the momentum
228 TVector3 ver(xyz[0],xyz[1],xyz[2]);
229 TVector3 mom(pxpypz[0],pxpypz[1],pxpypz[2]);
231 // avoid momenta along axis
232 if (TMath::Abs(mom[0])<kSafe) mom[0] = TMath::Sign(kSafe*TMath::Abs(mom[1]), mom[0]);
233 if (TMath::Abs(mom[1])<kSafe) mom[1] = TMath::Sign(kSafe*TMath::Abs(mom[0]), mom[1]);
235 // Rotate to the local coordinate system
236 ver.RotateZ(-fAlpha);
237 mom.RotateZ(-fAlpha);
239 // x of the reference plane
242 Double_t charge = (Double_t)sign;
246 fP[2] = TMath::Sin(mom.Phi());
247 fP[3] = mom.Pz()/mom.Pt();
248 fP[4] = TMath::Sign(1/mom.Pt(),charge);
250 // Covariance matrix (formulas to be simplified)
252 if (TMath::Abs( 1-fP[2]) < kSafe) fP[2] = 1.- kSafe; //Protection
253 else if (TMath::Abs(-1-fP[2]) < kSafe) fP[2] =-1.+ kSafe; //Protection
255 Double_t pt=1./TMath::Abs(fP[4]);
256 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
258 Double_t m00=-sn;// m10=cs;
259 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
260 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
261 Double_t m35=pt, m45=-pt*pt*fP[3];
267 Double_t cv34 = TMath::Sqrt(cv[3 ]*cv[3 ]+cv[4 ]*cv[4 ]);
268 Double_t a1=cv[13]-cv[9]*(m23*m44+m43*m24)/m23/m43;
269 Double_t a2=m23*m24-m23*(m23*m44+m43*m24)/m43;
270 Double_t a3=m43*m44-m43*(m23*m44+m43*m24)/m23;
271 Double_t a4=cv[14]-2.*cv[9]*m24*m44/m23/m43;
272 Double_t a5=m24*m24-2.*m24*m44*m23/m43;
273 Double_t a6=m44*m44-2.*m24*m44*m43/m23;
275 fC[0 ] = cv[0 ]+cv[2 ];
276 fC[1 ] = TMath::Sign(cv34,cv[3 ]/m00);
278 fC[3 ] = (cv[10]/m44-cv[6]/m43)/(m24/m44-m23/m43)/m00;
279 fC[10] = (cv[6]/m00-fC[3 ]*m23)/m43;
280 fC[6 ] = (cv[15]/m00-fC[10]*m45)/m35;
281 fC[4 ] = (cv[12]-cv[8]*m44/m43)/(m24-m23*m44/m43);
282 fC[11] = (cv[8]-fC[4]*m23)/m43;
283 fC[7 ] = cv[17]/m35-fC[11]*m45/m35;
284 fC[5 ] = TMath::Abs((a4-a6*a1/a3)/(a5-a6*a2/a3));
285 fC[14] = TMath::Abs(a1/a3-a2*fC[5]/a3);
286 fC[12] = (cv[9]-fC[5]*m23*m23-fC[14]*m43*m43)/m23/m43;
287 Double_t b1=cv[18]-fC[12]*m23*m45-fC[14]*m43*m45;
290 Double_t b4=cv[19]-fC[12]*m24*m45-fC[14]*m44*m45;
293 fC[8 ] = (b4-b6*b1/b3)/(b5-b6*b2/b3);
294 fC[13] = b1/b3-b2*fC[8]/b3;
295 fC[9 ] = TMath::Abs((cv[20]-fC[14]*(m45*m45)-fC[13]*2.*m35*m45)/(m35*m35));
302 //_____________________________________________________________________________
303 void AliExternalTrackParam::Reset() {
305 // Resets all the parameters to 0
308 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
309 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
312 //_____________________________________________________________________________
313 void AliExternalTrackParam::AddCovariance(const Double_t c[15]) {
315 // Add "something" to the track covarince matrix.
316 // May be needed to account for unknown mis-calibration/mis-alignment
319 fC[1] +=c[1]; fC[2] +=c[2];
320 fC[3] +=c[3]; fC[4] +=c[4]; fC[5] +=c[5];
321 fC[6] +=c[6]; fC[7] +=c[7]; fC[8] +=c[8]; fC[9] +=c[9];
322 fC[10]+=c[10]; fC[11]+=c[11]; fC[12]+=c[12]; fC[13]+=c[13]; fC[14]+=c[14];
327 Double_t AliExternalTrackParam::GetP() const {
328 //---------------------------------------------------------------------
329 // This function returns the track momentum
330 // Results for (nearly) straight tracks are meaningless !
331 //---------------------------------------------------------------------
332 if (TMath::Abs(fP[4])<=kAlmost0) return kVeryBig;
333 return TMath::Sqrt(1.+ fP[3]*fP[3])/TMath::Abs(fP[4]);
336 Double_t AliExternalTrackParam::Get1P() const {
337 //---------------------------------------------------------------------
338 // This function returns the 1/(track momentum)
339 //---------------------------------------------------------------------
340 return TMath::Abs(fP[4])/TMath::Sqrt(1.+ fP[3]*fP[3]);
343 //_______________________________________________________________________
344 Double_t AliExternalTrackParam::GetD(Double_t x,Double_t y,Double_t b) const {
345 //------------------------------------------------------------------
346 // This function calculates the transverse impact parameter
347 // with respect to a point with global coordinates (x,y)
348 // in the magnetic field "b" (kG)
349 //------------------------------------------------------------------
350 if (TMath::Abs(b) < kAlmost0Field) return GetLinearD(x,y);
351 Double_t rp4=GetC(b);
353 Double_t xt=fX, yt=fP[0];
355 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
356 Double_t a = x*cs + y*sn;
357 y = -x*sn + y*cs; x=a;
360 sn=rp4*xt - fP[2]; cs=rp4*yt + TMath::Sqrt((1.- fP[2])*(1.+fP[2]));
361 a=2*(xt*fP[2] - yt*TMath::Sqrt((1.-fP[2])*(1.+fP[2])))-rp4*(xt*xt + yt*yt);
362 return -a/(1 + TMath::Sqrt(sn*sn + cs*cs));
365 //_______________________________________________________________________
366 void AliExternalTrackParam::
367 GetDZ(Double_t x, Double_t y, Double_t z, Double_t b, Float_t dz[2]) const {
368 //------------------------------------------------------------------
369 // This function calculates the transverse and longitudinal impact parameters
370 // with respect to a point with global coordinates (x,y)
371 // in the magnetic field "b" (kG)
372 //------------------------------------------------------------------
373 Double_t f1 = fP[2], r1 = TMath::Sqrt((1.-f1)*(1.+f1));
374 Double_t xt=fX, yt=fP[0];
375 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
376 Double_t a = x*cs + y*sn;
377 y = -x*sn + y*cs; x=a;
380 Double_t rp4=GetC(b);
381 if ((TMath::Abs(b) < kAlmost0Field) || (TMath::Abs(rp4) < kAlmost0)) {
382 dz[0] = -(xt*f1 - yt*r1);
383 dz[1] = fP[1] + (dz[0]*f1 - xt)/r1*fP[3] - z;
387 sn=rp4*xt - f1; cs=rp4*yt + r1;
388 a=2*(xt*f1 - yt*r1)-rp4*(xt*xt + yt*yt);
389 Double_t rr=TMath::Sqrt(sn*sn + cs*cs);
391 Double_t f2 = -sn/rr, r2 = TMath::Sqrt((1.-f2)*(1.+f2));
392 dz[1] = fP[1] + fP[3]/rp4*TMath::ASin(f2*r1 - f1*r2) - z;
395 //_______________________________________________________________________
396 Double_t AliExternalTrackParam::GetLinearD(Double_t xv,Double_t yv) const {
397 //------------------------------------------------------------------
398 // This function calculates the transverse impact parameter
399 // with respect to a point with global coordinates (xv,yv)
400 // neglecting the track curvature.
401 //------------------------------------------------------------------
402 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
403 Double_t x= xv*cs + yv*sn;
404 Double_t y=-xv*sn + yv*cs;
406 Double_t d = (fX-x)*fP[2] - (fP[0]-y)*TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
411 Bool_t AliExternalTrackParam::CorrectForMeanMaterialdEdx
412 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
415 //------------------------------------------------------------------
416 // This function corrects the track parameters for the crossed material.
417 // "xOverX0" - X/X0, the thickness in units of the radiation length.
418 // "xTimesRho" - is the product length*density (g/cm^2).
419 // It should be passed as negative when propagating tracks
420 // from the intreaction point to the outside of the central barrel.
421 // "mass" - the mass of this particle (GeV/c^2).
422 // "dEdx" - mean enery loss (GeV/(g/cm^2)
423 // "anglecorr" - switch for the angular correction
424 //------------------------------------------------------------------
429 Double_t &fC22=fC[5];
430 Double_t &fC33=fC[9];
431 Double_t &fC43=fC[13];
432 Double_t &fC44=fC[14];
434 //Apply angle correction, if requested
436 Double_t angle=TMath::Sqrt((1.+ fP3*fP3)/((1-fP2)*(1.+fP2)));
443 Double_t beta2=p2/(p2 + mass*mass);
445 //Calculating the multiple scattering corrections******************
451 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
452 Double_t theta2=0.0136*0.0136/(beta2*p2)*TMath::Abs(xOverX0);
453 if (GetUseLogTermMS()) {
454 double lt = 1+0.038*TMath::Log(TMath::Abs(xOverX0));
455 if (lt>0) theta2 *= lt*lt;
457 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
458 cC22 = theta2*((1.-fP2)*(1.+fP2))*(1. + fP3*fP3);
459 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
460 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
461 cC44 = theta2*fP3*fP4*fP3*fP4;
464 //Calculating the energy loss corrections************************
466 if ((xTimesRho != 0.) && (beta2 < 1.)) {
467 Double_t dE=dEdx*xTimesRho;
468 Double_t e=TMath::Sqrt(p2 + mass*mass);
469 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
470 //cP4 = (1.- e/p2*dE);
471 if ( (1.+ dE/p2*(dE + 2*e)) < 0. ) return kFALSE;
472 cP4 = 1./TMath::Sqrt(1.+ dE/p2*(dE + 2*e)); //A precise formula by Ruben !
473 if (TMath::Abs(fP4*cP4)>100.) return kFALSE; //Do not track below 10 MeV/c
476 // Approximate energy loss fluctuation (M.Ivanov)
477 const Double_t knst=0.07; // To be tuned.
478 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
479 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
483 //Applying the corrections*****************************
495 Bool_t AliExternalTrackParam::CorrectForMeanMaterial
496 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
498 Double_t (*Bethe)(Double_t)) {
499 //------------------------------------------------------------------
500 // This function corrects the track parameters for the crossed material.
501 // "xOverX0" - X/X0, the thickness in units of the radiation length.
502 // "xTimesRho" - is the product length*density (g/cm^2).
503 // It should be passed as negative when propagating tracks
504 // from the intreaction point to the outside of the central barrel.
505 // "mass" - the mass of this particle (GeV/c^2).
506 // "anglecorr" - switch for the angular correction
507 // "Bethe" - function calculating the energy loss (GeV/(g/cm^2))
508 //------------------------------------------------------------------
510 Double_t bg=GetP()/mass;
511 Double_t dEdx=Bethe(bg);
513 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
516 Bool_t AliExternalTrackParam::CorrectForMeanMaterialZA
517 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
524 //------------------------------------------------------------------
525 // This function corrects the track parameters for the crossed material
526 // using the full Geant-like Bethe-Bloch formula parameterization
527 // "xOverX0" - X/X0, the thickness in units of the radiation length.
528 // "xTimesRho" - is the product length*density (g/cm^2).
529 // It should be passed as negative when propagating tracks
530 // from the intreaction point to the outside of the central barrel.
531 // "mass" - the mass of this particle (GeV/c^2).
532 // "density" - mean density (g/cm^3)
533 // "zOverA" - mean Z/A
534 // "exEnergy" - mean exitation energy (GeV)
535 // "jp1" - density effect first junction point
536 // "jp2" - density effect second junction point
537 // "anglecorr" - switch for the angular correction
539 // The default values of the parameters are for silicon
541 //------------------------------------------------------------------
543 Double_t bg=GetP()/mass;
544 Double_t dEdx=BetheBlochGeant(bg,density,jp1,jp2,exEnergy,zOverA);
546 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
551 Bool_t AliExternalTrackParam::CorrectForMaterial
552 (Double_t d, Double_t x0, Double_t mass, Double_t (*Bethe)(Double_t)) {
553 //------------------------------------------------------------------
554 // Deprecated function !
555 // Better use CorrectForMeanMaterial instead of it.
557 // This function corrects the track parameters for the crossed material
558 // "d" - the thickness (fraction of the radiation length)
559 // It should be passed as negative when propagating tracks
560 // from the intreaction point to the outside of the central barrel.
561 // "x0" - the radiation length (g/cm^2)
562 // "mass" - the mass of this particle (GeV/c^2)
563 //------------------------------------------------------------------
565 return CorrectForMeanMaterial(d,x0*d,mass,kTRUE,Bethe);
569 Double_t AliExternalTrackParam::BetheBlochAleph(Double_t bg,
576 // This is the empirical ALEPH parameterization of the Bethe-Bloch formula.
577 // It is normalized to 1 at the minimum.
581 // The default values for the kp* parameters are for ALICE TPC.
582 // The returned value is in MIP units
585 Double_t beta = bg/TMath::Sqrt(1.+ bg*bg);
587 Double_t aa = TMath::Power(beta,kp4);
588 Double_t bb = TMath::Power(1./bg,kp5);
590 bb=TMath::Log(kp3+bb);
592 return (kp2-aa-bb)*kp1/aa;
595 Double_t AliExternalTrackParam::BetheBlochGeant(Double_t bg,
602 // This is the parameterization of the Bethe-Bloch formula inspired by Geant.
605 // kp0 - density [g/cm^3]
606 // kp1 - density effect first junction point
607 // kp2 - density effect second junction point
608 // kp3 - mean excitation energy [GeV]
611 // The default values for the kp* parameters are for silicon.
612 // The returned value is in [GeV/(g/cm^2)].
615 const Double_t mK = 0.307075e-3; // [GeV*cm^2/g]
616 const Double_t me = 0.511e-3; // [GeV/c^2]
617 const Double_t rho = kp0;
618 const Double_t x0 = kp1*2.303;
619 const Double_t x1 = kp2*2.303;
620 const Double_t mI = kp3;
621 const Double_t mZA = kp4;
622 const Double_t bg2 = bg*bg;
623 const Double_t maxT= 2*me*bg2; // neglecting the electron mass
627 const Double_t x=TMath::Log(bg);
628 const Double_t lhwI=TMath::Log(28.816*1e-9*TMath::Sqrt(rho*mZA)/mI);
632 const Double_t r=(x1-x)/(x1-x0);
633 d2 = lhwI + x - 0.5 + (0.5 - lhwI - x0)*r*r*r;
636 return mK*mZA*(1+bg2)/bg2*
637 (0.5*TMath::Log(2*me*bg2*maxT/(mI*mI)) - bg2/(1+bg2) - d2);
640 Double_t AliExternalTrackParam::BetheBlochSolid(Double_t bg) {
641 //------------------------------------------------------------------
642 // This is an approximation of the Bethe-Bloch formula,
643 // reasonable for solid materials.
644 // All the parameters are, in fact, for Si.
645 // The returned value is in [GeV/(g/cm^2)]
646 //------------------------------------------------------------------
648 return BetheBlochGeant(bg);
651 Double_t AliExternalTrackParam::BetheBlochGas(Double_t bg) {
652 //------------------------------------------------------------------
653 // This is an approximation of the Bethe-Bloch formula,
654 // reasonable for gas materials.
655 // All the parameters are, in fact, for Ne.
656 // The returned value is in [GeV/(g/cm^2)]
657 //------------------------------------------------------------------
659 const Double_t rho = 0.9e-3;
660 const Double_t x0 = 2.;
661 const Double_t x1 = 4.;
662 const Double_t mI = 140.e-9;
663 const Double_t mZA = 0.49555;
665 return BetheBlochGeant(bg,rho,x0,x1,mI,mZA);
668 Bool_t AliExternalTrackParam::Rotate(Double_t alpha) {
669 //------------------------------------------------------------------
670 // Transform this track to the local coord. system rotated
671 // by angle "alpha" (rad) with respect to the global coord. system.
672 //------------------------------------------------------------------
673 if (TMath::Abs(fP[2]) >= kAlmost1) {
674 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
678 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
679 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
683 Double_t &fC00=fC[0];
684 Double_t &fC10=fC[1];
685 Double_t &fC20=fC[3];
686 Double_t &fC21=fC[4];
687 Double_t &fC22=fC[5];
688 Double_t &fC30=fC[6];
689 Double_t &fC32=fC[8];
690 Double_t &fC40=fC[10];
691 Double_t &fC42=fC[12];
694 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
695 Double_t sf=fP2, cf=TMath::Sqrt((1.- fP2)*(1.+fP2)); // Improve precision
696 // RS: check if rotation does no invalidate track model (cos(local_phi)>=0, i.e. particle
697 // direction in local frame is along the X axis
698 if ((cf*ca+sf*sa)<0) {
699 AliDebug(1,Form("Rotation failed: local cos(phi) would become %.2f",cf*ca+sf*sa));
703 Double_t tmp=sf*ca - cf*sa;
705 if (TMath::Abs(tmp) >= kAlmost1) {
706 if (TMath::Abs(tmp) > 1.+ Double_t(FLT_EPSILON))
707 AliWarning(Form("Rotation failed ! %.10e",tmp));
715 if (TMath::Abs(cf)<kAlmost0) {
716 AliError(Form("Too small cosine value %f",cf));
720 Double_t rr=(ca+sf/cf*sa);
737 Bool_t AliExternalTrackParam::Invert() {
738 //------------------------------------------------------------------
739 // Transform this track to the local coord. system rotated by 180 deg.
740 //------------------------------------------------------------------
742 fAlpha += TMath::Pi();
743 while (fAlpha < -TMath::Pi()) fAlpha += 2*TMath::Pi();
744 while (fAlpha >= TMath::Pi()) fAlpha -= 2*TMath::Pi();
751 fC[1] = -fC[1]; // since the fP1 and fP2 are not inverted, their covariances with others change sign
761 Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
762 //----------------------------------------------------------------
763 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
764 //----------------------------------------------------------------
766 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
768 Double_t crv=GetC(b);
769 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
771 Double_t x2r = crv*dx;
772 Double_t f1=fP[2], f2=f1 + x2r;
773 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
774 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
775 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
777 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
780 &fC10=fC[1], &fC11=fC[2],
781 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
782 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
783 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
785 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
786 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
787 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
790 double dy2dx = (f1+f2)/(r1+r2);
792 if (TMath::Abs(x2r)<0.05) {
793 fP1 += dx*(r2 + f2*dy2dx)*fP3; // Many thanks to P.Hristov !
797 // for small dx/R the linear apporximation of the arc by the segment is OK,
798 // but at large dx/R the error is very large and leads to incorrect Z propagation
799 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
800 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
801 // Similarly, the rotation angle in linear in dx only for dx<<R
802 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
803 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
805 fP2 = TMath::Sin(rot + TMath::ASin(fP2));
810 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
811 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
812 Double_t f12= dx*fP3*f1/(r1*r1*r1);
813 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
815 Double_t f24= dx; f24*=cc;
818 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
819 Double_t b02=f24*fC40;
820 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
821 Double_t b12=f24*fC41;
822 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
823 Double_t b22=f24*fC42;
824 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
825 Double_t b42=f24*fC44;
826 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
827 Double_t b32=f24*fC43;
830 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
831 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
832 Double_t a22=f24*b42;
834 //F*C*Ft = C + (b + bt + a)
835 fC00 += b00 + b00 + a00;
836 fC10 += b10 + b01 + a01;
837 fC20 += b20 + b02 + a02;
840 fC11 += b11 + b11 + a11;
841 fC21 += b21 + b12 + a12;
844 fC22 += b22 + b22 + a22;
853 Bool_t AliExternalTrackParam::PropagateParamOnlyTo(Double_t xk, Double_t b) {
854 //----------------------------------------------------------------
855 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
856 // Only parameters are propagated, not the matrix. To be used for small
857 // distances only (<mm, i.e. misalignment)
858 //----------------------------------------------------------------
860 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
862 Double_t crv=GetC(b);
863 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
865 Double_t x2r = crv*dx;
866 Double_t f1=fP[2], f2=f1 + x2r;
867 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
868 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
869 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
871 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
872 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
873 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
876 double dy2dx = (f1+f2)/(r1+r2);
878 fP[1] += dx*(r2 + f2*dy2dx)*fP[3]; // Many thanks to P.Hristov !
885 AliExternalTrackParam::Propagate(Double_t alpha, Double_t x, Double_t b) {
886 //------------------------------------------------------------------
887 // Transform this track to the local coord. system rotated
888 // by angle "alpha" (rad) with respect to the global coord. system,
889 // and propagate this track to the plane X=xk (cm) in the field "b" (kG)
890 //------------------------------------------------------------------
892 //Save the parameters
895 Double_t ps[5], cs[15];
896 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
897 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
900 if (PropagateTo(x,b)) return kTRUE;
902 //Restore the parameters, if the operation failed
905 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
906 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
910 Bool_t AliExternalTrackParam::PropagateBxByBz
911 (Double_t alpha, Double_t x, Double_t b[3]) {
912 //------------------------------------------------------------------
913 // Transform this track to the local coord. system rotated
914 // by angle "alpha" (rad) with respect to the global coord. system,
915 // and propagate this track to the plane X=xk (cm),
916 // taking into account all three components of the B field, "b[3]" (kG)
917 //------------------------------------------------------------------
919 //Save the parameters
922 Double_t ps[5], cs[15];
923 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
924 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
927 if (PropagateToBxByBz(x,b)) return kTRUE;
929 //Restore the parameters, if the operation failed
932 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
933 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
938 void AliExternalTrackParam::Propagate(Double_t len, Double_t x[3],
939 Double_t p[3], Double_t bz) const {
940 //+++++++++++++++++++++++++++++++++++++++++
941 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
942 // Extrapolate track along simple helix in magnetic field
943 // Arguments: len -distance alogn helix, [cm]
944 // bz - mag field, [kGaus]
945 // Returns: x and p contain extrapolated positon and momentum
946 // The momentum returned for straight-line tracks is meaningless !
947 //+++++++++++++++++++++++++++++++++++++++++
950 if (OneOverPt() < kAlmost0 || TMath::Abs(bz) < kAlmost0Field || GetC(bz) < kAlmost0){ //straight-line tracks
951 Double_t unit[3]; GetDirection(unit);
956 p[0]=unit[0]/kAlmost0;
957 p[1]=unit[1]/kAlmost0;
958 p[2]=unit[2]/kAlmost0;
962 Double_t a = -kB2C*bz*GetSign();
964 x[0] += p[0]*TMath::Sin(rho*len)/a - p[1]*(1-TMath::Cos(rho*len))/a;
965 x[1] += p[1]*TMath::Sin(rho*len)/a + p[0]*(1-TMath::Cos(rho*len))/a;
969 p[0] = p0 *TMath::Cos(rho*len) - p[1]*TMath::Sin(rho*len);
970 p[1] = p[1]*TMath::Cos(rho*len) + p0 *TMath::Sin(rho*len);
974 Bool_t AliExternalTrackParam::Intersect(Double_t pnt[3], Double_t norm[3],
976 //+++++++++++++++++++++++++++++++++++++++++
977 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
978 // Finds point of intersection (if exists) of the helix with the plane.
979 // Stores result in fX and fP.
980 // Arguments: planePoint,planeNorm - the plane defined by any plane's point
981 // and vector, normal to the plane
982 // Returns: kTrue if helix intersects the plane, kFALSE otherwise.
983 //+++++++++++++++++++++++++++++++++++++++++
984 Double_t x0[3]; GetXYZ(x0); //get track position in MARS
986 //estimates initial helix length up to plane
988 (pnt[0]-x0[0])*norm[0] + (pnt[1]-x0[1])*norm[1] + (pnt[2]-x0[2])*norm[2];
989 Double_t dist=99999,distPrev=dist;
991 while(TMath::Abs(dist)>0.00001){
992 //calculates helix at the distance s from x0 ALONG the helix
995 //distance between current helix position and plane
996 dist=(x[0]-pnt[0])*norm[0]+(x[1]-pnt[1])*norm[1]+(x[2]-pnt[2])*norm[2];
998 if(TMath::Abs(dist) >= TMath::Abs(distPrev)) {return kFALSE;}
1002 //on exit pnt is intersection point,norm is track vector at that point,
1004 for (Int_t i=0; i<3; i++) {pnt[i]=x[i]; norm[i]=p[i];}
1009 AliExternalTrackParam::GetPredictedChi2(Double_t p[2],Double_t cov[3]) const {
1010 //----------------------------------------------------------------
1011 // Estimate the chi2 of the space point "p" with the cov. matrix "cov"
1012 //----------------------------------------------------------------
1013 Double_t sdd = fC[0] + cov[0];
1014 Double_t sdz = fC[1] + cov[1];
1015 Double_t szz = fC[2] + cov[2];
1016 Double_t det = sdd*szz - sdz*sdz;
1018 if (TMath::Abs(det) < kAlmost0) return kVeryBig;
1020 Double_t d = fP[0] - p[0];
1021 Double_t z = fP[1] - p[1];
1023 return (d*szz*d - 2*d*sdz*z + z*sdd*z)/det;
1026 Double_t AliExternalTrackParam::
1027 GetPredictedChi2(Double_t p[3],Double_t covyz[3],Double_t covxyz[3]) const {
1028 //----------------------------------------------------------------
1029 // Estimate the chi2 of the 3D space point "p" and
1030 // the full covariance matrix "covyz" and "covxyz"
1032 // Cov(x,x) ... : covxyz[0]
1033 // Cov(y,x) ... : covxyz[1] covyz[0]
1034 // Cov(z,x) ... : covxyz[2] covyz[1] covyz[2]
1035 //----------------------------------------------------------------
1043 Double_t f=GetSnp();
1044 if (TMath::Abs(f) >= kAlmost1) return kVeryBig;
1045 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1046 Double_t a=f/r, b=GetTgl()/r;
1048 Double_t s2=333.*333.; //something reasonably big (cm^2)
1051 v(0,0)= s2; v(0,1)= a*s2; v(0,2)= b*s2;;
1052 v(1,0)=a*s2; v(1,1)=a*a*s2 + GetSigmaY2(); v(1,2)=a*b*s2 + GetSigmaZY();
1053 v(2,0)=b*s2; v(2,1)=a*b*s2 + GetSigmaZY(); v(2,2)=b*b*s2 + GetSigmaZ2();
1055 v(0,0)+=covxyz[0]; v(0,1)+=covxyz[1]; v(0,2)+=covxyz[2];
1056 v(1,0)+=covxyz[1]; v(1,1)+=covyz[0]; v(1,2)+=covyz[1];
1057 v(2,0)+=covxyz[2]; v(2,1)+=covyz[1]; v(2,2)+=covyz[2];
1060 if (!v.IsValid()) return kVeryBig;
1063 for (Int_t i = 0; i < 3; i++)
1064 for (Int_t j = 0; j < 3; j++) chi2 += res[i]*res[j]*v(i,j);
1069 Double_t AliExternalTrackParam::
1070 GetPredictedChi2(const AliExternalTrackParam *t) const {
1071 //----------------------------------------------------------------
1072 // Estimate the chi2 (5 dof) of this track with respect to the track
1073 // given by the argument.
1074 // The two tracks must be in the same reference system
1075 // and estimated at the same reference plane.
1076 //----------------------------------------------------------------
1078 if (TMath::Abs(1. - t->GetAlpha()/GetAlpha()) > FLT_EPSILON) {
1079 AliError("The reference systems of the tracks differ !");
1082 if (TMath::Abs(1. - t->GetX()/GetX()) > FLT_EPSILON) {
1083 AliError("The reference of the tracks planes differ !");
1088 c(0,0)=GetSigmaY2();
1089 c(1,0)=GetSigmaZY(); c(1,1)=GetSigmaZ2();
1090 c(2,0)=GetSigmaSnpY(); c(2,1)=GetSigmaSnpZ(); c(2,2)=GetSigmaSnp2();
1091 c(3,0)=GetSigmaTglY(); c(3,1)=GetSigmaTglZ(); c(3,2)=GetSigmaTglSnp(); c(3,3)=GetSigmaTgl2();
1092 c(4,0)=GetSigma1PtY(); c(4,1)=GetSigma1PtZ(); c(4,2)=GetSigma1PtSnp(); c(4,3)=GetSigma1PtTgl(); c(4,4)=GetSigma1Pt2();
1094 c(0,0)+=t->GetSigmaY2();
1095 c(1,0)+=t->GetSigmaZY(); c(1,1)+=t->GetSigmaZ2();
1096 c(2,0)+=t->GetSigmaSnpY();c(2,1)+=t->GetSigmaSnpZ();c(2,2)+=t->GetSigmaSnp2();
1097 c(3,0)+=t->GetSigmaTglY();c(3,1)+=t->GetSigmaTglZ();c(3,2)+=t->GetSigmaTglSnp();c(3,3)+=t->GetSigmaTgl2();
1098 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();
1100 c(0,2)=c(2,0); c(1,2)=c(2,1);
1101 c(0,3)=c(3,0); c(1,3)=c(3,1); c(2,3)=c(3,2);
1102 c(0,4)=c(4,0); c(1,4)=c(4,1); c(2,4)=c(4,2); c(3,4)=c(4,3);
1105 if (!c.IsValid()) return kVeryBig;
1111 GetSnp() - t->GetSnp(),
1112 GetTgl() - t->GetTgl(),
1113 GetSigned1Pt() - t->GetSigned1Pt()
1117 for (Int_t i = 0; i < 5; i++)
1118 for (Int_t j = 0; j < 5; j++) chi2 += res[i]*res[j]*c(i,j);
1123 Bool_t AliExternalTrackParam::
1124 PropagateTo(Double_t p[3],Double_t covyz[3],Double_t covxyz[3],Double_t bz) {
1125 //----------------------------------------------------------------
1126 // Propagate this track to the plane
1127 // the 3D space point "p" (with the covariance matrix "covyz" and "covxyz")
1129 // The magnetic field is "bz" (kG)
1131 // The track curvature and the change of the covariance matrix
1132 // of the track parameters are negleted !
1133 // (So the "step" should be small compared with 1/curvature)
1134 //----------------------------------------------------------------
1136 Double_t f=GetSnp();
1137 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1138 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1139 Double_t a=f/r, b=GetTgl()/r;
1141 Double_t s2=333.*333.; //something reasonably big (cm^2)
1144 tV(0,0)= s2; tV(0,1)= a*s2; tV(0,2)= b*s2;
1145 tV(1,0)=a*s2; tV(1,1)=a*a*s2; tV(1,2)=a*b*s2;
1146 tV(2,0)=b*s2; tV(2,1)=a*b*s2; tV(2,2)=b*b*s2;
1149 pV(0,0)=covxyz[0]; pV(0,1)=covxyz[1]; pV(0,2)=covxyz[2];
1150 pV(1,0)=covxyz[1]; pV(1,1)=covyz[0]; pV(1,2)=covyz[1];
1151 pV(2,0)=covxyz[2]; pV(2,1)=covyz[1]; pV(2,2)=covyz[2];
1153 TMatrixDSym tpV(tV);
1156 if (!tpV.IsValid()) return kFALSE;
1158 TMatrixDSym pW(3),tW(3);
1159 for (Int_t i=0; i<3; i++)
1160 for (Int_t j=0; j<3; j++) {
1162 for (Int_t k=0; k<3; k++) {
1163 pW(i,j) += tV(i,k)*tpV(k,j);
1164 tW(i,j) += pV(i,k)*tpV(k,j);
1168 Double_t t[3] = {GetX(), GetY(), GetZ()};
1171 for (Int_t i=0; i<3; i++) x += (tW(0,i)*t[i] + pW(0,i)*p[i]);
1172 Double_t crv=GetC(bz);
1173 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1175 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1179 for (Int_t i=0; i<3; i++) fP[0] += (tW(1,i)*t[i] + pW(1,i)*p[i]);
1181 for (Int_t i=0; i<3; i++) fP[1] += (tW(2,i)*t[i] + pW(2,i)*p[i]);
1186 Double_t *AliExternalTrackParam::GetResiduals(
1187 Double_t *p,Double_t *cov,Bool_t updated) const {
1188 //------------------------------------------------------------------
1189 // Returns the track residuals with the space point "p" having
1190 // the covariance matrix "cov".
1191 // If "updated" is kTRUE, the track parameters expected to be updated,
1192 // otherwise they must be predicted.
1193 //------------------------------------------------------------------
1194 static Double_t res[2];
1196 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1198 r00-=fC[0]; r01-=fC[1]; r11-=fC[2];
1200 r00+=fC[0]; r01+=fC[1]; r11+=fC[2];
1202 Double_t det=r00*r11 - r01*r01;
1204 if (TMath::Abs(det) < kAlmost0) return 0;
1206 Double_t tmp=r00; r00=r11/det; r11=tmp/det;
1208 if (r00 < 0.) return 0;
1209 if (r11 < 0.) return 0;
1211 Double_t dy = fP[0] - p[0];
1212 Double_t dz = fP[1] - p[1];
1214 res[0]=dy*TMath::Sqrt(r00);
1215 res[1]=dz*TMath::Sqrt(r11);
1220 Bool_t AliExternalTrackParam::Update(Double_t p[2], Double_t cov[3]) {
1221 //------------------------------------------------------------------
1222 // Update the track parameters with the space point "p" having
1223 // the covariance matrix "cov"
1224 //------------------------------------------------------------------
1225 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
1228 &fC10=fC[1], &fC11=fC[2],
1229 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1230 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1231 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1233 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1234 r00+=fC00; r01+=fC10; r11+=fC11;
1235 Double_t det=r00*r11 - r01*r01;
1237 if (TMath::Abs(det) < kAlmost0) return kFALSE;
1240 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
1242 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
1243 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
1244 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
1245 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
1246 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
1248 Double_t dy=p[0] - fP0, dz=p[1] - fP1;
1249 Double_t sf=fP2 + k20*dy + k21*dz;
1250 if (TMath::Abs(sf) > kAlmost1) return kFALSE;
1252 fP0 += k00*dy + k01*dz;
1253 fP1 += k10*dy + k11*dz;
1255 fP3 += k30*dy + k31*dz;
1256 fP4 += k40*dy + k41*dz;
1258 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
1259 Double_t c12=fC21, c13=fC31, c14=fC41;
1261 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
1262 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
1263 fC40-=k00*c04+k01*c14;
1265 fC11-=k10*c01+k11*fC11;
1266 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
1267 fC41-=k10*c04+k11*c14;
1269 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
1270 fC42-=k20*c04+k21*c14;
1272 fC33-=k30*c03+k31*c13;
1273 fC43-=k30*c04+k31*c14;
1275 fC44-=k40*c04+k41*c14;
1283 AliExternalTrackParam::GetHelixParameters(Double_t hlx[6], Double_t b) const {
1284 //--------------------------------------------------------------------
1285 // External track parameters -> helix parameters
1286 // "b" - magnetic field (kG)
1287 //--------------------------------------------------------------------
1288 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1290 hlx[0]=fP[0]; hlx[1]=fP[1]; hlx[2]=fP[2]; hlx[3]=fP[3];
1292 hlx[5]=fX*cs - hlx[0]*sn; // x0
1293 hlx[0]=fX*sn + hlx[0]*cs; // y0
1295 hlx[2]=TMath::ASin(hlx[2]) + fAlpha; // phi0
1297 hlx[4]=GetC(b); // C
1301 static void Evaluate(const Double_t *h, Double_t t,
1302 Double_t r[3], //radius vector
1303 Double_t g[3], //first defivatives
1304 Double_t gg[3]) //second derivatives
1306 //--------------------------------------------------------------------
1307 // Calculate position of a point on a track and some derivatives
1308 //--------------------------------------------------------------------
1309 Double_t phase=h[4]*t+h[2];
1310 Double_t sn=TMath::Sin(phase), cs=TMath::Cos(phase);
1314 if (TMath::Abs(h[4])>kAlmost0) {
1315 r[0] += (sn - h[6])/h[4];
1316 r[1] -= (cs - h[7])/h[4];
1318 r[2] = h[1] + h[3]*t;
1320 g[0] = cs; g[1]=sn; g[2]=h[3];
1322 gg[0]=-h[4]*sn; gg[1]=h[4]*cs; gg[2]=0.;
1325 Double_t AliExternalTrackParam::GetDCA(const AliExternalTrackParam *p,
1326 Double_t b, Double_t &xthis, Double_t &xp) const {
1327 //------------------------------------------------------------
1328 // Returns the (weighed !) distance of closest approach between
1329 // this track and the track "p".
1330 // Other returned values:
1331 // xthis, xt - coordinates of tracks' reference planes at the DCA
1332 //-----------------------------------------------------------
1333 Double_t dy2=GetSigmaY2() + p->GetSigmaY2();
1334 Double_t dz2=GetSigmaZ2() + p->GetSigmaZ2();
1337 Double_t p1[8]; GetHelixParameters(p1,b);
1338 p1[6]=TMath::Sin(p1[2]); p1[7]=TMath::Cos(p1[2]);
1339 Double_t p2[8]; p->GetHelixParameters(p2,b);
1340 p2[6]=TMath::Sin(p2[2]); p2[7]=TMath::Cos(p2[2]);
1343 Double_t r1[3],g1[3],gg1[3]; Double_t t1=0.;
1344 Evaluate(p1,t1,r1,g1,gg1);
1345 Double_t r2[3],g2[3],gg2[3]; Double_t t2=0.;
1346 Evaluate(p2,t2,r2,g2,gg2);
1348 Double_t dx=r2[0]-r1[0], dy=r2[1]-r1[1], dz=r2[2]-r1[2];
1349 Double_t dm=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1353 Double_t gt1=-(dx*g1[0]/dx2 + dy*g1[1]/dy2 + dz*g1[2]/dz2);
1354 Double_t gt2=+(dx*g2[0]/dx2 + dy*g2[1]/dy2 + dz*g2[2]/dz2);
1355 Double_t h11=(g1[0]*g1[0] - dx*gg1[0])/dx2 +
1356 (g1[1]*g1[1] - dy*gg1[1])/dy2 +
1357 (g1[2]*g1[2] - dz*gg1[2])/dz2;
1358 Double_t h22=(g2[0]*g2[0] + dx*gg2[0])/dx2 +
1359 (g2[1]*g2[1] + dy*gg2[1])/dy2 +
1360 (g2[2]*g2[2] + dz*gg2[2])/dz2;
1361 Double_t h12=-(g1[0]*g2[0]/dx2 + g1[1]*g2[1]/dy2 + g1[2]*g2[2]/dz2);
1363 Double_t det=h11*h22-h12*h12;
1366 if (TMath::Abs(det)<1.e-33) {
1367 //(quasi)singular Hessian
1370 dt1=-(gt1*h22 - gt2*h12)/det;
1371 dt2=-(h11*gt2 - h12*gt1)/det;
1374 if ((dt1*gt1+dt2*gt2)>0) {dt1=-dt1; dt2=-dt2;}
1376 //check delta(phase1) ?
1377 //check delta(phase2) ?
1379 if (TMath::Abs(dt1)/(TMath::Abs(t1)+1.e-3) < 1.e-4)
1380 if (TMath::Abs(dt2)/(TMath::Abs(t2)+1.e-3) < 1.e-4) {
1381 if ((gt1*gt1+gt2*gt2) > 1.e-4/dy2/dy2)
1382 AliDebug(1," stopped at not a stationary point !");
1383 Double_t lmb=h11+h22; lmb=lmb-TMath::Sqrt(lmb*lmb-4*det);
1385 AliDebug(1," stopped at not a minimum !");
1390 for (Int_t div=1 ; ; div*=2) {
1391 Evaluate(p1,t1+dt1,r1,g1,gg1);
1392 Evaluate(p2,t2+dt2,r2,g2,gg2);
1393 dx=r2[0]-r1[0]; dy=r2[1]-r1[1]; dz=r2[2]-r1[2];
1394 dd=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1398 AliDebug(1," overshoot !"); break;
1408 if (max<=0) AliDebug(1," too many iterations !");
1410 Double_t cs=TMath::Cos(GetAlpha());
1411 Double_t sn=TMath::Sin(GetAlpha());
1412 xthis=r1[0]*cs + r1[1]*sn;
1414 cs=TMath::Cos(p->GetAlpha());
1415 sn=TMath::Sin(p->GetAlpha());
1416 xp=r2[0]*cs + r2[1]*sn;
1418 return TMath::Sqrt(dm*TMath::Sqrt(dy2*dz2));
1421 Double_t AliExternalTrackParam::
1422 PropagateToDCA(AliExternalTrackParam *p, Double_t b) {
1423 //--------------------------------------------------------------
1424 // Propagates this track and the argument track to the position of the
1425 // distance of closest approach.
1426 // Returns the (weighed !) distance of closest approach.
1427 //--------------------------------------------------------------
1429 Double_t dca=GetDCA(p,b,xthis,xp);
1431 if (!PropagateTo(xthis,b)) {
1432 //AliWarning(" propagation failed !");
1436 if (!p->PropagateTo(xp,b)) {
1437 //AliWarning(" propagation failed !";
1445 Bool_t AliExternalTrackParam::PropagateToDCA(const AliVVertex *vtx,
1446 Double_t b, Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1448 // Propagate this track to the DCA to vertex "vtx",
1449 // if the (rough) transverse impact parameter is not bigger then "maxd".
1450 // Magnetic field is "b" (kG).
1452 // a) The track gets extapolated to the DCA to the vertex.
1453 // b) The impact parameters and their covariance matrix are calculated.
1455 // In the case of success, the returned value is kTRUE
1456 // (otherwise, it's kFALSE)
1458 Double_t alpha=GetAlpha();
1459 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1460 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1461 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1462 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1465 //Estimate the impact parameter neglecting the track curvature
1466 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1467 if (d > maxd) return kFALSE;
1469 //Propagate to the DCA
1470 Double_t crv=GetC(b);
1471 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1473 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1474 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1475 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1479 yv=-xv*sn + yv*cs; xv=x;
1481 if (!Propagate(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1483 if (dz==0) return kTRUE;
1484 dz[0] = GetParameter()[0] - yv;
1485 dz[1] = GetParameter()[1] - zv;
1487 if (covar==0) return kTRUE;
1488 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1490 //***** Improvements by A.Dainese
1491 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1492 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1493 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1494 covar[1] = GetCovariance()[1]; // between (x,y) and z
1495 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1501 Bool_t AliExternalTrackParam::PropagateToDCABxByBz(const AliVVertex *vtx,
1502 Double_t b[3], Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1504 // Propagate this track to the DCA to vertex "vtx",
1505 // if the (rough) transverse impact parameter is not bigger then "maxd".
1507 // This function takes into account all three components of the magnetic
1508 // field given by the b[3] arument (kG)
1510 // a) The track gets extapolated to the DCA to the vertex.
1511 // b) The impact parameters and their covariance matrix are calculated.
1513 // In the case of success, the returned value is kTRUE
1514 // (otherwise, it's kFALSE)
1516 Double_t alpha=GetAlpha();
1517 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1518 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1519 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1520 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1523 //Estimate the impact parameter neglecting the track curvature
1524 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1525 if (d > maxd) return kFALSE;
1527 //Propagate to the DCA
1528 Double_t crv=GetC(b[2]);
1529 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1531 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1532 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1533 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1537 yv=-xv*sn + yv*cs; xv=x;
1539 if (!PropagateBxByBz(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1541 if (dz==0) return kTRUE;
1542 dz[0] = GetParameter()[0] - yv;
1543 dz[1] = GetParameter()[1] - zv;
1545 if (covar==0) return kTRUE;
1546 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1548 //***** Improvements by A.Dainese
1549 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1550 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1551 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1552 covar[1] = GetCovariance()[1]; // between (x,y) and z
1553 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1559 void AliExternalTrackParam::GetDirection(Double_t d[3]) const {
1560 //----------------------------------------------------------------
1561 // This function returns a unit vector along the track direction
1562 // in the global coordinate system.
1563 //----------------------------------------------------------------
1564 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1566 Double_t csp =TMath::Sqrt((1.-snp)*(1.+snp));
1567 Double_t norm=TMath::Sqrt(1.+ fP[3]*fP[3]);
1568 d[0]=(csp*cs - snp*sn)/norm;
1569 d[1]=(snp*cs + csp*sn)/norm;
1573 Bool_t AliExternalTrackParam::GetPxPyPz(Double_t p[3]) const {
1574 //---------------------------------------------------------------------
1575 // This function returns the global track momentum components
1576 // Results for (nearly) straight tracks are meaningless !
1577 //---------------------------------------------------------------------
1578 p[0]=fP[4]; p[1]=fP[2]; p[2]=fP[3];
1579 return Local2GlobalMomentum(p,fAlpha);
1582 Double_t AliExternalTrackParam::Px() const {
1583 //---------------------------------------------------------------------
1584 // Returns x-component of momentum
1585 // Result for (nearly) straight tracks is meaningless !
1586 //---------------------------------------------------------------------
1588 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1594 Double_t AliExternalTrackParam::Py() const {
1595 //---------------------------------------------------------------------
1596 // Returns y-component of momentum
1597 // Result for (nearly) straight tracks is meaningless !
1598 //---------------------------------------------------------------------
1600 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1606 Double_t AliExternalTrackParam::Xv() const {
1607 //---------------------------------------------------------------------
1608 // Returns x-component of first track point
1609 //---------------------------------------------------------------------
1611 Double_t r[3]={0.,0.,0.};
1617 Double_t AliExternalTrackParam::Yv() const {
1618 //---------------------------------------------------------------------
1619 // Returns y-component of first track point
1620 //---------------------------------------------------------------------
1622 Double_t r[3]={0.,0.,0.};
1628 Double_t AliExternalTrackParam::Theta() const {
1629 // return theta angle of momentum
1631 return 0.5*TMath::Pi() - TMath::ATan(fP[3]);
1634 Double_t AliExternalTrackParam::Phi() const {
1635 //---------------------------------------------------------------------
1636 // Returns the azimuthal angle of momentum
1638 //---------------------------------------------------------------------
1640 Double_t phi=TMath::ASin(fP[2]) + fAlpha;
1641 if (phi<0.) phi+=2.*TMath::Pi();
1642 else if (phi>=2.*TMath::Pi()) phi-=2.*TMath::Pi();
1647 Double_t AliExternalTrackParam::M() const {
1648 // return particle mass
1650 // No mass information available so far.
1651 // Redifine in derived class!
1656 Double_t AliExternalTrackParam::E() const {
1657 // return particle energy
1659 // No PID information available so far.
1660 // Redifine in derived class!
1665 Double_t AliExternalTrackParam::Eta() const {
1666 // return pseudorapidity
1668 return -TMath::Log(TMath::Tan(0.5 * Theta()));
1671 Double_t AliExternalTrackParam::Y() const {
1674 // No PID information available so far.
1675 // Redifine in derived class!
1680 Bool_t AliExternalTrackParam::GetXYZ(Double_t *r) const {
1681 //---------------------------------------------------------------------
1682 // This function returns the global track position
1683 //---------------------------------------------------------------------
1684 r[0]=fX; r[1]=fP[0]; r[2]=fP[1];
1685 return Local2GlobalPosition(r,fAlpha);
1688 Bool_t AliExternalTrackParam::GetCovarianceXYZPxPyPz(Double_t cv[21]) const {
1689 //---------------------------------------------------------------------
1690 // This function returns the global covariance matrix of the track params
1692 // Cov(x,x) ... : cv[0]
1693 // Cov(y,x) ... : cv[1] cv[2]
1694 // Cov(z,x) ... : cv[3] cv[4] cv[5]
1695 // Cov(px,x)... : cv[6] cv[7] cv[8] cv[9]
1696 // Cov(py,x)... : cv[10] cv[11] cv[12] cv[13] cv[14]
1697 // Cov(pz,x)... : cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
1699 // Results for (nearly) straight tracks are meaningless !
1700 //---------------------------------------------------------------------
1701 if (TMath::Abs(fP[4])<=kAlmost0) {
1702 for (Int_t i=0; i<21; i++) cv[i]=0.;
1705 if (TMath::Abs(fP[2]) > kAlmost1) {
1706 for (Int_t i=0; i<21; i++) cv[i]=0.;
1709 Double_t pt=1./TMath::Abs(fP[4]);
1710 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1711 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
1713 Double_t m00=-sn, m10=cs;
1714 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
1715 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
1716 Double_t m35=pt, m45=-pt*pt*fP[3];
1722 cv[0 ] = fC[0]*m00*m00;
1723 cv[1 ] = fC[0]*m00*m10;
1724 cv[2 ] = fC[0]*m10*m10;
1728 cv[6 ] = m00*(fC[3]*m23 + fC[10]*m43);
1729 cv[7 ] = m10*(fC[3]*m23 + fC[10]*m43);
1730 cv[8 ] = fC[4]*m23 + fC[11]*m43;
1731 cv[9 ] = m23*(fC[5]*m23 + fC[12]*m43) + m43*(fC[12]*m23 + fC[14]*m43);
1732 cv[10] = m00*(fC[3]*m24 + fC[10]*m44);
1733 cv[11] = m10*(fC[3]*m24 + fC[10]*m44);
1734 cv[12] = fC[4]*m24 + fC[11]*m44;
1735 cv[13] = m23*(fC[5]*m24 + fC[12]*m44) + m43*(fC[12]*m24 + fC[14]*m44);
1736 cv[14] = m24*(fC[5]*m24 + fC[12]*m44) + m44*(fC[12]*m24 + fC[14]*m44);
1737 cv[15] = m00*(fC[6]*m35 + fC[10]*m45);
1738 cv[16] = m10*(fC[6]*m35 + fC[10]*m45);
1739 cv[17] = fC[7]*m35 + fC[11]*m45;
1740 cv[18] = m23*(fC[8]*m35 + fC[12]*m45) + m43*(fC[13]*m35 + fC[14]*m45);
1741 cv[19] = m24*(fC[8]*m35 + fC[12]*m45) + m44*(fC[13]*m35 + fC[14]*m45);
1742 cv[20] = m35*(fC[9]*m35 + fC[13]*m45) + m45*(fC[13]*m35 + fC[14]*m45);
1749 AliExternalTrackParam::GetPxPyPzAt(Double_t x, Double_t b, Double_t *p) const {
1750 //---------------------------------------------------------------------
1751 // This function returns the global track momentum extrapolated to
1752 // the radial position "x" (cm) in the magnetic field "b" (kG)
1753 //---------------------------------------------------------------------
1755 p[1]=fP[2]+(x-fX)*GetC(b);
1757 return Local2GlobalMomentum(p,fAlpha);
1761 AliExternalTrackParam::GetYAt(Double_t x, Double_t b, Double_t &y) const {
1762 //---------------------------------------------------------------------
1763 // This function returns the local Y-coordinate of the intersection
1764 // point between this track and the reference plane "x" (cm).
1765 // Magnetic field "b" (kG)
1766 //---------------------------------------------------------------------
1768 if(TMath::Abs(dx)<=kAlmost0) {y=fP[0]; return kTRUE;}
1770 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1772 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1773 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1775 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1776 y = fP[0] + dx*(f1+f2)/(r1+r2);
1781 AliExternalTrackParam::GetZAt(Double_t x, Double_t b, Double_t &z) const {
1782 //---------------------------------------------------------------------
1783 // This function returns the local Z-coordinate of the intersection
1784 // point between this track and the reference plane "x" (cm).
1785 // Magnetic field "b" (kG)
1786 //---------------------------------------------------------------------
1788 if(TMath::Abs(dx)<=kAlmost0) {z=fP[1]; return kTRUE;}
1790 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1792 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1793 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1795 Double_t r1=sqrt((1.-f1)*(1.+f1)), r2=sqrt((1.-f2)*(1.+f2));
1796 z = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3]; // Many thanks to P.Hristov !
1801 AliExternalTrackParam::GetXYZAt(Double_t x, Double_t b, Double_t *r) const {
1802 //---------------------------------------------------------------------
1803 // This function returns the global track position extrapolated to
1804 // the radial position "x" (cm) in the magnetic field "b" (kG)
1805 //---------------------------------------------------------------------
1807 if(TMath::Abs(dx)<=kAlmost0) return GetXYZ(r);
1809 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1811 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1812 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1814 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1816 r[1] = fP[0] + dx*(f1+f2)/(r1+r2);
1817 r[2] = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3];//Thanks to Andrea & Peter
1819 return Local2GlobalPosition(r,fAlpha);
1822 //_____________________________________________________________________________
1823 void AliExternalTrackParam::Print(Option_t* /*option*/) const
1825 // print the parameters and the covariance matrix
1827 printf("AliExternalTrackParam: x = %-12g alpha = %-12g\n", fX, fAlpha);
1828 printf(" parameters: %12g %12g %12g %12g %12g\n",
1829 fP[0], fP[1], fP[2], fP[3], fP[4]);
1830 printf(" covariance: %12g\n", fC[0]);
1831 printf(" %12g %12g\n", fC[1], fC[2]);
1832 printf(" %12g %12g %12g\n", fC[3], fC[4], fC[5]);
1833 printf(" %12g %12g %12g %12g\n",
1834 fC[6], fC[7], fC[8], fC[9]);
1835 printf(" %12g %12g %12g %12g %12g\n",
1836 fC[10], fC[11], fC[12], fC[13], fC[14]);
1839 Double_t AliExternalTrackParam::GetSnpAt(Double_t x,Double_t b) const {
1841 // Get sinus at given x
1843 Double_t crv=GetC(b);
1844 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1846 Double_t res = fP[2]+dx*crv;
1850 Bool_t AliExternalTrackParam::GetDistance(AliExternalTrackParam *param2, Double_t x, Double_t dist[3], Double_t bz){
1851 //------------------------------------------------------------------------
1852 // Get the distance between two tracks at the local position x
1853 // working in the local frame of this track.
1854 // Origin : Marian.Ivanov@cern.ch
1855 //-----------------------------------------------------------------------
1859 if (!GetYAt(x,bz,xyz[1])) return kFALSE;
1860 if (!GetZAt(x,bz,xyz[2])) return kFALSE;
1863 if (TMath::Abs(GetAlpha()-param2->GetAlpha())<kAlmost0){
1865 if (!param2->GetYAt(x,bz,xyz2[1])) return kFALSE;
1866 if (!param2->GetZAt(x,bz,xyz2[2])) return kFALSE;
1870 Double_t dfi = param2->GetAlpha()-GetAlpha();
1871 Double_t ca = TMath::Cos(dfi), sa = TMath::Sin(dfi);
1872 xyz2[0] = xyz[0]*ca+xyz[1]*sa;
1873 xyz2[1] = -xyz[0]*sa+xyz[1]*ca;
1876 if (!param2->GetYAt(xyz2[0],bz,xyz1[1])) return kFALSE;
1877 if (!param2->GetZAt(xyz2[0],bz,xyz1[2])) return kFALSE;
1879 xyz2[0] = xyz1[0]*ca-xyz1[1]*sa;
1880 xyz2[1] = +xyz1[0]*sa+xyz1[1]*ca;
1883 dist[0] = xyz[0]-xyz2[0];
1884 dist[1] = xyz[1]-xyz2[1];
1885 dist[2] = xyz[2]-xyz2[2];
1892 // Draw functionality.
1893 // Origin: Marian Ivanov, Marian.Ivanov@cern.ch
1896 void AliExternalTrackParam::DrawTrack(Float_t magf, Float_t minR, Float_t maxR, Float_t stepR){
1900 if (minR>maxR) return ;
1901 if (stepR<=0) return ;
1902 Int_t npoints = TMath::Nint((maxR-minR)/stepR)+1;
1903 if (npoints<1) return;
1904 TPolyMarker3D *polymarker = new TPolyMarker3D(npoints);
1905 FillPolymarker(polymarker, magf,minR,maxR,stepR);
1910 void AliExternalTrackParam::FillPolymarker(TPolyMarker3D *pol, Float_t magF, Float_t minR, Float_t maxR, Float_t stepR){
1912 // Fill points in the polymarker
1915 for (Double_t r=minR; r<maxR; r+=stepR){
1917 GetXYZAt(r,magF,point);
1918 pol->SetPoint(counter,point[0],point[1], point[2]);
1919 // printf("xyz\t%f\t%f\t%f\n",point[0], point[1],point[2]);
1924 Int_t AliExternalTrackParam::GetIndex(Int_t i, Int_t j) const {
1926 Int_t min = TMath::Min(i,j);
1927 Int_t max = TMath::Max(i,j);
1929 return min+(max+1)*max/2;
1933 void AliExternalTrackParam::g3helx3(Double_t qfield,
1936 /******************************************************************
1938 * GEANT3 tracking routine in a constant field oriented *
1940 * Tracking is performed with a conventional *
1941 * helix step method *
1943 * Authors R.Brun, M.Hansroul ********* *
1944 * Rewritten V.Perevoztchikov *
1946 * Rewritten in C++ by I.Belikov *
1948 * qfield (kG) - particle charge times magnetic field *
1949 * step (cm) - step length along the helix *
1950 * vect[7](cm,GeV/c) - input/output x, y, z, px/p, py/p ,pz/p, p *
1952 ******************************************************************/
1953 const Int_t ix=0, iy=1, iz=2, ipx=3, ipy=4, ipz=5, ipp=6;
1954 const Double_t kOvSqSix=TMath::Sqrt(1./6.);
1956 Double_t cosx=vect[ipx], cosy=vect[ipy], cosz=vect[ipz];
1958 Double_t rho = qfield*kB2C/vect[ipp];
1959 Double_t tet = rho*step;
1961 Double_t tsint, sintt, sint, cos1t;
1962 if (TMath::Abs(tet) > 0.03) {
1963 sint = TMath::Sin(tet);
1965 tsint = (tet - sint)/tet;
1966 Double_t t=TMath::Sin(0.5*tet);
1970 sintt = (1.-tet*kOvSqSix)*(1.+tet*kOvSqSix); // 1.- tsint;
1975 Double_t f1 = step*sintt;
1976 Double_t f2 = step*cos1t;
1977 Double_t f3 = step*tsint*cosz;
1978 Double_t f4 = -tet*cos1t;
1981 vect[ix] += f1*cosx - f2*cosy;
1982 vect[iy] += f1*cosy + f2*cosx;
1983 vect[iz] += f1*cosz + f3;
1985 vect[ipx] += f4*cosx - f5*cosy;
1986 vect[ipy] += f4*cosy + f5*cosx;
1990 Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]) {
1991 //----------------------------------------------------------------
1992 // Extrapolate this track to the plane X=xk in the field b[].
1994 // X [cm] is in the "tracking coordinate system" of this track.
1995 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
1996 //----------------------------------------------------------------
1999 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
2000 if (TMath::Abs(fP[4])<=kAlmost0) return kFALSE;
2001 // Do not propagate tracks outside the ALICE detector
2002 if (TMath::Abs(dx)>1e5 ||
2003 TMath::Abs(GetY())>1e5 ||
2004 TMath::Abs(GetZ())>1e5) {
2005 AliWarning(Form("Anomalous track, target X:%f",xk));
2010 Double_t crv=GetC(b[2]);
2011 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
2013 Double_t x2r = crv*dx;
2014 Double_t f1=fP[2], f2=f1 + x2r;
2015 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2016 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2019 // Estimate the covariance matrix
2020 Double_t &fP3=fP[3], &fP4=fP[4];
2023 &fC10=fC[1], &fC11=fC[2],
2024 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
2025 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
2026 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
2028 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
2031 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
2032 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
2033 Double_t f12= dx*fP3*f1/(r1*r1*r1);
2034 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
2035 Double_t f13= dx/r1;
2036 Double_t f24= dx; f24*=cc;
2039 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
2040 Double_t b02=f24*fC40;
2041 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
2042 Double_t b12=f24*fC41;
2043 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
2044 Double_t b22=f24*fC42;
2045 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
2046 Double_t b42=f24*fC44;
2047 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
2048 Double_t b32=f24*fC43;
2051 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
2052 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
2053 Double_t a22=f24*b42;
2055 //F*C*Ft = C + (b + bt + a)
2056 fC00 += b00 + b00 + a00;
2057 fC10 += b10 + b01 + a01;
2058 fC20 += b20 + b02 + a02;
2061 fC11 += b11 + b11 + a11;
2062 fC21 += b21 + b12 + a12;
2065 fC22 += b22 + b22 + a22;
2071 // Appoximate step length
2072 double dy2dx = (f1+f2)/(r1+r2);
2073 Double_t step = (TMath::Abs(x2r)<0.05) ? dx*TMath::Abs(r2 + f2*dy2dx) // chord
2074 : 2.*TMath::ASin(0.5*dx*TMath::Sqrt(1.+dy2dx*dy2dx)*crv)/crv; // arc
2075 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
2077 // Get the track's (x,y,z) and (px,py,pz) in the Global System
2078 Double_t r[3]; GetXYZ(r);
2079 Double_t p[3]; GetPxPyPz(p);
2086 // Rotate to the system where Bx=By=0.
2087 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
2088 Double_t cosphi=1., sinphi=0.;
2089 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
2090 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
2091 Double_t costet=1., sintet=0.;
2092 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
2095 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
2096 vect[1] = -sinphi*r[0] + cosphi*r[1];
2097 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
2099 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
2100 vect[4] = -sinphi*p[0] + cosphi*p[1];
2101 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
2106 // Do the helix step
2107 g3helx3(GetSign()*bb,step,vect);
2110 // Rotate back to the Global System
2111 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
2112 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
2113 r[2] = -sintet*vect[0] + costet*vect[2];
2115 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
2116 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
2117 p[2] = -sintet*vect[3] + costet*vect[5];
2120 // Rotate back to the Tracking System
2121 Double_t cosalp = TMath::Cos(fAlpha);
2122 Double_t sinalp =-TMath::Sin(fAlpha);
2125 t = cosalp*r[0] - sinalp*r[1];
2126 r[1] = sinalp*r[0] + cosalp*r[1];
2129 t = cosalp*p[0] - sinalp*p[1];
2130 p[1] = sinalp*p[0] + cosalp*p[1];
2134 // Do the final correcting step to the target plane (linear approximation)
2135 Double_t x=r[0], y=r[1], z=r[2];
2136 if (TMath::Abs(dx) > kAlmost0) {
2137 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
2145 // Calculate the track parameters
2146 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
2152 fP[4] = GetSign()/(t*pp);
2157 Bool_t AliExternalTrackParam::Translate(Double_t *vTrasl,Double_t *covV){
2159 //Translation: in the event mixing, the tracks can be shifted
2160 //of the difference among primary vertices (vTrasl) and
2161 //the covariance matrix is changed accordingly
2162 //(covV = covariance of the primary vertex).
2163 //Origin: "Romita, Rossella" <R.Romita@gsi.de>
2165 TVector3 translation;
2166 // vTrasl coordinates in the local system
2167 translation.SetXYZ(vTrasl[0],vTrasl[1],vTrasl[2]);
2168 translation.RotateZ(-fAlpha);
2169 translation.GetXYZ(vTrasl);
2171 //compute the new x,y,z of the track
2172 Double_t newX=fX-vTrasl[0];
2173 Double_t newY=fP[0]-vTrasl[1];
2174 Double_t newZ=fP[1]-vTrasl[2];
2176 //define the new parameters
2177 Double_t newParam[5];
2184 // recompute the covariance matrix:
2185 // 1. covV in the local system
2186 Double_t cosRot=TMath::Cos(fAlpha), sinRot=TMath::Sin(fAlpha);
2207 if(uUi.Determinant() <= 0.) {return kFALSE;}
2208 TMatrixD uUiQi(uUi,TMatrixD::kMult,qQi);
2209 TMatrixD m(qQi,TMatrixD::kTransposeMult,uUiQi);
2211 //2. compute the new covariance matrix of the track
2212 Double_t sigmaXX=m(0,0);
2213 Double_t sigmaXZ=m(2,0);
2214 Double_t sigmaXY=m(1,0);
2215 Double_t sigmaYY=GetSigmaY2()+m(1,1);
2216 Double_t sigmaYZ=fC[1]+m(1,2);
2217 Double_t sigmaZZ=fC[2]+m(2,2);
2218 Double_t covarianceYY=sigmaYY + (-1.)*((sigmaXY*sigmaXY)/sigmaXX);
2219 Double_t covarianceYZ=sigmaYZ-(sigmaXZ*sigmaXY/sigmaXX);
2220 Double_t covarianceZZ=sigmaZZ-((sigmaXZ*sigmaXZ)/sigmaXX);
2222 Double_t newCov[15];
2223 newCov[0]=covarianceYY;
2224 newCov[1]=covarianceYZ;
2225 newCov[2]=covarianceZZ;
2226 for(Int_t i=3;i<15;i++){
2230 // set the new parameters
2232 Set(newX,fAlpha,newParam,newCov);
2237 void AliExternalTrackParam::CheckCovariance() {
2239 // This function forces the diagonal elements of the covariance matrix to be positive.
2240 // In case the diagonal element is bigger than the maximal allowed value, it is set to
2241 // the limit and the off-diagonal elements that correspond to it are set to zero.
2243 fC[0] = TMath::Abs(fC[0]);
2251 fC[2] = TMath::Abs(fC[2]);
2259 fC[5] = TMath::Abs(fC[5]);
2267 fC[9] = TMath::Abs(fC[9]);
2275 fC[14] = TMath::Abs(fC[14]);
2276 if (fC[14]>kC14max) {
2284 // The part below is used for tests and normally is commented out
2285 // TMatrixDSym m(5);
2289 // m(1,0)=fC[1]; m(1,1)=fC[2];
2290 // m(2,0)=fC[3]; m(2,1)=fC[4]; m(2,2)=fC[5];
2291 // m(3,0)=fC[6]; m(3,1)=fC[7]; m(3,2)=fC[8]; m(3,3)=fC[9];
2292 // 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];
2295 // m(0,2)=m(2,0); m(1,2)=m(2,1);
2296 // m(0,3)=m(3,0); m(1,3)=m(3,1); m(2,3)=m(3,2);
2297 // m(0,4)=m(4,0); m(1,4)=m(4,1); m(2,4)=m(4,2); m(3,4)=m(4,3);
2298 // m.EigenVectors(eig);
2300 // // assert(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0);
2301 // if (!(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0)) {
2302 // AliWarning("Negative eigenvalues of the covariance matrix!");
2308 Bool_t AliExternalTrackParam::ConstrainToVertex(const AliVVertex* vtx, Double_t b[3])
2310 // Constrain TPC inner params constrained
2315 Double_t dz[2], cov[3];
2316 if (!PropagateToDCABxByBz(vtx, b, 3, dz, cov))
2320 vtx->GetCovarianceMatrix(covar);
2322 Double_t p[2]= { fP[0] - dz[0], fP[1] - dz[1] };
2323 Double_t c[3]= { covar[2], 0., covar[5] };
2325 Double_t chi2C = GetPredictedChi2(p,c);
2335 //___________________________________________________________________________________________
2336 Bool_t AliExternalTrackParam::GetXatLabR(Double_t r,Double_t &x, Double_t bz, Int_t dir) const
2338 // Get local X of the track position estimated at the radius lab radius r.
2339 // The track curvature is accounted exactly
2341 // The flag "dir" can be used to remove the ambiguity of which intersection to take (out of 2 possible)
2342 // 0 - take the intersection closest to the current track position
2343 // >0 - go along the track (increasing fX)
2344 // <0 - go backward (decreasing fX)
2346 const Double_t &fy=fP[0], &sn = fP[2];
2348 double crv = GetC(bz);
2349 if (TMath::Abs(crv)<=kAlmost0) { // this is a straight track
2350 if (TMath::Abs(sn)>=kAlmost1) { // || to Y axis
2351 double det = (r-fX)*(r+fX);
2352 if (det<0) return kFALSE; // does not reach raduis r
2354 if (dir==0) return kTRUE;
2355 det = TMath::Sqrt(det);
2356 if (dir>0) { // along the track direction
2357 if (sn>0) {if (fy>det) return kFALSE;} // track is along Y axis and above the circle
2358 else {if (fy<-det) return kFALSE;} // track is against Y axis amd belo the circle
2360 else if(dir>0) { // agains track direction
2361 if (sn>0) {if (fy<-det) return kFALSE;} // track is along Y axis
2362 else if (fy>det) return kFALSE; // track is against Y axis
2365 else if (TMath::Abs(sn)<=kAlmost0) { // || to X axis
2366 double det = (r-fy)*(r+fy);
2367 if (det<0) return kFALSE; // does not reach raduis r
2368 det = TMath::Sqrt(det);
2370 x = fX>0 ? det : -det; // choose the solution requiring the smalest step
2373 else if (dir>0) { // along the track direction
2374 if (fX > det) return kFALSE; // current point is in on the right from the circle
2375 else if (fX <-det) x = -det; // on the left
2376 else x = det; // within the circle
2378 else { // against the track direction
2379 if (fX <-det) return kFALSE;
2380 else if (fX > det) x = det;
2384 else { // general case of straight line
2385 double cs = TMath::Sqrt((1-sn)*(1+sn));
2386 double xsyc = fX*sn-fy*cs;
2387 double det = (r-xsyc)*(r+xsyc);
2388 if (det<0) return kFALSE; // does not reach raduis r
2389 det = TMath::Sqrt(det);
2390 double xcys = fX*cs+fy*sn;
2392 if (dir==0) t += t>0 ? -det:det; // chose the solution requiring the smalest step
2393 else if (dir>0) { // go in increasing fX direction. ( t+-det > 0)
2394 if (t>=-det) t += -det; // take minimal step giving t>0
2395 else return kFALSE; // both solutions have negative t
2397 else { // go in increasing fX direction. (t+-det < 0)
2398 if (t<det) t -= det; // take minimal step giving t<0
2399 else return kFALSE; // both solutions have positive t
2405 // get center of the track circle
2406 double tR = 1./crv; // track radius (for the moment signed)
2407 double cs = TMath::Sqrt((1-sn)*(1+sn));
2408 double x0 = fX - sn*tR;
2409 double y0 = fy + cs*tR;
2410 double r0 = TMath::Sqrt(x0*x0+y0*y0);
2411 // printf("Xc:%+e Yc:%+e\n",x0,y0);
2413 if (r0<=kAlmost0) return kFALSE; // the track is concentric to circle
2414 tR = TMath::Abs(tR);
2415 double tR2r0 = tR/r0;
2416 double g = 0.5*(r*r/(r0*tR) - tR2r0 - 1./tR2r0);
2417 double det = (1.-g)*(1.+g);
2418 if (det<0) return kFALSE; // does not reach raduis r
2419 det = TMath::Sqrt(det);
2421 // the intersection happens in 2 points: {x0+tR*C,y0+tR*S}
2422 // with C=f*c0+-|s0|*det and S=f*s0-+c0 sign(s0)*det
2423 // where s0 and c0 make direction for the circle center (=x0/r0 and y0/r0)
2425 double tmp = 1.+g*tR2r0;
2428 if (TMath::Abs(y0)>kAlmost0) { // when y0==0 the x,y is unique
2429 double dfx = tR2r0*TMath::Abs(y0)*det;
2430 double dfy = tR2r0*x0*TMath::Sign(det,y0);
2431 if (dir==0) { // chose the one which corresponds to smallest step
2432 double delta = (x-fX)*dfx-(y-fy)*dfy; // the choice of + in C will lead to smaller step if delta<0
2433 if (delta<0) x += dfx;
2436 else if (dir>0) { // along track direction: x must be > fX
2437 x -= dfx; // try the smallest step (dfx is positive)
2438 if (x<fX && (x+=dfx+dfx)<fX) return kFALSE;
2440 else { // backward: x must be < fX
2441 x += dfx; // try the smallest step (dfx is positive)
2442 if (x>fX && (x-=dfx+dfx)>fX) return kFALSE;
2445 else { // special case: track touching the circle just in 1 point
2446 if ( (dir>0&&x<fX) || (dir<0&&x>fX) ) return kFALSE;