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) {
217 cs=TMath::Cos(fAlpha);
218 sn=TMath::Sin(fAlpha);
221 fAlpha -= TMath::Sign(kSafe, fAlpha);
222 cs=TMath::Cos(fAlpha);
223 sn=TMath::Sin(fAlpha);
225 // Get the vertex of origin and the momentum
226 TVector3 ver(xyz[0],xyz[1],xyz[2]);
227 TVector3 mom(pxpypz[0],pxpypz[1],pxpypz[2]);
229 // avoid momenta along axis
230 if (TMath::Abs(mom[0])<kSafe) mom[0] = TMath::Sign(kSafe*TMath::Abs(mom[1]), mom[0]);
231 if (TMath::Abs(mom[1])<kSafe) mom[1] = TMath::Sign(kSafe*TMath::Abs(mom[0]), mom[1]);
233 // Rotate to the local coordinate system
234 ver.RotateZ(-fAlpha);
235 mom.RotateZ(-fAlpha);
237 // x of the reference plane
240 Double_t charge = (Double_t)sign;
244 fP[2] = TMath::Sin(mom.Phi());
245 fP[3] = mom.Pz()/mom.Pt();
246 fP[4] = TMath::Sign(1/mom.Pt(),charge);
248 // Covariance matrix (formulas to be simplified)
250 if (TMath::Abs( 1-fP[2]) < kSafe) fP[2] = 1.- kSafe; //Protection
251 else if (TMath::Abs(-1-fP[2]) < kSafe) fP[2] =-1.+ kSafe; //Protection
253 Double_t pt=1./TMath::Abs(fP[4]);
254 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
256 Double_t m00=-sn;// m10=cs;
257 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
258 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
259 Double_t m35=pt, m45=-pt*pt*fP[3];
265 Double_t cv34 = TMath::Sqrt(cv[3 ]*cv[3 ]+cv[4 ]*cv[4 ]);
266 Double_t a1=cv[13]-cv[9]*(m23*m44+m43*m24)/m23/m43;
267 Double_t a2=m23*m24-m23*(m23*m44+m43*m24)/m43;
268 Double_t a3=m43*m44-m43*(m23*m44+m43*m24)/m23;
269 Double_t a4=cv[14]-2.*cv[9]*m24*m44/m23/m43;
270 Double_t a5=m24*m24-2.*m24*m44*m23/m43;
271 Double_t a6=m44*m44-2.*m24*m44*m43/m23;
273 fC[0 ] = cv[0 ]+cv[2 ];
274 fC[1 ] = TMath::Sign(cv34,cv[3 ]/m00);
276 fC[3 ] = (cv[10]/m44-cv[6]/m43)/(m24/m44-m23/m43)/m00;
277 fC[10] = (cv[6]/m00-fC[3 ]*m23)/m43;
278 fC[6 ] = (cv[15]/m00-fC[10]*m45)/m35;
279 fC[4 ] = (cv[12]-cv[8]*m44/m43)/(m24-m23*m44/m43);
280 fC[11] = (cv[8]-fC[4]*m23)/m43;
281 fC[7 ] = cv[17]/m35-fC[11]*m45/m35;
282 fC[5 ] = TMath::Abs((a4-a6*a1/a3)/(a5-a6*a2/a3));
283 fC[14] = TMath::Abs(a1/a3-a2*fC[5]/a3);
284 fC[12] = (cv[9]-fC[5]*m23*m23-fC[14]*m43*m43)/m23/m43;
285 Double_t b1=cv[18]-fC[12]*m23*m45-fC[14]*m43*m45;
288 Double_t b4=cv[19]-fC[12]*m24*m45-fC[14]*m44*m45;
291 fC[8 ] = (b4-b6*b1/b3)/(b5-b6*b2/b3);
292 fC[13] = b1/b3-b2*fC[8]/b3;
293 fC[9 ] = TMath::Abs((cv[20]-fC[14]*(m45*m45)-fC[13]*2.*m35*m45)/(m35*m35));
300 //_____________________________________________________________________________
301 void AliExternalTrackParam::Reset() {
303 // Resets all the parameters to 0
306 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
307 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
310 //_____________________________________________________________________________
311 void AliExternalTrackParam::AddCovariance(const Double_t c[15]) {
313 // Add "something" to the track covarince matrix.
314 // May be needed to account for unknown mis-calibration/mis-alignment
317 fC[1] +=c[1]; fC[2] +=c[2];
318 fC[3] +=c[3]; fC[4] +=c[4]; fC[5] +=c[5];
319 fC[6] +=c[6]; fC[7] +=c[7]; fC[8] +=c[8]; fC[9] +=c[9];
320 fC[10]+=c[10]; fC[11]+=c[11]; fC[12]+=c[12]; fC[13]+=c[13]; fC[14]+=c[14];
325 Double_t AliExternalTrackParam::GetP() const {
326 //---------------------------------------------------------------------
327 // This function returns the track momentum
328 // Results for (nearly) straight tracks are meaningless !
329 //---------------------------------------------------------------------
330 if (TMath::Abs(fP[4])<=kAlmost0) return kVeryBig;
331 return TMath::Sqrt(1.+ fP[3]*fP[3])/TMath::Abs(fP[4]);
334 Double_t AliExternalTrackParam::Get1P() const {
335 //---------------------------------------------------------------------
336 // This function returns the 1/(track momentum)
337 //---------------------------------------------------------------------
338 return TMath::Abs(fP[4])/TMath::Sqrt(1.+ fP[3]*fP[3]);
341 //_______________________________________________________________________
342 Double_t AliExternalTrackParam::GetD(Double_t x,Double_t y,Double_t b) const {
343 //------------------------------------------------------------------
344 // This function calculates the transverse impact parameter
345 // with respect to a point with global coordinates (x,y)
346 // in the magnetic field "b" (kG)
347 //------------------------------------------------------------------
348 if (TMath::Abs(b) < kAlmost0Field) return GetLinearD(x,y);
349 Double_t rp4=GetC(b);
351 Double_t xt=fX, yt=fP[0];
353 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
354 Double_t a = x*cs + y*sn;
355 y = -x*sn + y*cs; x=a;
358 sn=rp4*xt - fP[2]; cs=rp4*yt + TMath::Sqrt((1.- fP[2])*(1.+fP[2]));
359 a=2*(xt*fP[2] - yt*TMath::Sqrt((1.-fP[2])*(1.+fP[2])))-rp4*(xt*xt + yt*yt);
360 return -a/(1 + TMath::Sqrt(sn*sn + cs*cs));
363 //_______________________________________________________________________
364 void AliExternalTrackParam::
365 GetDZ(Double_t x, Double_t y, Double_t z, Double_t b, Float_t dz[2]) const {
366 //------------------------------------------------------------------
367 // This function calculates the transverse and longitudinal impact parameters
368 // with respect to a point with global coordinates (x,y)
369 // in the magnetic field "b" (kG)
370 //------------------------------------------------------------------
371 Double_t f1 = fP[2], r1 = TMath::Sqrt((1.-f1)*(1.+f1));
372 Double_t xt=fX, yt=fP[0];
373 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
374 Double_t a = x*cs + y*sn;
375 y = -x*sn + y*cs; x=a;
378 Double_t rp4=GetC(b);
379 if ((TMath::Abs(b) < kAlmost0Field) || (TMath::Abs(rp4) < kAlmost0)) {
380 dz[0] = -(xt*f1 - yt*r1);
381 dz[1] = fP[1] + (dz[0]*f1 - xt)/r1*fP[3] - z;
385 sn=rp4*xt - f1; cs=rp4*yt + r1;
386 a=2*(xt*f1 - yt*r1)-rp4*(xt*xt + yt*yt);
387 Double_t rr=TMath::Sqrt(sn*sn + cs*cs);
389 Double_t f2 = -sn/rr, r2 = TMath::Sqrt((1.-f2)*(1.+f2));
390 dz[1] = fP[1] + fP[3]/rp4*TMath::ASin(f2*r1 - f1*r2) - z;
393 //_______________________________________________________________________
394 Double_t AliExternalTrackParam::GetLinearD(Double_t xv,Double_t yv) const {
395 //------------------------------------------------------------------
396 // This function calculates the transverse impact parameter
397 // with respect to a point with global coordinates (xv,yv)
398 // neglecting the track curvature.
399 //------------------------------------------------------------------
400 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
401 Double_t x= xv*cs + yv*sn;
402 Double_t y=-xv*sn + yv*cs;
404 Double_t d = (fX-x)*fP[2] - (fP[0]-y)*TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
409 Bool_t AliExternalTrackParam::CorrectForMeanMaterialdEdx
410 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
413 //------------------------------------------------------------------
414 // This function corrects the track parameters for the crossed material.
415 // "xOverX0" - X/X0, the thickness in units of the radiation length.
416 // "xTimesRho" - is the product length*density (g/cm^2).
417 // It should be passed as negative when propagating tracks
418 // from the intreaction point to the outside of the central barrel.
419 // "mass" - the mass of this particle (GeV/c^2).
420 // "dEdx" - mean enery loss (GeV/(g/cm^2)
421 // "anglecorr" - switch for the angular correction
422 //------------------------------------------------------------------
427 Double_t &fC22=fC[5];
428 Double_t &fC33=fC[9];
429 Double_t &fC43=fC[13];
430 Double_t &fC44=fC[14];
432 //Apply angle correction, if requested
434 Double_t angle=TMath::Sqrt((1.+ fP3*fP3)/((1-fP2)*(1.+fP2)));
441 Double_t beta2=p2/(p2 + mass*mass);
443 //Calculating the multiple scattering corrections******************
449 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
450 Double_t theta2=0.0136*0.0136/(beta2*p2)*TMath::Abs(xOverX0);
451 if (GetUseLogTermMS()) {
452 double lt = 1+0.038*TMath::Log(TMath::Abs(xOverX0));
453 if (lt>0) theta2 *= lt*lt;
455 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
456 cC22 = theta2*((1.-fP2)*(1.+fP2))*(1. + fP3*fP3);
457 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
458 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
459 cC44 = theta2*fP3*fP4*fP3*fP4;
462 //Calculating the energy loss corrections************************
464 if ((xTimesRho != 0.) && (beta2 < 1.)) {
465 Double_t dE=dEdx*xTimesRho;
466 Double_t e=TMath::Sqrt(p2 + mass*mass);
467 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
468 //cP4 = (1.- e/p2*dE);
469 if ( (1.+ dE/p2*(dE + 2*e)) < 0. ) return kFALSE;
470 cP4 = 1./TMath::Sqrt(1.+ dE/p2*(dE + 2*e)); //A precise formula by Ruben !
471 if (TMath::Abs(fP4*cP4)>100.) return kFALSE; //Do not track below 10 MeV/c
474 // Approximate energy loss fluctuation (M.Ivanov)
475 const Double_t knst=0.07; // To be tuned.
476 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
477 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
481 //Applying the corrections*****************************
493 Bool_t AliExternalTrackParam::CorrectForMeanMaterial
494 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
496 Double_t (*Bethe)(Double_t)) {
497 //------------------------------------------------------------------
498 // This function corrects the track parameters for the crossed material.
499 // "xOverX0" - X/X0, the thickness in units of the radiation length.
500 // "xTimesRho" - is the product length*density (g/cm^2).
501 // It should be passed as negative when propagating tracks
502 // from the intreaction point to the outside of the central barrel.
503 // "mass" - the mass of this particle (GeV/c^2).
504 // "anglecorr" - switch for the angular correction
505 // "Bethe" - function calculating the energy loss (GeV/(g/cm^2))
506 //------------------------------------------------------------------
508 Double_t bg=GetP()/mass;
509 Double_t dEdx=Bethe(bg);
511 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
514 Bool_t AliExternalTrackParam::CorrectForMeanMaterialZA
515 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
522 //------------------------------------------------------------------
523 // This function corrects the track parameters for the crossed material
524 // using the full Geant-like Bethe-Bloch formula parameterization
525 // "xOverX0" - X/X0, the thickness in units of the radiation length.
526 // "xTimesRho" - is the product length*density (g/cm^2).
527 // It should be passed as negative when propagating tracks
528 // from the intreaction point to the outside of the central barrel.
529 // "mass" - the mass of this particle (GeV/c^2).
530 // "density" - mean density (g/cm^3)
531 // "zOverA" - mean Z/A
532 // "exEnergy" - mean exitation energy (GeV)
533 // "jp1" - density effect first junction point
534 // "jp2" - density effect second junction point
535 // "anglecorr" - switch for the angular correction
537 // The default values of the parameters are for silicon
539 //------------------------------------------------------------------
541 Double_t bg=GetP()/mass;
542 Double_t dEdx=BetheBlochGeant(bg,density,jp1,jp2,exEnergy,zOverA);
544 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
549 Bool_t AliExternalTrackParam::CorrectForMaterial
550 (Double_t d, Double_t x0, Double_t mass, Double_t (*Bethe)(Double_t)) {
551 //------------------------------------------------------------------
552 // Deprecated function !
553 // Better use CorrectForMeanMaterial instead of it.
555 // This function corrects the track parameters for the crossed material
556 // "d" - the thickness (fraction of the radiation length)
557 // It should be passed as negative when propagating tracks
558 // from the intreaction point to the outside of the central barrel.
559 // "x0" - the radiation length (g/cm^2)
560 // "mass" - the mass of this particle (GeV/c^2)
561 //------------------------------------------------------------------
563 return CorrectForMeanMaterial(d,x0*d,mass,kTRUE,Bethe);
567 Double_t AliExternalTrackParam::BetheBlochAleph(Double_t bg,
574 // This is the empirical ALEPH parameterization of the Bethe-Bloch formula.
575 // It is normalized to 1 at the minimum.
579 // The default values for the kp* parameters are for ALICE TPC.
580 // The returned value is in MIP units
583 Double_t beta = bg/TMath::Sqrt(1.+ bg*bg);
585 Double_t aa = TMath::Power(beta,kp4);
586 Double_t bb = TMath::Power(1./bg,kp5);
588 bb=TMath::Log(kp3+bb);
590 return (kp2-aa-bb)*kp1/aa;
593 Double_t AliExternalTrackParam::BetheBlochGeant(Double_t bg,
600 // This is the parameterization of the Bethe-Bloch formula inspired by Geant.
603 // kp0 - density [g/cm^3]
604 // kp1 - density effect first junction point
605 // kp2 - density effect second junction point
606 // kp3 - mean excitation energy [GeV]
609 // The default values for the kp* parameters are for silicon.
610 // The returned value is in [GeV/(g/cm^2)].
613 const Double_t mK = 0.307075e-3; // [GeV*cm^2/g]
614 const Double_t me = 0.511e-3; // [GeV/c^2]
615 const Double_t rho = kp0;
616 const Double_t x0 = kp1*2.303;
617 const Double_t x1 = kp2*2.303;
618 const Double_t mI = kp3;
619 const Double_t mZA = kp4;
620 const Double_t bg2 = bg*bg;
621 const Double_t maxT= 2*me*bg2; // neglecting the electron mass
625 const Double_t x=TMath::Log(bg);
626 const Double_t lhwI=TMath::Log(28.816*1e-9*TMath::Sqrt(rho*mZA)/mI);
630 const Double_t r=(x1-x)/(x1-x0);
631 d2 = lhwI + x - 0.5 + (0.5 - lhwI - x0)*r*r*r;
634 return mK*mZA*(1+bg2)/bg2*
635 (0.5*TMath::Log(2*me*bg2*maxT/(mI*mI)) - bg2/(1+bg2) - d2);
638 Double_t AliExternalTrackParam::BetheBlochSolid(Double_t bg) {
639 //------------------------------------------------------------------
640 // This is an approximation of the Bethe-Bloch formula,
641 // reasonable for solid materials.
642 // All the parameters are, in fact, for Si.
643 // The returned value is in [GeV/(g/cm^2)]
644 //------------------------------------------------------------------
646 return BetheBlochGeant(bg);
649 Double_t AliExternalTrackParam::BetheBlochGas(Double_t bg) {
650 //------------------------------------------------------------------
651 // This is an approximation of the Bethe-Bloch formula,
652 // reasonable for gas materials.
653 // All the parameters are, in fact, for Ne.
654 // The returned value is in [GeV/(g/cm^2)]
655 //------------------------------------------------------------------
657 const Double_t rho = 0.9e-3;
658 const Double_t x0 = 2.;
659 const Double_t x1 = 4.;
660 const Double_t mI = 140.e-9;
661 const Double_t mZA = 0.49555;
663 return BetheBlochGeant(bg,rho,x0,x1,mI,mZA);
666 Bool_t AliExternalTrackParam::Rotate(Double_t alpha) {
667 //------------------------------------------------------------------
668 // Transform this track to the local coord. system rotated
669 // by angle "alpha" (rad) with respect to the global coord. system.
670 //------------------------------------------------------------------
671 if (TMath::Abs(fP[2]) >= kAlmost1) {
672 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
676 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
677 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
681 Double_t &fC00=fC[0];
682 Double_t &fC10=fC[1];
683 Double_t &fC20=fC[3];
684 Double_t &fC21=fC[4];
685 Double_t &fC22=fC[5];
686 Double_t &fC30=fC[6];
687 Double_t &fC32=fC[8];
688 Double_t &fC40=fC[10];
689 Double_t &fC42=fC[12];
692 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
693 Double_t sf=fP2, cf=TMath::Sqrt((1.- fP2)*(1.+fP2)); // Improve precision
694 // RS: check if rotation does no invalidate track model (cos(local_phi)>=0, i.e. particle
695 // direction in local frame is along the X axis
696 if ((cf*ca+sf*sa)<0) {
697 AliDebug(1,Form("Rotation failed: local cos(phi) would become %.2f",cf*ca+sf*sa));
701 Double_t tmp=sf*ca - cf*sa;
703 if (TMath::Abs(tmp) >= kAlmost1) {
704 if (TMath::Abs(tmp) > 1.+ Double_t(FLT_EPSILON))
705 AliWarning(Form("Rotation failed ! %.10e",tmp));
713 if (TMath::Abs(cf)<kAlmost0) {
714 AliError(Form("Too small cosine value %f",cf));
718 Double_t rr=(ca+sf/cf*sa);
735 Bool_t AliExternalTrackParam::Invert() {
736 //------------------------------------------------------------------
737 // Transform this track to the local coord. system rotated by 180 deg.
738 //------------------------------------------------------------------
740 fAlpha += TMath::Pi();
741 while (fAlpha < -TMath::Pi()) fAlpha += 2*TMath::Pi();
742 while (fAlpha >= TMath::Pi()) fAlpha -= 2*TMath::Pi();
749 fC[1] = -fC[1]; // since the fP1 and fP2 are not inverted, their covariances with others change sign
759 Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
760 //----------------------------------------------------------------
761 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
762 //----------------------------------------------------------------
764 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
766 Double_t crv=GetC(b);
767 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
769 Double_t x2r = crv*dx;
770 Double_t f1=fP[2], f2=f1 + x2r;
771 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
772 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
773 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
775 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
778 &fC10=fC[1], &fC11=fC[2],
779 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
780 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
781 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
783 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
784 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
785 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
788 double dy2dx = (f1+f2)/(r1+r2);
790 if (TMath::Abs(x2r)<0.05) {
791 fP1 += dx*(r2 + f2*dy2dx)*fP3; // Many thanks to P.Hristov !
795 // for small dx/R the linear apporximation of the arc by the segment is OK,
796 // but at large dx/R the error is very large and leads to incorrect Z propagation
797 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
798 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
799 // Similarly, the rotation angle in linear in dx only for dx<<R
800 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
801 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
803 fP2 = TMath::Sin(rot + TMath::ASin(fP2));
808 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
809 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
810 Double_t f12= dx*fP3*f1/(r1*r1*r1);
811 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
813 Double_t f24= dx; f24*=cc;
816 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
817 Double_t b02=f24*fC40;
818 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
819 Double_t b12=f24*fC41;
820 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
821 Double_t b22=f24*fC42;
822 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
823 Double_t b42=f24*fC44;
824 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
825 Double_t b32=f24*fC43;
828 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
829 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
830 Double_t a22=f24*b42;
832 //F*C*Ft = C + (b + bt + a)
833 fC00 += b00 + b00 + a00;
834 fC10 += b10 + b01 + a01;
835 fC20 += b20 + b02 + a02;
838 fC11 += b11 + b11 + a11;
839 fC21 += b21 + b12 + a12;
842 fC22 += b22 + b22 + a22;
852 AliExternalTrackParam::Propagate(Double_t alpha, Double_t x, Double_t b) {
853 //------------------------------------------------------------------
854 // Transform this track to the local coord. system rotated
855 // by angle "alpha" (rad) with respect to the global coord. system,
856 // and propagate this track to the plane X=xk (cm) in the field "b" (kG)
857 //------------------------------------------------------------------
859 //Save the parameters
862 Double_t ps[5], cs[15];
863 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
864 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
867 if (PropagateTo(x,b)) return kTRUE;
869 //Restore the parameters, if the operation failed
872 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
873 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
877 Bool_t AliExternalTrackParam::PropagateBxByBz
878 (Double_t alpha, Double_t x, Double_t b[3]) {
879 //------------------------------------------------------------------
880 // Transform this track to the local coord. system rotated
881 // by angle "alpha" (rad) with respect to the global coord. system,
882 // and propagate this track to the plane X=xk (cm),
883 // taking into account all three components of the B field, "b[3]" (kG)
884 //------------------------------------------------------------------
886 //Save the parameters
889 Double_t ps[5], cs[15];
890 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
891 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
894 if (PropagateToBxByBz(x,b)) return kTRUE;
896 //Restore the parameters, if the operation failed
899 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
900 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
905 void AliExternalTrackParam::Propagate(Double_t len, Double_t x[3],
906 Double_t p[3], Double_t bz) const {
907 //+++++++++++++++++++++++++++++++++++++++++
908 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
909 // Extrapolate track along simple helix in magnetic field
910 // Arguments: len -distance alogn helix, [cm]
911 // bz - mag field, [kGaus]
912 // Returns: x and p contain extrapolated positon and momentum
913 // The momentum returned for straight-line tracks is meaningless !
914 //+++++++++++++++++++++++++++++++++++++++++
917 if (OneOverPt() < kAlmost0 || TMath::Abs(bz) < kAlmost0Field || GetC(bz) < kAlmost0){ //straight-line tracks
918 Double_t unit[3]; GetDirection(unit);
923 p[0]=unit[0]/kAlmost0;
924 p[1]=unit[1]/kAlmost0;
925 p[2]=unit[2]/kAlmost0;
929 Double_t a = -kB2C*bz*GetSign();
931 x[0] += p[0]*TMath::Sin(rho*len)/a - p[1]*(1-TMath::Cos(rho*len))/a;
932 x[1] += p[1]*TMath::Sin(rho*len)/a + p[0]*(1-TMath::Cos(rho*len))/a;
936 p[0] = p0 *TMath::Cos(rho*len) - p[1]*TMath::Sin(rho*len);
937 p[1] = p[1]*TMath::Cos(rho*len) + p0 *TMath::Sin(rho*len);
941 Bool_t AliExternalTrackParam::Intersect(Double_t pnt[3], Double_t norm[3],
943 //+++++++++++++++++++++++++++++++++++++++++
944 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
945 // Finds point of intersection (if exists) of the helix with the plane.
946 // Stores result in fX and fP.
947 // Arguments: planePoint,planeNorm - the plane defined by any plane's point
948 // and vector, normal to the plane
949 // Returns: kTrue if helix intersects the plane, kFALSE otherwise.
950 //+++++++++++++++++++++++++++++++++++++++++
951 Double_t x0[3]; GetXYZ(x0); //get track position in MARS
953 //estimates initial helix length up to plane
955 (pnt[0]-x0[0])*norm[0] + (pnt[1]-x0[1])*norm[1] + (pnt[2]-x0[2])*norm[2];
956 Double_t dist=99999,distPrev=dist;
958 while(TMath::Abs(dist)>0.00001){
959 //calculates helix at the distance s from x0 ALONG the helix
962 //distance between current helix position and plane
963 dist=(x[0]-pnt[0])*norm[0]+(x[1]-pnt[1])*norm[1]+(x[2]-pnt[2])*norm[2];
965 if(TMath::Abs(dist) >= TMath::Abs(distPrev)) {return kFALSE;}
969 //on exit pnt is intersection point,norm is track vector at that point,
971 for (Int_t i=0; i<3; i++) {pnt[i]=x[i]; norm[i]=p[i];}
976 AliExternalTrackParam::GetPredictedChi2(Double_t p[2],Double_t cov[3]) const {
977 //----------------------------------------------------------------
978 // Estimate the chi2 of the space point "p" with the cov. matrix "cov"
979 //----------------------------------------------------------------
980 Double_t sdd = fC[0] + cov[0];
981 Double_t sdz = fC[1] + cov[1];
982 Double_t szz = fC[2] + cov[2];
983 Double_t det = sdd*szz - sdz*sdz;
985 if (TMath::Abs(det) < kAlmost0) return kVeryBig;
987 Double_t d = fP[0] - p[0];
988 Double_t z = fP[1] - p[1];
990 return (d*szz*d - 2*d*sdz*z + z*sdd*z)/det;
993 Double_t AliExternalTrackParam::
994 GetPredictedChi2(Double_t p[3],Double_t covyz[3],Double_t covxyz[3]) const {
995 //----------------------------------------------------------------
996 // Estimate the chi2 of the 3D space point "p" and
997 // the full covariance matrix "covyz" and "covxyz"
999 // Cov(x,x) ... : covxyz[0]
1000 // Cov(y,x) ... : covxyz[1] covyz[0]
1001 // Cov(z,x) ... : covxyz[2] covyz[1] covyz[2]
1002 //----------------------------------------------------------------
1010 Double_t f=GetSnp();
1011 if (TMath::Abs(f) >= kAlmost1) return kVeryBig;
1012 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1013 Double_t a=f/r, b=GetTgl()/r;
1015 Double_t s2=333.*333.; //something reasonably big (cm^2)
1018 v(0,0)= s2; v(0,1)= a*s2; v(0,2)= b*s2;;
1019 v(1,0)=a*s2; v(1,1)=a*a*s2 + GetSigmaY2(); v(1,2)=a*b*s2 + GetSigmaZY();
1020 v(2,0)=b*s2; v(2,1)=a*b*s2 + GetSigmaZY(); v(2,2)=b*b*s2 + GetSigmaZ2();
1022 v(0,0)+=covxyz[0]; v(0,1)+=covxyz[1]; v(0,2)+=covxyz[2];
1023 v(1,0)+=covxyz[1]; v(1,1)+=covyz[0]; v(1,2)+=covyz[1];
1024 v(2,0)+=covxyz[2]; v(2,1)+=covyz[1]; v(2,2)+=covyz[2];
1027 if (!v.IsValid()) return kVeryBig;
1030 for (Int_t i = 0; i < 3; i++)
1031 for (Int_t j = 0; j < 3; j++) chi2 += res[i]*res[j]*v(i,j);
1036 Double_t AliExternalTrackParam::
1037 GetPredictedChi2(const AliExternalTrackParam *t) const {
1038 //----------------------------------------------------------------
1039 // Estimate the chi2 (5 dof) of this track with respect to the track
1040 // given by the argument.
1041 // The two tracks must be in the same reference system
1042 // and estimated at the same reference plane.
1043 //----------------------------------------------------------------
1045 if (TMath::Abs(1. - t->GetAlpha()/GetAlpha()) > FLT_EPSILON) {
1046 AliError("The reference systems of the tracks differ !");
1049 if (TMath::Abs(1. - t->GetX()/GetX()) > FLT_EPSILON) {
1050 AliError("The reference of the tracks planes differ !");
1055 c(0,0)=GetSigmaY2();
1056 c(1,0)=GetSigmaZY(); c(1,1)=GetSigmaZ2();
1057 c(2,0)=GetSigmaSnpY(); c(2,1)=GetSigmaSnpZ(); c(2,2)=GetSigmaSnp2();
1058 c(3,0)=GetSigmaTglY(); c(3,1)=GetSigmaTglZ(); c(3,2)=GetSigmaTglSnp(); c(3,3)=GetSigmaTgl2();
1059 c(4,0)=GetSigma1PtY(); c(4,1)=GetSigma1PtZ(); c(4,2)=GetSigma1PtSnp(); c(4,3)=GetSigma1PtTgl(); c(4,4)=GetSigma1Pt2();
1061 c(0,0)+=t->GetSigmaY2();
1062 c(1,0)+=t->GetSigmaZY(); c(1,1)+=t->GetSigmaZ2();
1063 c(2,0)+=t->GetSigmaSnpY();c(2,1)+=t->GetSigmaSnpZ();c(2,2)+=t->GetSigmaSnp2();
1064 c(3,0)+=t->GetSigmaTglY();c(3,1)+=t->GetSigmaTglZ();c(3,2)+=t->GetSigmaTglSnp();c(3,3)+=t->GetSigmaTgl2();
1065 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();
1067 c(0,2)=c(2,0); c(1,2)=c(2,1);
1068 c(0,3)=c(3,0); c(1,3)=c(3,1); c(2,3)=c(3,2);
1069 c(0,4)=c(4,0); c(1,4)=c(4,1); c(2,4)=c(4,2); c(3,4)=c(4,3);
1072 if (!c.IsValid()) return kVeryBig;
1078 GetSnp() - t->GetSnp(),
1079 GetTgl() - t->GetTgl(),
1080 GetSigned1Pt() - t->GetSigned1Pt()
1084 for (Int_t i = 0; i < 5; i++)
1085 for (Int_t j = 0; j < 5; j++) chi2 += res[i]*res[j]*c(i,j);
1090 Bool_t AliExternalTrackParam::
1091 PropagateTo(Double_t p[3],Double_t covyz[3],Double_t covxyz[3],Double_t bz) {
1092 //----------------------------------------------------------------
1093 // Propagate this track to the plane
1094 // the 3D space point "p" (with the covariance matrix "covyz" and "covxyz")
1096 // The magnetic field is "bz" (kG)
1098 // The track curvature and the change of the covariance matrix
1099 // of the track parameters are negleted !
1100 // (So the "step" should be small compared with 1/curvature)
1101 //----------------------------------------------------------------
1103 Double_t f=GetSnp();
1104 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1105 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1106 Double_t a=f/r, b=GetTgl()/r;
1108 Double_t s2=333.*333.; //something reasonably big (cm^2)
1111 tV(0,0)= s2; tV(0,1)= a*s2; tV(0,2)= b*s2;
1112 tV(1,0)=a*s2; tV(1,1)=a*a*s2; tV(1,2)=a*b*s2;
1113 tV(2,0)=b*s2; tV(2,1)=a*b*s2; tV(2,2)=b*b*s2;
1116 pV(0,0)=covxyz[0]; pV(0,1)=covxyz[1]; pV(0,2)=covxyz[2];
1117 pV(1,0)=covxyz[1]; pV(1,1)=covyz[0]; pV(1,2)=covyz[1];
1118 pV(2,0)=covxyz[2]; pV(2,1)=covyz[1]; pV(2,2)=covyz[2];
1120 TMatrixDSym tpV(tV);
1123 if (!tpV.IsValid()) return kFALSE;
1125 TMatrixDSym pW(3),tW(3);
1126 for (Int_t i=0; i<3; i++)
1127 for (Int_t j=0; j<3; j++) {
1129 for (Int_t k=0; k<3; k++) {
1130 pW(i,j) += tV(i,k)*tpV(k,j);
1131 tW(i,j) += pV(i,k)*tpV(k,j);
1135 Double_t t[3] = {GetX(), GetY(), GetZ()};
1138 for (Int_t i=0; i<3; i++) x += (tW(0,i)*t[i] + pW(0,i)*p[i]);
1139 Double_t crv=GetC(bz);
1140 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1142 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1146 for (Int_t i=0; i<3; i++) fP[0] += (tW(1,i)*t[i] + pW(1,i)*p[i]);
1148 for (Int_t i=0; i<3; i++) fP[1] += (tW(2,i)*t[i] + pW(2,i)*p[i]);
1153 Double_t *AliExternalTrackParam::GetResiduals(
1154 Double_t *p,Double_t *cov,Bool_t updated) const {
1155 //------------------------------------------------------------------
1156 // Returns the track residuals with the space point "p" having
1157 // the covariance matrix "cov".
1158 // If "updated" is kTRUE, the track parameters expected to be updated,
1159 // otherwise they must be predicted.
1160 //------------------------------------------------------------------
1161 static Double_t res[2];
1163 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1165 r00-=fC[0]; r01-=fC[1]; r11-=fC[2];
1167 r00+=fC[0]; r01+=fC[1]; r11+=fC[2];
1169 Double_t det=r00*r11 - r01*r01;
1171 if (TMath::Abs(det) < kAlmost0) return 0;
1173 Double_t tmp=r00; r00=r11/det; r11=tmp/det;
1175 if (r00 < 0.) return 0;
1176 if (r11 < 0.) return 0;
1178 Double_t dy = fP[0] - p[0];
1179 Double_t dz = fP[1] - p[1];
1181 res[0]=dy*TMath::Sqrt(r00);
1182 res[1]=dz*TMath::Sqrt(r11);
1187 Bool_t AliExternalTrackParam::Update(Double_t p[2], Double_t cov[3]) {
1188 //------------------------------------------------------------------
1189 // Update the track parameters with the space point "p" having
1190 // the covariance matrix "cov"
1191 //------------------------------------------------------------------
1192 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
1195 &fC10=fC[1], &fC11=fC[2],
1196 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1197 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1198 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1200 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1201 r00+=fC00; r01+=fC10; r11+=fC11;
1202 Double_t det=r00*r11 - r01*r01;
1204 if (TMath::Abs(det) < kAlmost0) return kFALSE;
1207 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
1209 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
1210 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
1211 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
1212 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
1213 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
1215 Double_t dy=p[0] - fP0, dz=p[1] - fP1;
1216 Double_t sf=fP2 + k20*dy + k21*dz;
1217 if (TMath::Abs(sf) > kAlmost1) return kFALSE;
1219 fP0 += k00*dy + k01*dz;
1220 fP1 += k10*dy + k11*dz;
1222 fP3 += k30*dy + k31*dz;
1223 fP4 += k40*dy + k41*dz;
1225 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
1226 Double_t c12=fC21, c13=fC31, c14=fC41;
1228 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
1229 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
1230 fC40-=k00*c04+k01*c14;
1232 fC11-=k10*c01+k11*fC11;
1233 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
1234 fC41-=k10*c04+k11*c14;
1236 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
1237 fC42-=k20*c04+k21*c14;
1239 fC33-=k30*c03+k31*c13;
1240 fC43-=k30*c04+k31*c14;
1242 fC44-=k40*c04+k41*c14;
1250 AliExternalTrackParam::GetHelixParameters(Double_t hlx[6], Double_t b) const {
1251 //--------------------------------------------------------------------
1252 // External track parameters -> helix parameters
1253 // "b" - magnetic field (kG)
1254 //--------------------------------------------------------------------
1255 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1257 hlx[0]=fP[0]; hlx[1]=fP[1]; hlx[2]=fP[2]; hlx[3]=fP[3];
1259 hlx[5]=fX*cs - hlx[0]*sn; // x0
1260 hlx[0]=fX*sn + hlx[0]*cs; // y0
1262 hlx[2]=TMath::ASin(hlx[2]) + fAlpha; // phi0
1264 hlx[4]=GetC(b); // C
1268 static void Evaluate(const Double_t *h, Double_t t,
1269 Double_t r[3], //radius vector
1270 Double_t g[3], //first defivatives
1271 Double_t gg[3]) //second derivatives
1273 //--------------------------------------------------------------------
1274 // Calculate position of a point on a track and some derivatives
1275 //--------------------------------------------------------------------
1276 Double_t phase=h[4]*t+h[2];
1277 Double_t sn=TMath::Sin(phase), cs=TMath::Cos(phase);
1281 if (TMath::Abs(h[4])>kAlmost0) {
1282 r[0] += (sn - h[6])/h[4];
1283 r[1] -= (cs - h[7])/h[4];
1285 r[2] = h[1] + h[3]*t;
1287 g[0] = cs; g[1]=sn; g[2]=h[3];
1289 gg[0]=-h[4]*sn; gg[1]=h[4]*cs; gg[2]=0.;
1292 Double_t AliExternalTrackParam::GetDCA(const AliExternalTrackParam *p,
1293 Double_t b, Double_t &xthis, Double_t &xp) const {
1294 //------------------------------------------------------------
1295 // Returns the (weighed !) distance of closest approach between
1296 // this track and the track "p".
1297 // Other returned values:
1298 // xthis, xt - coordinates of tracks' reference planes at the DCA
1299 //-----------------------------------------------------------
1300 Double_t dy2=GetSigmaY2() + p->GetSigmaY2();
1301 Double_t dz2=GetSigmaZ2() + p->GetSigmaZ2();
1304 Double_t p1[8]; GetHelixParameters(p1,b);
1305 p1[6]=TMath::Sin(p1[2]); p1[7]=TMath::Cos(p1[2]);
1306 Double_t p2[8]; p->GetHelixParameters(p2,b);
1307 p2[6]=TMath::Sin(p2[2]); p2[7]=TMath::Cos(p2[2]);
1310 Double_t r1[3],g1[3],gg1[3]; Double_t t1=0.;
1311 Evaluate(p1,t1,r1,g1,gg1);
1312 Double_t r2[3],g2[3],gg2[3]; Double_t t2=0.;
1313 Evaluate(p2,t2,r2,g2,gg2);
1315 Double_t dx=r2[0]-r1[0], dy=r2[1]-r1[1], dz=r2[2]-r1[2];
1316 Double_t dm=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1320 Double_t gt1=-(dx*g1[0]/dx2 + dy*g1[1]/dy2 + dz*g1[2]/dz2);
1321 Double_t gt2=+(dx*g2[0]/dx2 + dy*g2[1]/dy2 + dz*g2[2]/dz2);
1322 Double_t h11=(g1[0]*g1[0] - dx*gg1[0])/dx2 +
1323 (g1[1]*g1[1] - dy*gg1[1])/dy2 +
1324 (g1[2]*g1[2] - dz*gg1[2])/dz2;
1325 Double_t h22=(g2[0]*g2[0] + dx*gg2[0])/dx2 +
1326 (g2[1]*g2[1] + dy*gg2[1])/dy2 +
1327 (g2[2]*g2[2] + dz*gg2[2])/dz2;
1328 Double_t h12=-(g1[0]*g2[0]/dx2 + g1[1]*g2[1]/dy2 + g1[2]*g2[2]/dz2);
1330 Double_t det=h11*h22-h12*h12;
1333 if (TMath::Abs(det)<1.e-33) {
1334 //(quasi)singular Hessian
1337 dt1=-(gt1*h22 - gt2*h12)/det;
1338 dt2=-(h11*gt2 - h12*gt1)/det;
1341 if ((dt1*gt1+dt2*gt2)>0) {dt1=-dt1; dt2=-dt2;}
1343 //check delta(phase1) ?
1344 //check delta(phase2) ?
1346 if (TMath::Abs(dt1)/(TMath::Abs(t1)+1.e-3) < 1.e-4)
1347 if (TMath::Abs(dt2)/(TMath::Abs(t2)+1.e-3) < 1.e-4) {
1348 if ((gt1*gt1+gt2*gt2) > 1.e-4/dy2/dy2)
1349 AliDebug(1," stopped at not a stationary point !");
1350 Double_t lmb=h11+h22; lmb=lmb-TMath::Sqrt(lmb*lmb-4*det);
1352 AliDebug(1," stopped at not a minimum !");
1357 for (Int_t div=1 ; ; div*=2) {
1358 Evaluate(p1,t1+dt1,r1,g1,gg1);
1359 Evaluate(p2,t2+dt2,r2,g2,gg2);
1360 dx=r2[0]-r1[0]; dy=r2[1]-r1[1]; dz=r2[2]-r1[2];
1361 dd=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1365 AliDebug(1," overshoot !"); break;
1375 if (max<=0) AliDebug(1," too many iterations !");
1377 Double_t cs=TMath::Cos(GetAlpha());
1378 Double_t sn=TMath::Sin(GetAlpha());
1379 xthis=r1[0]*cs + r1[1]*sn;
1381 cs=TMath::Cos(p->GetAlpha());
1382 sn=TMath::Sin(p->GetAlpha());
1383 xp=r2[0]*cs + r2[1]*sn;
1385 return TMath::Sqrt(dm*TMath::Sqrt(dy2*dz2));
1388 Double_t AliExternalTrackParam::
1389 PropagateToDCA(AliExternalTrackParam *p, Double_t b) {
1390 //--------------------------------------------------------------
1391 // Propagates this track and the argument track to the position of the
1392 // distance of closest approach.
1393 // Returns the (weighed !) distance of closest approach.
1394 //--------------------------------------------------------------
1396 Double_t dca=GetDCA(p,b,xthis,xp);
1398 if (!PropagateTo(xthis,b)) {
1399 //AliWarning(" propagation failed !");
1403 if (!p->PropagateTo(xp,b)) {
1404 //AliWarning(" propagation failed !";
1412 Bool_t AliExternalTrackParam::PropagateToDCA(const AliVVertex *vtx,
1413 Double_t b, Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1415 // Propagate this track to the DCA to vertex "vtx",
1416 // if the (rough) transverse impact parameter is not bigger then "maxd".
1417 // Magnetic field is "b" (kG).
1419 // a) The track gets extapolated to the DCA to the vertex.
1420 // b) The impact parameters and their covariance matrix are calculated.
1422 // In the case of success, the returned value is kTRUE
1423 // (otherwise, it's kFALSE)
1425 Double_t alpha=GetAlpha();
1426 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1427 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1428 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1429 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1432 //Estimate the impact parameter neglecting the track curvature
1433 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1434 if (d > maxd) return kFALSE;
1436 //Propagate to the DCA
1437 Double_t crv=GetC(b);
1438 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1440 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1441 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1442 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1446 yv=-xv*sn + yv*cs; xv=x;
1448 if (!Propagate(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1450 if (dz==0) return kTRUE;
1451 dz[0] = GetParameter()[0] - yv;
1452 dz[1] = GetParameter()[1] - zv;
1454 if (covar==0) return kTRUE;
1455 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1457 //***** Improvements by A.Dainese
1458 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1459 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1460 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1461 covar[1] = GetCovariance()[1]; // between (x,y) and z
1462 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1468 Bool_t AliExternalTrackParam::PropagateToDCABxByBz(const AliVVertex *vtx,
1469 Double_t b[3], Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1471 // Propagate this track to the DCA to vertex "vtx",
1472 // if the (rough) transverse impact parameter is not bigger then "maxd".
1474 // This function takes into account all three components of the magnetic
1475 // field given by the b[3] arument (kG)
1477 // a) The track gets extapolated to the DCA to the vertex.
1478 // b) The impact parameters and their covariance matrix are calculated.
1480 // In the case of success, the returned value is kTRUE
1481 // (otherwise, it's kFALSE)
1483 Double_t alpha=GetAlpha();
1484 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1485 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1486 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1487 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1490 //Estimate the impact parameter neglecting the track curvature
1491 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1492 if (d > maxd) return kFALSE;
1494 //Propagate to the DCA
1495 Double_t crv=GetC(b[2]);
1496 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1498 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1499 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1500 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1504 yv=-xv*sn + yv*cs; xv=x;
1506 if (!PropagateBxByBz(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1508 if (dz==0) return kTRUE;
1509 dz[0] = GetParameter()[0] - yv;
1510 dz[1] = GetParameter()[1] - zv;
1512 if (covar==0) return kTRUE;
1513 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1515 //***** Improvements by A.Dainese
1516 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1517 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1518 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1519 covar[1] = GetCovariance()[1]; // between (x,y) and z
1520 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1526 void AliExternalTrackParam::GetDirection(Double_t d[3]) const {
1527 //----------------------------------------------------------------
1528 // This function returns a unit vector along the track direction
1529 // in the global coordinate system.
1530 //----------------------------------------------------------------
1531 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1533 Double_t csp =TMath::Sqrt((1.-snp)*(1.+snp));
1534 Double_t norm=TMath::Sqrt(1.+ fP[3]*fP[3]);
1535 d[0]=(csp*cs - snp*sn)/norm;
1536 d[1]=(snp*cs + csp*sn)/norm;
1540 Bool_t AliExternalTrackParam::GetPxPyPz(Double_t p[3]) const {
1541 //---------------------------------------------------------------------
1542 // This function returns the global track momentum components
1543 // Results for (nearly) straight tracks are meaningless !
1544 //---------------------------------------------------------------------
1545 p[0]=fP[4]; p[1]=fP[2]; p[2]=fP[3];
1546 return Local2GlobalMomentum(p,fAlpha);
1549 Double_t AliExternalTrackParam::Px() const {
1550 //---------------------------------------------------------------------
1551 // Returns x-component of momentum
1552 // Result for (nearly) straight tracks is meaningless !
1553 //---------------------------------------------------------------------
1555 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1561 Double_t AliExternalTrackParam::Py() const {
1562 //---------------------------------------------------------------------
1563 // Returns y-component of momentum
1564 // Result for (nearly) straight tracks is meaningless !
1565 //---------------------------------------------------------------------
1567 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1573 Double_t AliExternalTrackParam::Xv() const {
1574 //---------------------------------------------------------------------
1575 // Returns x-component of first track point
1576 //---------------------------------------------------------------------
1578 Double_t r[3]={0.,0.,0.};
1584 Double_t AliExternalTrackParam::Yv() const {
1585 //---------------------------------------------------------------------
1586 // Returns y-component of first track point
1587 //---------------------------------------------------------------------
1589 Double_t r[3]={0.,0.,0.};
1595 Double_t AliExternalTrackParam::Theta() const {
1596 // return theta angle of momentum
1598 return 0.5*TMath::Pi() - TMath::ATan(fP[3]);
1601 Double_t AliExternalTrackParam::Phi() const {
1602 //---------------------------------------------------------------------
1603 // Returns the azimuthal angle of momentum
1605 //---------------------------------------------------------------------
1607 Double_t phi=TMath::ASin(fP[2]) + fAlpha;
1608 if (phi<0.) phi+=2.*TMath::Pi();
1609 else if (phi>=2.*TMath::Pi()) phi-=2.*TMath::Pi();
1614 Double_t AliExternalTrackParam::M() const {
1615 // return particle mass
1617 // No mass information available so far.
1618 // Redifine in derived class!
1623 Double_t AliExternalTrackParam::E() const {
1624 // return particle energy
1626 // No PID information available so far.
1627 // Redifine in derived class!
1632 Double_t AliExternalTrackParam::Eta() const {
1633 // return pseudorapidity
1635 return -TMath::Log(TMath::Tan(0.5 * Theta()));
1638 Double_t AliExternalTrackParam::Y() const {
1641 // No PID information available so far.
1642 // Redifine in derived class!
1647 Bool_t AliExternalTrackParam::GetXYZ(Double_t *r) const {
1648 //---------------------------------------------------------------------
1649 // This function returns the global track position
1650 //---------------------------------------------------------------------
1651 r[0]=fX; r[1]=fP[0]; r[2]=fP[1];
1652 return Local2GlobalPosition(r,fAlpha);
1655 Bool_t AliExternalTrackParam::GetCovarianceXYZPxPyPz(Double_t cv[21]) const {
1656 //---------------------------------------------------------------------
1657 // This function returns the global covariance matrix of the track params
1659 // Cov(x,x) ... : cv[0]
1660 // Cov(y,x) ... : cv[1] cv[2]
1661 // Cov(z,x) ... : cv[3] cv[4] cv[5]
1662 // Cov(px,x)... : cv[6] cv[7] cv[8] cv[9]
1663 // Cov(py,x)... : cv[10] cv[11] cv[12] cv[13] cv[14]
1664 // Cov(pz,x)... : cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
1666 // Results for (nearly) straight tracks are meaningless !
1667 //---------------------------------------------------------------------
1668 if (TMath::Abs(fP[4])<=kAlmost0) {
1669 for (Int_t i=0; i<21; i++) cv[i]=0.;
1672 if (TMath::Abs(fP[2]) > kAlmost1) {
1673 for (Int_t i=0; i<21; i++) cv[i]=0.;
1676 Double_t pt=1./TMath::Abs(fP[4]);
1677 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1678 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
1680 Double_t m00=-sn, m10=cs;
1681 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
1682 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
1683 Double_t m35=pt, m45=-pt*pt*fP[3];
1689 cv[0 ] = fC[0]*m00*m00;
1690 cv[1 ] = fC[0]*m00*m10;
1691 cv[2 ] = fC[0]*m10*m10;
1695 cv[6 ] = m00*(fC[3]*m23 + fC[10]*m43);
1696 cv[7 ] = m10*(fC[3]*m23 + fC[10]*m43);
1697 cv[8 ] = fC[4]*m23 + fC[11]*m43;
1698 cv[9 ] = m23*(fC[5]*m23 + fC[12]*m43) + m43*(fC[12]*m23 + fC[14]*m43);
1699 cv[10] = m00*(fC[3]*m24 + fC[10]*m44);
1700 cv[11] = m10*(fC[3]*m24 + fC[10]*m44);
1701 cv[12] = fC[4]*m24 + fC[11]*m44;
1702 cv[13] = m23*(fC[5]*m24 + fC[12]*m44) + m43*(fC[12]*m24 + fC[14]*m44);
1703 cv[14] = m24*(fC[5]*m24 + fC[12]*m44) + m44*(fC[12]*m24 + fC[14]*m44);
1704 cv[15] = m00*(fC[6]*m35 + fC[10]*m45);
1705 cv[16] = m10*(fC[6]*m35 + fC[10]*m45);
1706 cv[17] = fC[7]*m35 + fC[11]*m45;
1707 cv[18] = m23*(fC[8]*m35 + fC[12]*m45) + m43*(fC[13]*m35 + fC[14]*m45);
1708 cv[19] = m24*(fC[8]*m35 + fC[12]*m45) + m44*(fC[13]*m35 + fC[14]*m45);
1709 cv[20] = m35*(fC[9]*m35 + fC[13]*m45) + m45*(fC[13]*m35 + fC[14]*m45);
1716 AliExternalTrackParam::GetPxPyPzAt(Double_t x, Double_t b, Double_t *p) const {
1717 //---------------------------------------------------------------------
1718 // This function returns the global track momentum extrapolated to
1719 // the radial position "x" (cm) in the magnetic field "b" (kG)
1720 //---------------------------------------------------------------------
1722 p[1]=fP[2]+(x-fX)*GetC(b);
1724 return Local2GlobalMomentum(p,fAlpha);
1728 AliExternalTrackParam::GetYAt(Double_t x, Double_t b, Double_t &y) const {
1729 //---------------------------------------------------------------------
1730 // This function returns the local Y-coordinate of the intersection
1731 // point between this track and the reference plane "x" (cm).
1732 // Magnetic field "b" (kG)
1733 //---------------------------------------------------------------------
1735 if(TMath::Abs(dx)<=kAlmost0) {y=fP[0]; return kTRUE;}
1737 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1739 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1740 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1742 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1743 y = fP[0] + dx*(f1+f2)/(r1+r2);
1748 AliExternalTrackParam::GetZAt(Double_t x, Double_t b, Double_t &z) const {
1749 //---------------------------------------------------------------------
1750 // This function returns the local Z-coordinate of the intersection
1751 // point between this track and the reference plane "x" (cm).
1752 // Magnetic field "b" (kG)
1753 //---------------------------------------------------------------------
1755 if(TMath::Abs(dx)<=kAlmost0) {z=fP[1]; return kTRUE;}
1757 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1759 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1760 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1762 Double_t r1=sqrt((1.-f1)*(1.+f1)), r2=sqrt((1.-f2)*(1.+f2));
1763 z = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3]; // Many thanks to P.Hristov !
1768 AliExternalTrackParam::GetXYZAt(Double_t x, Double_t b, Double_t *r) const {
1769 //---------------------------------------------------------------------
1770 // This function returns the global track position extrapolated to
1771 // the radial position "x" (cm) in the magnetic field "b" (kG)
1772 //---------------------------------------------------------------------
1774 if(TMath::Abs(dx)<=kAlmost0) return GetXYZ(r);
1776 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1778 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1779 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1781 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1783 r[1] = fP[0] + dx*(f1+f2)/(r1+r2);
1784 r[2] = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3];//Thanks to Andrea & Peter
1786 return Local2GlobalPosition(r,fAlpha);
1789 //_____________________________________________________________________________
1790 void AliExternalTrackParam::Print(Option_t* /*option*/) const
1792 // print the parameters and the covariance matrix
1794 printf("AliExternalTrackParam: x = %-12g alpha = %-12g\n", fX, fAlpha);
1795 printf(" parameters: %12g %12g %12g %12g %12g\n",
1796 fP[0], fP[1], fP[2], fP[3], fP[4]);
1797 printf(" covariance: %12g\n", fC[0]);
1798 printf(" %12g %12g\n", fC[1], fC[2]);
1799 printf(" %12g %12g %12g\n", fC[3], fC[4], fC[5]);
1800 printf(" %12g %12g %12g %12g\n",
1801 fC[6], fC[7], fC[8], fC[9]);
1802 printf(" %12g %12g %12g %12g %12g\n",
1803 fC[10], fC[11], fC[12], fC[13], fC[14]);
1806 Double_t AliExternalTrackParam::GetSnpAt(Double_t x,Double_t b) const {
1808 // Get sinus at given x
1810 Double_t crv=GetC(b);
1811 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1813 Double_t res = fP[2]+dx*crv;
1817 Bool_t AliExternalTrackParam::GetDistance(AliExternalTrackParam *param2, Double_t x, Double_t dist[3], Double_t bz){
1818 //------------------------------------------------------------------------
1819 // Get the distance between two tracks at the local position x
1820 // working in the local frame of this track.
1821 // Origin : Marian.Ivanov@cern.ch
1822 //-----------------------------------------------------------------------
1826 if (!GetYAt(x,bz,xyz[1])) return kFALSE;
1827 if (!GetZAt(x,bz,xyz[2])) return kFALSE;
1830 if (TMath::Abs(GetAlpha()-param2->GetAlpha())<kAlmost0){
1832 if (!param2->GetYAt(x,bz,xyz2[1])) return kFALSE;
1833 if (!param2->GetZAt(x,bz,xyz2[2])) return kFALSE;
1837 Double_t dfi = param2->GetAlpha()-GetAlpha();
1838 Double_t ca = TMath::Cos(dfi), sa = TMath::Sin(dfi);
1839 xyz2[0] = xyz[0]*ca+xyz[1]*sa;
1840 xyz2[1] = -xyz[0]*sa+xyz[1]*ca;
1843 if (!param2->GetYAt(xyz2[0],bz,xyz1[1])) return kFALSE;
1844 if (!param2->GetZAt(xyz2[0],bz,xyz1[2])) return kFALSE;
1846 xyz2[0] = xyz1[0]*ca-xyz1[1]*sa;
1847 xyz2[1] = +xyz1[0]*sa+xyz1[1]*ca;
1850 dist[0] = xyz[0]-xyz2[0];
1851 dist[1] = xyz[1]-xyz2[1];
1852 dist[2] = xyz[2]-xyz2[2];
1859 // Draw functionality.
1860 // Origin: Marian Ivanov, Marian.Ivanov@cern.ch
1863 void AliExternalTrackParam::DrawTrack(Float_t magf, Float_t minR, Float_t maxR, Float_t stepR){
1867 if (minR>maxR) return ;
1868 if (stepR<=0) return ;
1869 Int_t npoints = TMath::Nint((maxR-minR)/stepR)+1;
1870 if (npoints<1) return;
1871 TPolyMarker3D *polymarker = new TPolyMarker3D(npoints);
1872 FillPolymarker(polymarker, magf,minR,maxR,stepR);
1877 void AliExternalTrackParam::FillPolymarker(TPolyMarker3D *pol, Float_t magF, Float_t minR, Float_t maxR, Float_t stepR){
1879 // Fill points in the polymarker
1882 for (Double_t r=minR; r<maxR; r+=stepR){
1884 GetXYZAt(r,magF,point);
1885 pol->SetPoint(counter,point[0],point[1], point[2]);
1886 // printf("xyz\t%f\t%f\t%f\n",point[0], point[1],point[2]);
1891 Int_t AliExternalTrackParam::GetIndex(Int_t i, Int_t j) const {
1893 Int_t min = TMath::Min(i,j);
1894 Int_t max = TMath::Max(i,j);
1896 return min+(max+1)*max/2;
1900 void AliExternalTrackParam::g3helx3(Double_t qfield,
1903 /******************************************************************
1905 * GEANT3 tracking routine in a constant field oriented *
1907 * Tracking is performed with a conventional *
1908 * helix step method *
1910 * Authors R.Brun, M.Hansroul ********* *
1911 * Rewritten V.Perevoztchikov *
1913 * Rewritten in C++ by I.Belikov *
1915 * qfield (kG) - particle charge times magnetic field *
1916 * step (cm) - step length along the helix *
1917 * vect[7](cm,GeV/c) - input/output x, y, z, px/p, py/p ,pz/p, p *
1919 ******************************************************************/
1920 const Int_t ix=0, iy=1, iz=2, ipx=3, ipy=4, ipz=5, ipp=6;
1921 const Double_t kOvSqSix=TMath::Sqrt(1./6.);
1923 Double_t cosx=vect[ipx], cosy=vect[ipy], cosz=vect[ipz];
1925 Double_t rho = qfield*kB2C/vect[ipp];
1926 Double_t tet = rho*step;
1928 Double_t tsint, sintt, sint, cos1t;
1929 if (TMath::Abs(tet) > 0.03) {
1930 sint = TMath::Sin(tet);
1932 tsint = (tet - sint)/tet;
1933 Double_t t=TMath::Sin(0.5*tet);
1937 sintt = (1.-tet*kOvSqSix)*(1.+tet*kOvSqSix); // 1.- tsint;
1942 Double_t f1 = step*sintt;
1943 Double_t f2 = step*cos1t;
1944 Double_t f3 = step*tsint*cosz;
1945 Double_t f4 = -tet*cos1t;
1948 vect[ix] += f1*cosx - f2*cosy;
1949 vect[iy] += f1*cosy + f2*cosx;
1950 vect[iz] += f1*cosz + f3;
1952 vect[ipx] += f4*cosx - f5*cosy;
1953 vect[ipy] += f4*cosy + f5*cosx;
1957 Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]) {
1958 //----------------------------------------------------------------
1959 // Extrapolate this track to the plane X=xk in the field b[].
1961 // X [cm] is in the "tracking coordinate system" of this track.
1962 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
1963 //----------------------------------------------------------------
1966 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
1967 if (TMath::Abs(fP[4])<=kAlmost0) return kFALSE;
1968 // Do not propagate tracks outside the ALICE detector
1969 if (TMath::Abs(dx)>1e5 ||
1970 TMath::Abs(GetY())>1e5 ||
1971 TMath::Abs(GetZ())>1e5) {
1972 AliWarning(Form("Anomalous track, target X:%f",xk));
1977 Double_t crv=GetC(b[2]);
1978 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1980 Double_t x2r = crv*dx;
1981 Double_t f1=fP[2], f2=f1 + x2r;
1982 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1983 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1986 // Estimate the covariance matrix
1987 Double_t &fP3=fP[3], &fP4=fP[4];
1990 &fC10=fC[1], &fC11=fC[2],
1991 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1992 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1993 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1995 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1998 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
1999 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
2000 Double_t f12= dx*fP3*f1/(r1*r1*r1);
2001 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
2002 Double_t f13= dx/r1;
2003 Double_t f24= dx; f24*=cc;
2006 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
2007 Double_t b02=f24*fC40;
2008 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
2009 Double_t b12=f24*fC41;
2010 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
2011 Double_t b22=f24*fC42;
2012 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
2013 Double_t b42=f24*fC44;
2014 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
2015 Double_t b32=f24*fC43;
2018 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
2019 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
2020 Double_t a22=f24*b42;
2022 //F*C*Ft = C + (b + bt + a)
2023 fC00 += b00 + b00 + a00;
2024 fC10 += b10 + b01 + a01;
2025 fC20 += b20 + b02 + a02;
2028 fC11 += b11 + b11 + a11;
2029 fC21 += b21 + b12 + a12;
2032 fC22 += b22 + b22 + a22;
2038 // Appoximate step length
2039 double dy2dx = (f1+f2)/(r1+r2);
2040 Double_t step = (TMath::Abs(x2r)<0.05) ? dx*TMath::Abs(r2 + f2*dy2dx) // chord
2041 : 2.*TMath::ASin(0.5*dx*TMath::Sqrt(1.+dy2dx*dy2dx)*crv)/crv; // arc
2042 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
2044 // Get the track's (x,y,z) and (px,py,pz) in the Global System
2045 Double_t r[3]; GetXYZ(r);
2046 Double_t p[3]; GetPxPyPz(p);
2053 // Rotate to the system where Bx=By=0.
2054 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
2055 Double_t cosphi=1., sinphi=0.;
2056 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
2057 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
2058 Double_t costet=1., sintet=0.;
2059 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
2062 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
2063 vect[1] = -sinphi*r[0] + cosphi*r[1];
2064 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
2066 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
2067 vect[4] = -sinphi*p[0] + cosphi*p[1];
2068 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
2073 // Do the helix step
2074 g3helx3(GetSign()*bb,step,vect);
2077 // Rotate back to the Global System
2078 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
2079 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
2080 r[2] = -sintet*vect[0] + costet*vect[2];
2082 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
2083 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
2084 p[2] = -sintet*vect[3] + costet*vect[5];
2087 // Rotate back to the Tracking System
2088 Double_t cosalp = TMath::Cos(fAlpha);
2089 Double_t sinalp =-TMath::Sin(fAlpha);
2092 t = cosalp*r[0] - sinalp*r[1];
2093 r[1] = sinalp*r[0] + cosalp*r[1];
2096 t = cosalp*p[0] - sinalp*p[1];
2097 p[1] = sinalp*p[0] + cosalp*p[1];
2101 // Do the final correcting step to the target plane (linear approximation)
2102 Double_t x=r[0], y=r[1], z=r[2];
2103 if (TMath::Abs(dx) > kAlmost0) {
2104 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
2112 // Calculate the track parameters
2113 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
2119 fP[4] = GetSign()/(t*pp);
2124 Bool_t AliExternalTrackParam::Translate(Double_t *vTrasl,Double_t *covV){
2126 //Translation: in the event mixing, the tracks can be shifted
2127 //of the difference among primary vertices (vTrasl) and
2128 //the covariance matrix is changed accordingly
2129 //(covV = covariance of the primary vertex).
2130 //Origin: "Romita, Rossella" <R.Romita@gsi.de>
2132 TVector3 translation;
2133 // vTrasl coordinates in the local system
2134 translation.SetXYZ(vTrasl[0],vTrasl[1],vTrasl[2]);
2135 translation.RotateZ(-fAlpha);
2136 translation.GetXYZ(vTrasl);
2138 //compute the new x,y,z of the track
2139 Double_t newX=fX-vTrasl[0];
2140 Double_t newY=fP[0]-vTrasl[1];
2141 Double_t newZ=fP[1]-vTrasl[2];
2143 //define the new parameters
2144 Double_t newParam[5];
2151 // recompute the covariance matrix:
2152 // 1. covV in the local system
2153 Double_t cosRot=TMath::Cos(fAlpha), sinRot=TMath::Sin(fAlpha);
2174 if(uUi.Determinant() <= 0.) {return kFALSE;}
2175 TMatrixD uUiQi(uUi,TMatrixD::kMult,qQi);
2176 TMatrixD m(qQi,TMatrixD::kTransposeMult,uUiQi);
2178 //2. compute the new covariance matrix of the track
2179 Double_t sigmaXX=m(0,0);
2180 Double_t sigmaXZ=m(2,0);
2181 Double_t sigmaXY=m(1,0);
2182 Double_t sigmaYY=GetSigmaY2()+m(1,1);
2183 Double_t sigmaYZ=fC[1]+m(1,2);
2184 Double_t sigmaZZ=fC[2]+m(2,2);
2185 Double_t covarianceYY=sigmaYY + (-1.)*((sigmaXY*sigmaXY)/sigmaXX);
2186 Double_t covarianceYZ=sigmaYZ-(sigmaXZ*sigmaXY/sigmaXX);
2187 Double_t covarianceZZ=sigmaZZ-((sigmaXZ*sigmaXZ)/sigmaXX);
2189 Double_t newCov[15];
2190 newCov[0]=covarianceYY;
2191 newCov[1]=covarianceYZ;
2192 newCov[2]=covarianceZZ;
2193 for(Int_t i=3;i<15;i++){
2197 // set the new parameters
2199 Set(newX,fAlpha,newParam,newCov);
2204 void AliExternalTrackParam::CheckCovariance() {
2206 // This function forces the diagonal elements of the covariance matrix to be positive.
2207 // In case the diagonal element is bigger than the maximal allowed value, it is set to
2208 // the limit and the off-diagonal elements that correspond to it are set to zero.
2210 fC[0] = TMath::Abs(fC[0]);
2218 fC[2] = TMath::Abs(fC[2]);
2226 fC[5] = TMath::Abs(fC[5]);
2234 fC[9] = TMath::Abs(fC[9]);
2242 fC[14] = TMath::Abs(fC[14]);
2243 if (fC[14]>kC14max) {
2251 // The part below is used for tests and normally is commented out
2252 // TMatrixDSym m(5);
2256 // m(1,0)=fC[1]; m(1,1)=fC[2];
2257 // m(2,0)=fC[3]; m(2,1)=fC[4]; m(2,2)=fC[5];
2258 // m(3,0)=fC[6]; m(3,1)=fC[7]; m(3,2)=fC[8]; m(3,3)=fC[9];
2259 // 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];
2262 // m(0,2)=m(2,0); m(1,2)=m(2,1);
2263 // m(0,3)=m(3,0); m(1,3)=m(3,1); m(2,3)=m(3,2);
2264 // m(0,4)=m(4,0); m(1,4)=m(4,1); m(2,4)=m(4,2); m(3,4)=m(4,3);
2265 // m.EigenVectors(eig);
2267 // // assert(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0);
2268 // if (!(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0)) {
2269 // AliWarning("Negative eigenvalues of the covariance matrix!");
2275 Bool_t AliExternalTrackParam::ConstrainToVertex(const AliVVertex* vtx, Double_t b[3])
2277 // Constrain TPC inner params constrained
2282 Double_t dz[2], cov[3];
2283 if (!PropagateToDCABxByBz(vtx, b, 3, dz, cov))
2287 vtx->GetCovarianceMatrix(covar);
2289 Double_t p[2]= { fP[0] - dz[0], fP[1] - dz[1] };
2290 Double_t c[3]= { covar[2], 0., covar[5] };
2292 Double_t chi2C = GetPredictedChi2(p,c);