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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11 * appear in the supporting documentation. The authors make no claims *
12 * about the suitability of this software for any purpose. It is *
13 * provided "as is" without express or implied warranty. *
14 **************************************************************************/
18 ///////////////////////////////////////////////////////////////////////////////
20 // Implementation of the external track parameterisation class. //
22 // This parameterisation is used to exchange tracks between the detectors. //
23 // A set of functions returning the position and the momentum of tracks //
24 // in the global coordinate system as well as the track impact parameters //
26 // Origin: I.Belikov, CERN, Jouri.Belikov@cern.ch //
27 ///////////////////////////////////////////////////////////////////////////////
31 #include <TMatrixDSym.h>
32 #include <TPolyMarker3D.h>
36 #include "AliExternalTrackParam.h"
37 #include "AliVVertex.h"
40 ClassImp(AliExternalTrackParam)
42 Double32_t AliExternalTrackParam::fgMostProbablePt=kMostProbablePt;
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)<2*kSafe) {
216 if (fAlpha>0) fAlpha += fAlpha< TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
217 else fAlpha += fAlpha>-TMath::Pi()/2. ? -2*kSafe : 2*kSafe;
218 cs=TMath::Cos(fAlpha);
219 sn=TMath::Sin(fAlpha);
221 else if (TMath::Abs(cs)<2*kSafe) {
222 if (fAlpha>0) fAlpha += fAlpha> TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
223 else fAlpha += fAlpha>-TMath::Pi()/2. ? 2*kSafe : -2*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]*m43-cv[6]*m44)/(m24*m43-m23*m44)/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]*m43-cv[8]*m44)/(m24*m43-m23*m44);
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*a3-a6*a1)/(a5*a3-a6*a2));
285 fC[14] = TMath::Abs((a1-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). Negative mass means charge=2 particle
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)));
442 if (mass<0) p += p; // q=2 particle
444 Double_t beta2=p2/(p2 + mass*mass);
446 //Calculating the multiple scattering corrections******************
452 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
453 Double_t theta2=0.0136*0.0136/(beta2*p2)*TMath::Abs(xOverX0);
454 if (GetUseLogTermMS()) {
455 double lt = 1+0.038*TMath::Log(TMath::Abs(xOverX0));
456 if (lt>0) theta2 *= lt*lt;
458 if (mass<0) theta2 *= 4; // q=2 particle
459 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
460 cC22 = theta2*((1.-fP2)*(1.+fP2))*(1. + fP3*fP3);
461 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
462 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
463 cC44 = theta2*fP3*fP4*fP3*fP4;
466 //Calculating the energy loss corrections************************
468 if ((xTimesRho != 0.) && (beta2 < 1.)) {
469 Double_t dE=dEdx*xTimesRho;
470 Double_t e=TMath::Sqrt(p2 + mass*mass);
471 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
472 if ( (1.+ dE/p2*(dE + 2*e)) < 0. ) return kFALSE;
473 cP4 = 1./TMath::Sqrt(1.+ dE/p2*(dE + 2*e)); //A precise formula by Ruben !
474 if (TMath::Abs(fP4*cP4)>100.) return kFALSE; //Do not track below 10 MeV/c
477 // Approximate energy loss fluctuation (M.Ivanov)
478 const Double_t knst=0.07; // To be tuned.
479 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
480 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
484 //Applying the corrections*****************************
496 Bool_t AliExternalTrackParam::CorrectForMeanMaterial
497 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
499 Double_t (*Bethe)(Double_t)) {
500 //------------------------------------------------------------------
501 // This function corrects the track parameters for the crossed material.
502 // "xOverX0" - X/X0, the thickness in units of the radiation length.
503 // "xTimesRho" - is the product length*density (g/cm^2).
504 // It should be passed as negative when propagating tracks
505 // from the intreaction point to the outside of the central barrel.
506 // "mass" - the mass of this particle (GeV/c^2). mass<0 means charge=2
507 // "anglecorr" - switch for the angular correction
508 // "Bethe" - function calculating the energy loss (GeV/(g/cm^2))
509 //------------------------------------------------------------------
510 Double_t bg=GetP()/mass;
513 AliDebug(2,Form("Mass %f corresponds to unknown PID particle",mass));
518 Double_t dEdx=Bethe(bg);
519 if (mass<0) dEdx *= 4;
521 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
524 Bool_t AliExternalTrackParam::CorrectForMeanMaterialZA
525 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
532 //------------------------------------------------------------------
533 // This function corrects the track parameters for the crossed material
534 // using the full Geant-like Bethe-Bloch formula parameterization
535 // "xOverX0" - X/X0, the thickness in units of the radiation length.
536 // "xTimesRho" - is the product length*density (g/cm^2).
537 // It should be passed as negative when propagating tracks
538 // from the intreaction point to the outside of the central barrel.
539 // "mass" - the mass of this particle (GeV/c^2). mass<0 means charge=2 particle
540 // "density" - mean density (g/cm^3)
541 // "zOverA" - mean Z/A
542 // "exEnergy" - mean exitation energy (GeV)
543 // "jp1" - density effect first junction point
544 // "jp2" - density effect second junction point
545 // "anglecorr" - switch for the angular correction
547 // The default values of the parameters are for silicon
549 //------------------------------------------------------------------
551 Double_t bg=GetP()/mass;
554 AliDebug(2,Form("Mass %f corresponds to unknown PID particle",mass));
559 Double_t dEdx=BetheBlochGeant(bg,density,jp1,jp2,exEnergy,zOverA);
560 if (mass<0) dEdx *= 4;
561 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
566 Bool_t AliExternalTrackParam::CorrectForMaterial
567 (Double_t d, Double_t x0, Double_t mass, Double_t (*Bethe)(Double_t)) {
568 //------------------------------------------------------------------
569 // Deprecated function !
570 // Better use CorrectForMeanMaterial instead of it.
572 // This function corrects the track parameters for the crossed material
573 // "d" - the thickness (fraction of the radiation length)
574 // It should be passed as negative when propagating tracks
575 // from the intreaction point to the outside of the central barrel.
576 // "x0" - the radiation length (g/cm^2)
577 // "mass" - the mass of this particle (GeV/c^2)
578 //------------------------------------------------------------------
580 return CorrectForMeanMaterial(d,x0*d,mass,kTRUE,Bethe);
584 Double_t AliExternalTrackParam::BetheBlochAleph(Double_t bg,
591 // This is the empirical ALEPH parameterization of the Bethe-Bloch formula.
592 // It is normalized to 1 at the minimum.
596 // The default values for the kp* parameters are for ALICE TPC.
597 // The returned value is in MIP units
600 Double_t beta = bg/TMath::Sqrt(1.+ bg*bg);
602 Double_t aa = TMath::Power(beta,kp4);
603 Double_t bb = TMath::Power(1./bg,kp5);
605 bb=TMath::Log(kp3+bb);
607 return (kp2-aa-bb)*kp1/aa;
610 Double_t AliExternalTrackParam::BetheBlochGeant(Double_t bg,
617 // This is the parameterization of the Bethe-Bloch formula inspired by Geant.
620 // kp0 - density [g/cm^3]
621 // kp1 - density effect first junction point
622 // kp2 - density effect second junction point
623 // kp3 - mean excitation energy [GeV]
626 // The default values for the kp* parameters are for silicon.
627 // The returned value is in [GeV/(g/cm^2)].
630 const Double_t mK = 0.307075e-3; // [GeV*cm^2/g]
631 const Double_t me = 0.511e-3; // [GeV/c^2]
632 const Double_t rho = kp0;
633 const Double_t x0 = kp1*2.303;
634 const Double_t x1 = kp2*2.303;
635 const Double_t mI = kp3;
636 const Double_t mZA = kp4;
637 const Double_t bg2 = bg*bg;
638 const Double_t maxT= 2*me*bg2; // neglecting the electron mass
642 const Double_t x=TMath::Log(bg);
643 const Double_t lhwI=TMath::Log(28.816*1e-9*TMath::Sqrt(rho*mZA)/mI);
647 const Double_t r=(x1-x)/(x1-x0);
648 d2 = lhwI + x - 0.5 + (0.5 - lhwI - x0)*r*r*r;
651 return mK*mZA*(1+bg2)/bg2*
652 (0.5*TMath::Log(2*me*bg2*maxT/(mI*mI)) - bg2/(1+bg2) - d2);
655 Double_t AliExternalTrackParam::BetheBlochSolid(Double_t bg) {
656 //------------------------------------------------------------------
657 // This is an approximation of the Bethe-Bloch formula,
658 // reasonable for solid materials.
659 // All the parameters are, in fact, for Si.
660 // The returned value is in [GeV/(g/cm^2)]
661 //------------------------------------------------------------------
663 return BetheBlochGeant(bg);
666 Double_t AliExternalTrackParam::BetheBlochGas(Double_t bg) {
667 //------------------------------------------------------------------
668 // This is an approximation of the Bethe-Bloch formula,
669 // reasonable for gas materials.
670 // All the parameters are, in fact, for Ne.
671 // The returned value is in [GeV/(g/cm^2)]
672 //------------------------------------------------------------------
674 const Double_t rho = 0.9e-3;
675 const Double_t x0 = 2.;
676 const Double_t x1 = 4.;
677 const Double_t mI = 140.e-9;
678 const Double_t mZA = 0.49555;
680 return BetheBlochGeant(bg,rho,x0,x1,mI,mZA);
683 Bool_t AliExternalTrackParam::Rotate(Double_t alpha) {
684 //------------------------------------------------------------------
685 // Transform this track to the local coord. system rotated
686 // by angle "alpha" (rad) with respect to the global coord. system.
687 //------------------------------------------------------------------
688 if (TMath::Abs(fP[2]) >= kAlmost1) {
689 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
693 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
694 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
698 Double_t &fC00=fC[0];
699 Double_t &fC10=fC[1];
700 Double_t &fC20=fC[3];
701 Double_t &fC21=fC[4];
702 Double_t &fC22=fC[5];
703 Double_t &fC30=fC[6];
704 Double_t &fC32=fC[8];
705 Double_t &fC40=fC[10];
706 Double_t &fC42=fC[12];
709 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
710 Double_t sf=fP2, cf=TMath::Sqrt((1.- fP2)*(1.+fP2)); // Improve precision
711 // RS: check if rotation does no invalidate track model (cos(local_phi)>=0, i.e. particle
712 // direction in local frame is along the X axis
713 if ((cf*ca+sf*sa)<0) {
714 AliDebug(1,Form("Rotation failed: local cos(phi) would become %.2f",cf*ca+sf*sa));
718 Double_t tmp=sf*ca - cf*sa;
720 if (TMath::Abs(tmp) >= kAlmost1) {
721 if (TMath::Abs(tmp) > 1.+ Double_t(FLT_EPSILON))
722 AliWarning(Form("Rotation failed ! %.10e",tmp));
730 if (TMath::Abs(cf)<kAlmost0) {
731 AliError(Form("Too small cosine value %f",cf));
735 Double_t rr=(ca+sf/cf*sa);
752 Bool_t AliExternalTrackParam::Invert() {
753 //------------------------------------------------------------------
754 // Transform this track to the local coord. system rotated by 180 deg.
755 //------------------------------------------------------------------
757 fAlpha += TMath::Pi();
758 while (fAlpha < -TMath::Pi()) fAlpha += 2*TMath::Pi();
759 while (fAlpha >= TMath::Pi()) fAlpha -= 2*TMath::Pi();
766 fC[1] = -fC[1]; // since the fP1 and fP2 are not inverted, their covariances with others change sign
776 Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
777 //----------------------------------------------------------------
778 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
779 //----------------------------------------------------------------
781 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
783 Double_t crv=GetC(b);
784 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
786 Double_t x2r = crv*dx;
787 Double_t f1=fP[2], f2=f1 + x2r;
788 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
789 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
790 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
792 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
795 &fC10=fC[1], &fC11=fC[2],
796 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
797 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
798 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
800 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
801 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
802 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
805 double dy2dx = (f1+f2)/(r1+r2);
807 if (TMath::Abs(x2r)<0.05) {
808 fP1 += dx*(r2 + f2*dy2dx)*fP3; // Many thanks to P.Hristov !
812 // for small dx/R the linear apporximation of the arc by the segment is OK,
813 // but at large dx/R the error is very large and leads to incorrect Z propagation
814 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
815 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
816 // Similarly, the rotation angle in linear in dx only for dx<<R
817 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
818 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
820 fP2 = TMath::Sin(rot + TMath::ASin(fP2));
825 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
826 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
827 Double_t f12= dx*fP3*f1/(r1*r1*r1);
828 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
830 Double_t f24= dx; f24*=cc;
832 Double_t rinv = 1./r1;
833 Double_t r3inv = rinv*rinv*rinv;
834 Double_t f24= x2r/fP4;
835 Double_t f02= dx*r3inv;
836 Double_t f04=0.5*f24*f02;
837 Double_t f12= f02*fP3*f1;
838 Double_t f14=0.5*f24*f02*fP3*f1;
839 Double_t f13= dx*rinv;
842 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
843 Double_t b02=f24*fC40;
844 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
845 Double_t b12=f24*fC41;
846 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
847 Double_t b22=f24*fC42;
848 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
849 Double_t b42=f24*fC44;
850 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
851 Double_t b32=f24*fC43;
854 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
855 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
856 Double_t a22=f24*b42;
858 //F*C*Ft = C + (b + bt + a)
859 fC00 += b00 + b00 + a00;
860 fC10 += b10 + b01 + a01;
861 fC20 += b20 + b02 + a02;
864 fC11 += b11 + b11 + a11;
865 fC21 += b21 + b12 + a12;
868 fC22 += b22 + b22 + a22;
877 Bool_t AliExternalTrackParam::PropagateParamOnlyTo(Double_t xk, Double_t b) {
878 //----------------------------------------------------------------
879 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
880 // Only parameters are propagated, not the matrix. To be used for small
881 // distances only (<mm, i.e. misalignment)
882 //----------------------------------------------------------------
884 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
886 Double_t crv=GetC(b);
887 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
889 Double_t x2r = crv*dx;
890 Double_t f1=fP[2], f2=f1 + x2r;
891 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
892 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
893 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
895 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
896 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
897 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
900 double dy2dx = (f1+f2)/(r1+r2);
902 fP[1] += dx*(r2 + f2*dy2dx)*fP[3]; // Many thanks to P.Hristov !
909 AliExternalTrackParam::Propagate(Double_t alpha, Double_t x, Double_t b) {
910 //------------------------------------------------------------------
911 // Transform this track to the local coord. system rotated
912 // by angle "alpha" (rad) with respect to the global coord. system,
913 // and propagate this track to the plane X=xk (cm) in the field "b" (kG)
914 //------------------------------------------------------------------
916 //Save the parameters
919 Double_t ps[5], cs[15];
920 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
921 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
924 if (PropagateTo(x,b)) return kTRUE;
926 //Restore the parameters, if the operation failed
929 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
930 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
934 Bool_t AliExternalTrackParam::PropagateBxByBz
935 (Double_t alpha, Double_t x, Double_t b[3]) {
936 //------------------------------------------------------------------
937 // Transform this track to the local coord. system rotated
938 // by angle "alpha" (rad) with respect to the global coord. system,
939 // and propagate this track to the plane X=xk (cm),
940 // taking into account all three components of the B field, "b[3]" (kG)
941 //------------------------------------------------------------------
943 //Save the parameters
946 Double_t ps[5], cs[15];
947 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
948 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
951 if (PropagateToBxByBz(x,b)) return kTRUE;
953 //Restore the parameters, if the operation failed
956 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
957 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
962 void AliExternalTrackParam::Propagate(Double_t len, Double_t x[3],
963 Double_t p[3], Double_t bz) const {
964 //+++++++++++++++++++++++++++++++++++++++++
965 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
966 // Extrapolate track along simple helix in magnetic field
967 // Arguments: len -distance alogn helix, [cm]
968 // bz - mag field, [kGaus]
969 // Returns: x and p contain extrapolated positon and momentum
970 // The momentum returned for straight-line tracks is meaningless !
971 //+++++++++++++++++++++++++++++++++++++++++
974 if (OneOverPt() < kAlmost0 || TMath::Abs(bz) < kAlmost0Field || GetC(bz) < kAlmost0){ //straight-line tracks
975 Double_t unit[3]; GetDirection(unit);
980 p[0]=unit[0]/kAlmost0;
981 p[1]=unit[1]/kAlmost0;
982 p[2]=unit[2]/kAlmost0;
986 Double_t a = -kB2C*bz*GetSign();
988 x[0] += p[0]*TMath::Sin(rho*len)/a - p[1]*(1-TMath::Cos(rho*len))/a;
989 x[1] += p[1]*TMath::Sin(rho*len)/a + p[0]*(1-TMath::Cos(rho*len))/a;
993 p[0] = p0 *TMath::Cos(rho*len) - p[1]*TMath::Sin(rho*len);
994 p[1] = p[1]*TMath::Cos(rho*len) + p0 *TMath::Sin(rho*len);
998 Bool_t AliExternalTrackParam::Intersect(Double_t pnt[3], Double_t norm[3],
1000 //+++++++++++++++++++++++++++++++++++++++++
1001 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
1002 // Finds point of intersection (if exists) of the helix with the plane.
1003 // Stores result in fX and fP.
1004 // Arguments: planePoint,planeNorm - the plane defined by any plane's point
1005 // and vector, normal to the plane
1006 // Returns: kTrue if helix intersects the plane, kFALSE otherwise.
1007 //+++++++++++++++++++++++++++++++++++++++++
1008 Double_t x0[3]; GetXYZ(x0); //get track position in MARS
1010 //estimates initial helix length up to plane
1012 (pnt[0]-x0[0])*norm[0] + (pnt[1]-x0[1])*norm[1] + (pnt[2]-x0[2])*norm[2];
1013 Double_t dist=99999,distPrev=dist;
1015 while(TMath::Abs(dist)>0.00001){
1016 //calculates helix at the distance s from x0 ALONG the helix
1017 Propagate(s,x,p,bz);
1019 //distance between current helix position and plane
1020 dist=(x[0]-pnt[0])*norm[0]+(x[1]-pnt[1])*norm[1]+(x[2]-pnt[2])*norm[2];
1022 if(TMath::Abs(dist) >= TMath::Abs(distPrev)) {return kFALSE;}
1026 //on exit pnt is intersection point,norm is track vector at that point,
1028 for (Int_t i=0; i<3; i++) {pnt[i]=x[i]; norm[i]=p[i];}
1033 AliExternalTrackParam::GetPredictedChi2(Double_t p[2],Double_t cov[3]) const {
1034 //----------------------------------------------------------------
1035 // Estimate the chi2 of the space point "p" with the cov. matrix "cov"
1036 //----------------------------------------------------------------
1037 Double_t sdd = fC[0] + cov[0];
1038 Double_t sdz = fC[1] + cov[1];
1039 Double_t szz = fC[2] + cov[2];
1040 Double_t det = sdd*szz - sdz*sdz;
1042 if (TMath::Abs(det) < kAlmost0) return kVeryBig;
1044 Double_t d = fP[0] - p[0];
1045 Double_t z = fP[1] - p[1];
1047 return (d*szz*d - 2*d*sdz*z + z*sdd*z)/det;
1050 Double_t AliExternalTrackParam::
1051 GetPredictedChi2(Double_t p[3],Double_t covyz[3],Double_t covxyz[3]) const {
1052 //----------------------------------------------------------------
1053 // Estimate the chi2 of the 3D space point "p" and
1054 // the full covariance matrix "covyz" and "covxyz"
1056 // Cov(x,x) ... : covxyz[0]
1057 // Cov(y,x) ... : covxyz[1] covyz[0]
1058 // Cov(z,x) ... : covxyz[2] covyz[1] covyz[2]
1059 //----------------------------------------------------------------
1067 Double_t f=GetSnp();
1068 if (TMath::Abs(f) >= kAlmost1) return kVeryBig;
1069 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1070 Double_t a=f/r, b=GetTgl()/r;
1072 Double_t s2=333.*333.; //something reasonably big (cm^2)
1075 v(0,0)= s2; v(0,1)= a*s2; v(0,2)= b*s2;;
1076 v(1,0)=a*s2; v(1,1)=a*a*s2 + GetSigmaY2(); v(1,2)=a*b*s2 + GetSigmaZY();
1077 v(2,0)=b*s2; v(2,1)=a*b*s2 + GetSigmaZY(); v(2,2)=b*b*s2 + GetSigmaZ2();
1079 v(0,0)+=covxyz[0]; v(0,1)+=covxyz[1]; v(0,2)+=covxyz[2];
1080 v(1,0)+=covxyz[1]; v(1,1)+=covyz[0]; v(1,2)+=covyz[1];
1081 v(2,0)+=covxyz[2]; v(2,1)+=covyz[1]; v(2,2)+=covyz[2];
1084 if (!v.IsValid()) return kVeryBig;
1087 for (Int_t i = 0; i < 3; i++)
1088 for (Int_t j = 0; j < 3; j++) chi2 += res[i]*res[j]*v(i,j);
1093 Double_t AliExternalTrackParam::
1094 GetPredictedChi2(const AliExternalTrackParam *t) const {
1095 //----------------------------------------------------------------
1096 // Estimate the chi2 (5 dof) of this track with respect to the track
1097 // given by the argument.
1098 // The two tracks must be in the same reference system
1099 // and estimated at the same reference plane.
1100 //----------------------------------------------------------------
1102 if (TMath::Abs(1. - t->GetAlpha()/GetAlpha()) > FLT_EPSILON) {
1103 AliError("The reference systems of the tracks differ !");
1106 if (TMath::Abs(1. - t->GetX()/GetX()) > FLT_EPSILON) {
1107 AliError("The reference of the tracks planes differ !");
1112 c(0,0)=GetSigmaY2();
1113 c(1,0)=GetSigmaZY(); c(1,1)=GetSigmaZ2();
1114 c(2,0)=GetSigmaSnpY(); c(2,1)=GetSigmaSnpZ(); c(2,2)=GetSigmaSnp2();
1115 c(3,0)=GetSigmaTglY(); c(3,1)=GetSigmaTglZ(); c(3,2)=GetSigmaTglSnp(); c(3,3)=GetSigmaTgl2();
1116 c(4,0)=GetSigma1PtY(); c(4,1)=GetSigma1PtZ(); c(4,2)=GetSigma1PtSnp(); c(4,3)=GetSigma1PtTgl(); c(4,4)=GetSigma1Pt2();
1118 c(0,0)+=t->GetSigmaY2();
1119 c(1,0)+=t->GetSigmaZY(); c(1,1)+=t->GetSigmaZ2();
1120 c(2,0)+=t->GetSigmaSnpY();c(2,1)+=t->GetSigmaSnpZ();c(2,2)+=t->GetSigmaSnp2();
1121 c(3,0)+=t->GetSigmaTglY();c(3,1)+=t->GetSigmaTglZ();c(3,2)+=t->GetSigmaTglSnp();c(3,3)+=t->GetSigmaTgl2();
1122 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();
1124 c(0,2)=c(2,0); c(1,2)=c(2,1);
1125 c(0,3)=c(3,0); c(1,3)=c(3,1); c(2,3)=c(3,2);
1126 c(0,4)=c(4,0); c(1,4)=c(4,1); c(2,4)=c(4,2); c(3,4)=c(4,3);
1129 if (!c.IsValid()) return kVeryBig;
1135 GetSnp() - t->GetSnp(),
1136 GetTgl() - t->GetTgl(),
1137 GetSigned1Pt() - t->GetSigned1Pt()
1141 for (Int_t i = 0; i < 5; i++)
1142 for (Int_t j = 0; j < 5; j++) chi2 += res[i]*res[j]*c(i,j);
1147 Bool_t AliExternalTrackParam::
1148 PropagateTo(Double_t p[3],Double_t covyz[3],Double_t covxyz[3],Double_t bz) {
1149 //----------------------------------------------------------------
1150 // Propagate this track to the plane
1151 // the 3D space point "p" (with the covariance matrix "covyz" and "covxyz")
1153 // The magnetic field is "bz" (kG)
1155 // The track curvature and the change of the covariance matrix
1156 // of the track parameters are negleted !
1157 // (So the "step" should be small compared with 1/curvature)
1158 //----------------------------------------------------------------
1160 Double_t f=GetSnp();
1161 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1162 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1163 Double_t a=f/r, b=GetTgl()/r;
1165 Double_t s2=333.*333.; //something reasonably big (cm^2)
1168 tV(0,0)= s2; tV(0,1)= a*s2; tV(0,2)= b*s2;
1169 tV(1,0)=a*s2; tV(1,1)=a*a*s2; tV(1,2)=a*b*s2;
1170 tV(2,0)=b*s2; tV(2,1)=a*b*s2; tV(2,2)=b*b*s2;
1173 pV(0,0)=covxyz[0]; pV(0,1)=covxyz[1]; pV(0,2)=covxyz[2];
1174 pV(1,0)=covxyz[1]; pV(1,1)=covyz[0]; pV(1,2)=covyz[1];
1175 pV(2,0)=covxyz[2]; pV(2,1)=covyz[1]; pV(2,2)=covyz[2];
1177 TMatrixDSym tpV(tV);
1180 if (!tpV.IsValid()) return kFALSE;
1182 TMatrixDSym pW(3),tW(3);
1183 for (Int_t i=0; i<3; i++)
1184 for (Int_t j=0; j<3; j++) {
1186 for (Int_t k=0; k<3; k++) {
1187 pW(i,j) += tV(i,k)*tpV(k,j);
1188 tW(i,j) += pV(i,k)*tpV(k,j);
1192 Double_t t[3] = {GetX(), GetY(), GetZ()};
1195 for (Int_t i=0; i<3; i++) x += (tW(0,i)*t[i] + pW(0,i)*p[i]);
1196 Double_t crv=GetC(bz);
1197 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1199 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1203 for (Int_t i=0; i<3; i++) fP[0] += (tW(1,i)*t[i] + pW(1,i)*p[i]);
1205 for (Int_t i=0; i<3; i++) fP[1] += (tW(2,i)*t[i] + pW(2,i)*p[i]);
1210 Double_t *AliExternalTrackParam::GetResiduals(
1211 Double_t *p,Double_t *cov,Bool_t updated) const {
1212 //------------------------------------------------------------------
1213 // Returns the track residuals with the space point "p" having
1214 // the covariance matrix "cov".
1215 // If "updated" is kTRUE, the track parameters expected to be updated,
1216 // otherwise they must be predicted.
1217 //------------------------------------------------------------------
1218 static Double_t res[2];
1220 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1222 r00-=fC[0]; r01-=fC[1]; r11-=fC[2];
1224 r00+=fC[0]; r01+=fC[1]; r11+=fC[2];
1226 Double_t det=r00*r11 - r01*r01;
1228 if (TMath::Abs(det) < kAlmost0) return 0;
1230 Double_t tmp=r00; r00=r11/det; r11=tmp/det;
1232 if (r00 < 0.) return 0;
1233 if (r11 < 0.) return 0;
1235 Double_t dy = fP[0] - p[0];
1236 Double_t dz = fP[1] - p[1];
1238 res[0]=dy*TMath::Sqrt(r00);
1239 res[1]=dz*TMath::Sqrt(r11);
1244 Bool_t AliExternalTrackParam::Update(Double_t p[2], Double_t cov[3]) {
1245 //------------------------------------------------------------------
1246 // Update the track parameters with the space point "p" having
1247 // the covariance matrix "cov"
1248 //------------------------------------------------------------------
1249 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
1252 &fC10=fC[1], &fC11=fC[2],
1253 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1254 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1255 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1257 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1258 r00+=fC00; r01+=fC10; r11+=fC11;
1259 Double_t det=r00*r11 - r01*r01;
1261 if (TMath::Abs(det) < kAlmost0) return kFALSE;
1264 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
1266 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
1267 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
1268 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
1269 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
1270 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
1272 Double_t dy=p[0] - fP0, dz=p[1] - fP1;
1273 Double_t sf=fP2 + k20*dy + k21*dz;
1274 if (TMath::Abs(sf) > kAlmost1) return kFALSE;
1276 fP0 += k00*dy + k01*dz;
1277 fP1 += k10*dy + k11*dz;
1279 fP3 += k30*dy + k31*dz;
1280 fP4 += k40*dy + k41*dz;
1282 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
1283 Double_t c12=fC21, c13=fC31, c14=fC41;
1285 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
1286 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
1287 fC40-=k00*c04+k01*c14;
1289 fC11-=k10*c01+k11*fC11;
1290 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
1291 fC41-=k10*c04+k11*c14;
1293 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
1294 fC42-=k20*c04+k21*c14;
1296 fC33-=k30*c03+k31*c13;
1297 fC43-=k30*c04+k31*c14;
1299 fC44-=k40*c04+k41*c14;
1307 AliExternalTrackParam::GetHelixParameters(Double_t hlx[6], Double_t b) const {
1308 //--------------------------------------------------------------------
1309 // External track parameters -> helix parameters
1310 // "b" - magnetic field (kG)
1311 //--------------------------------------------------------------------
1312 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1314 hlx[0]=fP[0]; hlx[1]=fP[1]; hlx[2]=fP[2]; hlx[3]=fP[3];
1316 hlx[5]=fX*cs - hlx[0]*sn; // x0
1317 hlx[0]=fX*sn + hlx[0]*cs; // y0
1319 hlx[2]=TMath::ASin(hlx[2]) + fAlpha; // phi0
1321 hlx[4]=GetC(b); // C
1325 static void Evaluate(const Double_t *h, Double_t t,
1326 Double_t r[3], //radius vector
1327 Double_t g[3], //first defivatives
1328 Double_t gg[3]) //second derivatives
1330 //--------------------------------------------------------------------
1331 // Calculate position of a point on a track and some derivatives
1332 //--------------------------------------------------------------------
1333 Double_t phase=h[4]*t+h[2];
1334 Double_t sn=TMath::Sin(phase), cs=TMath::Cos(phase);
1338 if (TMath::Abs(h[4])>kAlmost0) {
1339 r[0] += (sn - h[6])/h[4];
1340 r[1] -= (cs - h[7])/h[4];
1342 r[2] = h[1] + h[3]*t;
1344 g[0] = cs; g[1]=sn; g[2]=h[3];
1346 gg[0]=-h[4]*sn; gg[1]=h[4]*cs; gg[2]=0.;
1349 Double_t AliExternalTrackParam::GetDCA(const AliExternalTrackParam *p,
1350 Double_t b, Double_t &xthis, Double_t &xp) const {
1351 //------------------------------------------------------------
1352 // Returns the (weighed !) distance of closest approach between
1353 // this track and the track "p".
1354 // Other returned values:
1355 // xthis, xt - coordinates of tracks' reference planes at the DCA
1356 //-----------------------------------------------------------
1357 Double_t dy2=GetSigmaY2() + p->GetSigmaY2();
1358 Double_t dz2=GetSigmaZ2() + p->GetSigmaZ2();
1361 Double_t p1[8]; GetHelixParameters(p1,b);
1362 p1[6]=TMath::Sin(p1[2]); p1[7]=TMath::Cos(p1[2]);
1363 Double_t p2[8]; p->GetHelixParameters(p2,b);
1364 p2[6]=TMath::Sin(p2[2]); p2[7]=TMath::Cos(p2[2]);
1367 Double_t r1[3],g1[3],gg1[3]; Double_t t1=0.;
1368 Evaluate(p1,t1,r1,g1,gg1);
1369 Double_t r2[3],g2[3],gg2[3]; Double_t t2=0.;
1370 Evaluate(p2,t2,r2,g2,gg2);
1372 Double_t dx=r2[0]-r1[0], dy=r2[1]-r1[1], dz=r2[2]-r1[2];
1373 Double_t dm=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1377 Double_t gt1=-(dx*g1[0]/dx2 + dy*g1[1]/dy2 + dz*g1[2]/dz2);
1378 Double_t gt2=+(dx*g2[0]/dx2 + dy*g2[1]/dy2 + dz*g2[2]/dz2);
1379 Double_t h11=(g1[0]*g1[0] - dx*gg1[0])/dx2 +
1380 (g1[1]*g1[1] - dy*gg1[1])/dy2 +
1381 (g1[2]*g1[2] - dz*gg1[2])/dz2;
1382 Double_t h22=(g2[0]*g2[0] + dx*gg2[0])/dx2 +
1383 (g2[1]*g2[1] + dy*gg2[1])/dy2 +
1384 (g2[2]*g2[2] + dz*gg2[2])/dz2;
1385 Double_t h12=-(g1[0]*g2[0]/dx2 + g1[1]*g2[1]/dy2 + g1[2]*g2[2]/dz2);
1387 Double_t det=h11*h22-h12*h12;
1390 if (TMath::Abs(det)<1.e-33) {
1391 //(quasi)singular Hessian
1394 dt1=-(gt1*h22 - gt2*h12)/det;
1395 dt2=-(h11*gt2 - h12*gt1)/det;
1398 if ((dt1*gt1+dt2*gt2)>0) {dt1=-dt1; dt2=-dt2;}
1400 //check delta(phase1) ?
1401 //check delta(phase2) ?
1403 if (TMath::Abs(dt1)/(TMath::Abs(t1)+1.e-3) < 1.e-4)
1404 if (TMath::Abs(dt2)/(TMath::Abs(t2)+1.e-3) < 1.e-4) {
1405 if ((gt1*gt1+gt2*gt2) > 1.e-4/dy2/dy2)
1406 AliDebug(1," stopped at not a stationary point !");
1407 Double_t lmb=h11+h22; lmb=lmb-TMath::Sqrt(lmb*lmb-4*det);
1409 AliDebug(1," stopped at not a minimum !");
1414 for (Int_t div=1 ; ; div*=2) {
1415 Evaluate(p1,t1+dt1,r1,g1,gg1);
1416 Evaluate(p2,t2+dt2,r2,g2,gg2);
1417 dx=r2[0]-r1[0]; dy=r2[1]-r1[1]; dz=r2[2]-r1[2];
1418 dd=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1422 AliDebug(1," overshoot !"); break;
1432 if (max<=0) AliDebug(1," too many iterations !");
1434 Double_t cs=TMath::Cos(GetAlpha());
1435 Double_t sn=TMath::Sin(GetAlpha());
1436 xthis=r1[0]*cs + r1[1]*sn;
1438 cs=TMath::Cos(p->GetAlpha());
1439 sn=TMath::Sin(p->GetAlpha());
1440 xp=r2[0]*cs + r2[1]*sn;
1442 return TMath::Sqrt(dm*TMath::Sqrt(dy2*dz2));
1445 Double_t AliExternalTrackParam::
1446 PropagateToDCA(AliExternalTrackParam *p, Double_t b) {
1447 //--------------------------------------------------------------
1448 // Propagates this track and the argument track to the position of the
1449 // distance of closest approach.
1450 // Returns the (weighed !) distance of closest approach.
1451 //--------------------------------------------------------------
1453 Double_t dca=GetDCA(p,b,xthis,xp);
1455 if (!PropagateTo(xthis,b)) {
1456 //AliWarning(" propagation failed !");
1460 if (!p->PropagateTo(xp,b)) {
1461 //AliWarning(" propagation failed !";
1469 Bool_t AliExternalTrackParam::PropagateToDCA(const AliVVertex *vtx,
1470 Double_t b, Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1472 // Propagate this track to the DCA to vertex "vtx",
1473 // if the (rough) transverse impact parameter is not bigger then "maxd".
1474 // Magnetic field is "b" (kG).
1476 // a) The track gets extapolated to the DCA to the vertex.
1477 // b) The impact parameters and their covariance matrix are calculated.
1479 // In the case of success, the returned value is kTRUE
1480 // (otherwise, it's kFALSE)
1482 Double_t alpha=GetAlpha();
1483 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1484 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1485 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1486 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1489 //Estimate the impact parameter neglecting the track curvature
1490 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1491 if (d > maxd) return kFALSE;
1493 //Propagate to the DCA
1494 Double_t crv=GetC(b);
1495 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1497 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1498 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1499 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1503 yv=-xv*sn + yv*cs; xv=x;
1505 if (!Propagate(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1507 if (dz==0) return kTRUE;
1508 dz[0] = GetParameter()[0] - yv;
1509 dz[1] = GetParameter()[1] - zv;
1511 if (covar==0) return kTRUE;
1512 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1514 //***** Improvements by A.Dainese
1515 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1516 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1517 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1518 covar[1] = GetCovariance()[1]; // between (x,y) and z
1519 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1525 Bool_t AliExternalTrackParam::PropagateToDCABxByBz(const AliVVertex *vtx,
1526 Double_t b[3], Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1528 // Propagate this track to the DCA to vertex "vtx",
1529 // if the (rough) transverse impact parameter is not bigger then "maxd".
1531 // This function takes into account all three components of the magnetic
1532 // field given by the b[3] arument (kG)
1534 // a) The track gets extapolated to the DCA to the vertex.
1535 // b) The impact parameters and their covariance matrix are calculated.
1537 // In the case of success, the returned value is kTRUE
1538 // (otherwise, it's kFALSE)
1540 Double_t alpha=GetAlpha();
1541 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1542 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1543 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1544 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1547 //Estimate the impact parameter neglecting the track curvature
1548 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1549 if (d > maxd) return kFALSE;
1551 //Propagate to the DCA
1552 Double_t crv=GetC(b[2]);
1553 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1555 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1556 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1557 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1561 yv=-xv*sn + yv*cs; xv=x;
1563 if (!PropagateBxByBz(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1565 if (dz==0) return kTRUE;
1566 dz[0] = GetParameter()[0] - yv;
1567 dz[1] = GetParameter()[1] - zv;
1569 if (covar==0) return kTRUE;
1570 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1572 //***** Improvements by A.Dainese
1573 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1574 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1575 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1576 covar[1] = GetCovariance()[1]; // between (x,y) and z
1577 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1583 void AliExternalTrackParam::GetDirection(Double_t d[3]) const {
1584 //----------------------------------------------------------------
1585 // This function returns a unit vector along the track direction
1586 // in the global coordinate system.
1587 //----------------------------------------------------------------
1588 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1590 Double_t csp =TMath::Sqrt((1.-snp)*(1.+snp));
1591 Double_t norm=TMath::Sqrt(1.+ fP[3]*fP[3]);
1592 d[0]=(csp*cs - snp*sn)/norm;
1593 d[1]=(snp*cs + csp*sn)/norm;
1597 Bool_t AliExternalTrackParam::GetPxPyPz(Double_t p[3]) const {
1598 //---------------------------------------------------------------------
1599 // This function returns the global track momentum components
1600 // Results for (nearly) straight tracks are meaningless !
1601 //---------------------------------------------------------------------
1602 p[0]=fP[4]; p[1]=fP[2]; p[2]=fP[3];
1603 return Local2GlobalMomentum(p,fAlpha);
1606 Double_t AliExternalTrackParam::Px() const {
1607 //---------------------------------------------------------------------
1608 // Returns x-component of momentum
1609 // Result for (nearly) straight tracks is meaningless !
1610 //---------------------------------------------------------------------
1612 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1618 Double_t AliExternalTrackParam::Py() const {
1619 //---------------------------------------------------------------------
1620 // Returns y-component of momentum
1621 // Result for (nearly) straight tracks is meaningless !
1622 //---------------------------------------------------------------------
1624 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1630 Double_t AliExternalTrackParam::Xv() const {
1631 //---------------------------------------------------------------------
1632 // Returns x-component of first track point
1633 //---------------------------------------------------------------------
1635 Double_t r[3]={0.,0.,0.};
1641 Double_t AliExternalTrackParam::Yv() const {
1642 //---------------------------------------------------------------------
1643 // Returns y-component of first track point
1644 //---------------------------------------------------------------------
1646 Double_t r[3]={0.,0.,0.};
1652 Double_t AliExternalTrackParam::Theta() const {
1653 // return theta angle of momentum
1655 return 0.5*TMath::Pi() - TMath::ATan(fP[3]);
1658 Double_t AliExternalTrackParam::Phi() const {
1659 //---------------------------------------------------------------------
1660 // Returns the azimuthal angle of momentum
1662 //---------------------------------------------------------------------
1664 Double_t phi=TMath::ASin(fP[2]) + fAlpha;
1665 if (phi<0.) phi+=2.*TMath::Pi();
1666 else if (phi>=2.*TMath::Pi()) phi-=2.*TMath::Pi();
1671 Double_t AliExternalTrackParam::M() const {
1672 // return particle mass
1674 // No mass information available so far.
1675 // Redifine in derived class!
1680 Double_t AliExternalTrackParam::E() const {
1681 // return particle energy
1683 // No PID information available so far.
1684 // Redifine in derived class!
1689 Double_t AliExternalTrackParam::Eta() const {
1690 // return pseudorapidity
1692 return -TMath::Log(TMath::Tan(0.5 * Theta()));
1695 Double_t AliExternalTrackParam::Y() const {
1698 // No PID information available so far.
1699 // Redifine in derived class!
1704 Bool_t AliExternalTrackParam::GetXYZ(Double_t *r) const {
1705 //---------------------------------------------------------------------
1706 // This function returns the global track position
1707 //---------------------------------------------------------------------
1708 r[0]=fX; r[1]=fP[0]; r[2]=fP[1];
1709 return Local2GlobalPosition(r,fAlpha);
1712 Bool_t AliExternalTrackParam::GetCovarianceXYZPxPyPz(Double_t cv[21]) const {
1713 //---------------------------------------------------------------------
1714 // This function returns the global covariance matrix of the track params
1716 // Cov(x,x) ... : cv[0]
1717 // Cov(y,x) ... : cv[1] cv[2]
1718 // Cov(z,x) ... : cv[3] cv[4] cv[5]
1719 // Cov(px,x)... : cv[6] cv[7] cv[8] cv[9]
1720 // Cov(py,x)... : cv[10] cv[11] cv[12] cv[13] cv[14]
1721 // Cov(pz,x)... : cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
1723 // Results for (nearly) straight tracks are meaningless !
1724 //---------------------------------------------------------------------
1725 if (TMath::Abs(fP[4])<=kAlmost0) {
1726 for (Int_t i=0; i<21; i++) cv[i]=0.;
1729 if (TMath::Abs(fP[2]) > kAlmost1) {
1730 for (Int_t i=0; i<21; i++) cv[i]=0.;
1733 Double_t pt=1./TMath::Abs(fP[4]);
1734 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1735 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
1737 Double_t m00=-sn, m10=cs;
1738 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
1739 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
1740 Double_t m35=pt, m45=-pt*pt*fP[3];
1746 cv[0 ] = fC[0]*m00*m00;
1747 cv[1 ] = fC[0]*m00*m10;
1748 cv[2 ] = fC[0]*m10*m10;
1752 cv[6 ] = m00*(fC[3]*m23 + fC[10]*m43);
1753 cv[7 ] = m10*(fC[3]*m23 + fC[10]*m43);
1754 cv[8 ] = fC[4]*m23 + fC[11]*m43;
1755 cv[9 ] = m23*(fC[5]*m23 + fC[12]*m43) + m43*(fC[12]*m23 + fC[14]*m43);
1756 cv[10] = m00*(fC[3]*m24 + fC[10]*m44);
1757 cv[11] = m10*(fC[3]*m24 + fC[10]*m44);
1758 cv[12] = fC[4]*m24 + fC[11]*m44;
1759 cv[13] = m23*(fC[5]*m24 + fC[12]*m44) + m43*(fC[12]*m24 + fC[14]*m44);
1760 cv[14] = m24*(fC[5]*m24 + fC[12]*m44) + m44*(fC[12]*m24 + fC[14]*m44);
1761 cv[15] = m00*(fC[6]*m35 + fC[10]*m45);
1762 cv[16] = m10*(fC[6]*m35 + fC[10]*m45);
1763 cv[17] = fC[7]*m35 + fC[11]*m45;
1764 cv[18] = m23*(fC[8]*m35 + fC[12]*m45) + m43*(fC[13]*m35 + fC[14]*m45);
1765 cv[19] = m24*(fC[8]*m35 + fC[12]*m45) + m44*(fC[13]*m35 + fC[14]*m45);
1766 cv[20] = m35*(fC[9]*m35 + fC[13]*m45) + m45*(fC[13]*m35 + fC[14]*m45);
1773 AliExternalTrackParam::GetPxPyPzAt(Double_t x, Double_t b, Double_t *p) const {
1774 //---------------------------------------------------------------------
1775 // This function returns the global track momentum extrapolated to
1776 // the radial position "x" (cm) in the magnetic field "b" (kG)
1777 //---------------------------------------------------------------------
1779 p[1]=fP[2]+(x-fX)*GetC(b);
1781 return Local2GlobalMomentum(p,fAlpha);
1785 AliExternalTrackParam::GetYAt(Double_t x, Double_t b, Double_t &y) const {
1786 //---------------------------------------------------------------------
1787 // This function returns the local Y-coordinate of the intersection
1788 // point between this track and the reference plane "x" (cm).
1789 // Magnetic field "b" (kG)
1790 //---------------------------------------------------------------------
1792 if(TMath::Abs(dx)<=kAlmost0) {y=fP[0]; return kTRUE;}
1794 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1796 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1797 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1799 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1800 y = fP[0] + dx*(f1+f2)/(r1+r2);
1805 AliExternalTrackParam::GetZAt(Double_t x, Double_t b, Double_t &z) const {
1806 //---------------------------------------------------------------------
1807 // This function returns the local Z-coordinate of the intersection
1808 // point between this track and the reference plane "x" (cm).
1809 // Magnetic field "b" (kG)
1810 //---------------------------------------------------------------------
1812 if(TMath::Abs(dx)<=kAlmost0) {z=fP[1]; return kTRUE;}
1813 Double_t crv=GetC(b);
1814 Double_t x2r = crv*dx;
1815 Double_t f1=fP[2], f2=f1 + x2r;
1817 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1818 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1820 Double_t r1=sqrt((1.-f1)*(1.+f1)), r2=sqrt((1.-f2)*(1.+f2));
1821 double dy2dx = (f1+f2)/(r1+r2);
1822 if (TMath::Abs(x2r)<0.05) {
1823 z = fP[1] + dx*(r2 + f2*dy2dx)*fP[3]; // Many thanks to P.Hristov !
1826 // for small dx/R the linear apporximation of the arc by the segment is OK,
1827 // but at large dx/R the error is very large and leads to incorrect Z propagation
1828 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
1829 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
1830 // Similarly, the rotation angle in linear in dx only for dx<<R
1831 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
1832 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
1833 z = fP[1] + rot/crv*fP[3];
1839 AliExternalTrackParam::GetXYZAt(Double_t x, Double_t b, Double_t *r) const {
1840 //---------------------------------------------------------------------
1841 // This function returns the global track position extrapolated to
1842 // the radial position "x" (cm) in the magnetic field "b" (kG)
1843 //---------------------------------------------------------------------
1845 if(TMath::Abs(dx)<=kAlmost0) return GetXYZ(r);
1846 Double_t crv=GetC(b);
1847 Double_t x2r = crv*dx;
1848 Double_t f1=fP[2], f2=f1 + dx*crv;
1849 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1850 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1852 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1853 double dy2dx = (f1+f2)/(r1+r2);
1855 r[1] = fP[0] + dx*dy2dx;
1856 if (TMath::Abs(x2r)<0.05) {
1857 r[2] = fP[1] + dx*(r2 + f2*dy2dx)*fP[3];//Thanks to Andrea & Peter
1860 // for small dx/R the linear apporximation of the arc by the segment is OK,
1861 // but at large dx/R the error is very large and leads to incorrect Z propagation
1862 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
1863 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
1864 // Similarly, the rotation angle in linear in dx only for dx<<R
1865 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
1866 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
1867 r[2] = fP[1] + rot/crv*fP[3];
1869 return Local2GlobalPosition(r,fAlpha);
1872 //_____________________________________________________________________________
1873 void AliExternalTrackParam::Print(Option_t* /*option*/) const
1875 // print the parameters and the covariance matrix
1877 printf("AliExternalTrackParam: x = %-12g alpha = %-12g\n", fX, fAlpha);
1878 printf(" parameters: %12g %12g %12g %12g %12g\n",
1879 fP[0], fP[1], fP[2], fP[3], fP[4]);
1880 printf(" covariance: %12g\n", fC[0]);
1881 printf(" %12g %12g\n", fC[1], fC[2]);
1882 printf(" %12g %12g %12g\n", fC[3], fC[4], fC[5]);
1883 printf(" %12g %12g %12g %12g\n",
1884 fC[6], fC[7], fC[8], fC[9]);
1885 printf(" %12g %12g %12g %12g %12g\n",
1886 fC[10], fC[11], fC[12], fC[13], fC[14]);
1889 Double_t AliExternalTrackParam::GetSnpAt(Double_t x,Double_t b) const {
1891 // Get sinus at given x
1893 Double_t crv=GetC(b);
1894 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1896 Double_t res = fP[2]+dx*crv;
1900 Bool_t AliExternalTrackParam::GetDistance(AliExternalTrackParam *param2, Double_t x, Double_t dist[3], Double_t bz){
1901 //------------------------------------------------------------------------
1902 // Get the distance between two tracks at the local position x
1903 // working in the local frame of this track.
1904 // Origin : Marian.Ivanov@cern.ch
1905 //-----------------------------------------------------------------------
1909 if (!GetYAt(x,bz,xyz[1])) return kFALSE;
1910 if (!GetZAt(x,bz,xyz[2])) return kFALSE;
1913 if (TMath::Abs(GetAlpha()-param2->GetAlpha())<kAlmost0){
1915 if (!param2->GetYAt(x,bz,xyz2[1])) return kFALSE;
1916 if (!param2->GetZAt(x,bz,xyz2[2])) return kFALSE;
1920 Double_t dfi = param2->GetAlpha()-GetAlpha();
1921 Double_t ca = TMath::Cos(dfi), sa = TMath::Sin(dfi);
1922 xyz2[0] = xyz[0]*ca+xyz[1]*sa;
1923 xyz2[1] = -xyz[0]*sa+xyz[1]*ca;
1926 if (!param2->GetYAt(xyz2[0],bz,xyz1[1])) return kFALSE;
1927 if (!param2->GetZAt(xyz2[0],bz,xyz1[2])) return kFALSE;
1929 xyz2[0] = xyz1[0]*ca-xyz1[1]*sa;
1930 xyz2[1] = +xyz1[0]*sa+xyz1[1]*ca;
1933 dist[0] = xyz[0]-xyz2[0];
1934 dist[1] = xyz[1]-xyz2[1];
1935 dist[2] = xyz[2]-xyz2[2];
1942 // Draw functionality.
1943 // Origin: Marian Ivanov, Marian.Ivanov@cern.ch
1946 void AliExternalTrackParam::DrawTrack(Float_t magf, Float_t minR, Float_t maxR, Float_t stepR){
1950 if (minR>maxR) return ;
1951 if (stepR<=0) return ;
1952 Int_t npoints = TMath::Nint((maxR-minR)/stepR)+1;
1953 if (npoints<1) return;
1954 TPolyMarker3D *polymarker = new TPolyMarker3D(npoints);
1955 FillPolymarker(polymarker, magf,minR,maxR,stepR);
1960 void AliExternalTrackParam::FillPolymarker(TPolyMarker3D *pol, Float_t magF, Float_t minR, Float_t maxR, Float_t stepR){
1962 // Fill points in the polymarker
1965 for (Double_t r=minR; r<maxR; r+=stepR){
1967 GetXYZAt(r,magF,point);
1968 pol->SetPoint(counter,point[0],point[1], point[2]);
1969 // printf("xyz\t%f\t%f\t%f\n",point[0], point[1],point[2]);
1974 Int_t AliExternalTrackParam::GetIndex(Int_t i, Int_t j) const {
1976 Int_t min = TMath::Min(i,j);
1977 Int_t max = TMath::Max(i,j);
1979 return min+(max+1)*max/2;
1983 void AliExternalTrackParam::g3helx3(Double_t qfield,
1986 /******************************************************************
1988 * GEANT3 tracking routine in a constant field oriented *
1990 * Tracking is performed with a conventional *
1991 * helix step method *
1993 * Authors R.Brun, M.Hansroul ********* *
1994 * Rewritten V.Perevoztchikov *
1996 * Rewritten in C++ by I.Belikov *
1998 * qfield (kG) - particle charge times magnetic field *
1999 * step (cm) - step length along the helix *
2000 * vect[7](cm,GeV/c) - input/output x, y, z, px/p, py/p ,pz/p, p *
2002 ******************************************************************/
2003 const Int_t ix=0, iy=1, iz=2, ipx=3, ipy=4, ipz=5, ipp=6;
2004 const Double_t kOvSqSix=TMath::Sqrt(1./6.);
2006 Double_t cosx=vect[ipx], cosy=vect[ipy], cosz=vect[ipz];
2008 Double_t rho = qfield*kB2C/vect[ipp];
2009 Double_t tet = rho*step;
2011 Double_t tsint, sintt, sint, cos1t;
2012 if (TMath::Abs(tet) > 0.03) {
2013 sint = TMath::Sin(tet);
2015 tsint = (tet - sint)/tet;
2016 Double_t t=TMath::Sin(0.5*tet);
2020 sintt = (1.-tet*kOvSqSix)*(1.+tet*kOvSqSix); // 1.- tsint;
2025 Double_t f1 = step*sintt;
2026 Double_t f2 = step*cos1t;
2027 Double_t f3 = step*tsint*cosz;
2028 Double_t f4 = -tet*cos1t;
2031 vect[ix] += f1*cosx - f2*cosy;
2032 vect[iy] += f1*cosy + f2*cosx;
2033 vect[iz] += f1*cosz + f3;
2035 vect[ipx] += f4*cosx - f5*cosy;
2036 vect[ipy] += f4*cosy + f5*cosx;
2040 Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]) {
2041 //----------------------------------------------------------------
2042 // Extrapolate this track to the plane X=xk in the field b[].
2044 // X [cm] is in the "tracking coordinate system" of this track.
2045 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
2046 //----------------------------------------------------------------
2049 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
2050 if (TMath::Abs(fP[4])<=kAlmost0) return kFALSE;
2051 // Do not propagate tracks outside the ALICE detector
2052 if (TMath::Abs(dx)>1e5 ||
2053 TMath::Abs(GetY())>1e5 ||
2054 TMath::Abs(GetZ())>1e5) {
2055 AliWarning(Form("Anomalous track, target X:%f",xk));
2060 Double_t crv=GetC(b[2]);
2061 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
2063 Double_t x2r = crv*dx;
2064 Double_t f1=fP[2], f2=f1 + x2r;
2065 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2066 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2069 // Estimate the covariance matrix
2070 Double_t &fP3=fP[3], &fP4=fP[4];
2073 &fC10=fC[1], &fC11=fC[2],
2074 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
2075 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
2076 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
2078 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
2082 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
2083 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
2084 Double_t f12= dx*fP3*f1/(r1*r1*r1);
2085 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
2086 Double_t f13= dx/r1;
2087 Double_t f24= dx; f24*=cc;
2089 Double_t rinv = 1./r1;
2090 Double_t r3inv = rinv*rinv*rinv;
2091 Double_t f24= x2r/fP4;
2092 Double_t f02= dx*r3inv;
2093 Double_t f04=0.5*f24*f02;
2094 Double_t f12= f02*fP3*f1;
2095 Double_t f14=0.5*f24*f02*fP3*f1;
2096 Double_t f13= dx*rinv;
2099 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
2100 Double_t b02=f24*fC40;
2101 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
2102 Double_t b12=f24*fC41;
2103 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
2104 Double_t b22=f24*fC42;
2105 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
2106 Double_t b42=f24*fC44;
2107 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
2108 Double_t b32=f24*fC43;
2111 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
2112 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
2113 Double_t a22=f24*b42;
2115 //F*C*Ft = C + (b + bt + a)
2116 fC00 += b00 + b00 + a00;
2117 fC10 += b10 + b01 + a01;
2118 fC20 += b20 + b02 + a02;
2121 fC11 += b11 + b11 + a11;
2122 fC21 += b21 + b12 + a12;
2125 fC22 += b22 + b22 + a22;
2131 // Appoximate step length
2132 double dy2dx = (f1+f2)/(r1+r2);
2133 Double_t step = (TMath::Abs(x2r)<0.05) ? dx*TMath::Abs(r2 + f2*dy2dx) // chord
2134 : 2.*TMath::ASin(0.5*dx*TMath::Sqrt(1.+dy2dx*dy2dx)*crv)/crv; // arc
2135 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
2137 // Get the track's (x,y,z) and (px,py,pz) in the Global System
2138 Double_t r[3]; GetXYZ(r);
2139 Double_t p[3]; GetPxPyPz(p);
2146 // Rotate to the system where Bx=By=0.
2147 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
2148 Double_t cosphi=1., sinphi=0.;
2149 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
2150 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
2151 Double_t costet=1., sintet=0.;
2152 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
2155 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
2156 vect[1] = -sinphi*r[0] + cosphi*r[1];
2157 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
2159 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
2160 vect[4] = -sinphi*p[0] + cosphi*p[1];
2161 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
2166 // Do the helix step
2167 g3helx3(GetSign()*bb,step,vect);
2170 // Rotate back to the Global System
2171 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
2172 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
2173 r[2] = -sintet*vect[0] + costet*vect[2];
2175 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
2176 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
2177 p[2] = -sintet*vect[3] + costet*vect[5];
2180 // Rotate back to the Tracking System
2181 Double_t cosalp = TMath::Cos(fAlpha);
2182 Double_t sinalp =-TMath::Sin(fAlpha);
2185 t = cosalp*r[0] - sinalp*r[1];
2186 r[1] = sinalp*r[0] + cosalp*r[1];
2189 t = cosalp*p[0] - sinalp*p[1];
2190 p[1] = sinalp*p[0] + cosalp*p[1];
2194 // Do the final correcting step to the target plane (linear approximation)
2195 Double_t x=r[0], y=r[1], z=r[2];
2196 if (TMath::Abs(dx) > kAlmost0) {
2197 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
2205 // Calculate the track parameters
2206 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
2212 fP[4] = GetSign()/(t*pp);
2217 Bool_t AliExternalTrackParam::Translate(Double_t *vTrasl,Double_t *covV){
2219 //Translation: in the event mixing, the tracks can be shifted
2220 //of the difference among primary vertices (vTrasl) and
2221 //the covariance matrix is changed accordingly
2222 //(covV = covariance of the primary vertex).
2223 //Origin: "Romita, Rossella" <R.Romita@gsi.de>
2225 TVector3 translation;
2226 // vTrasl coordinates in the local system
2227 translation.SetXYZ(vTrasl[0],vTrasl[1],vTrasl[2]);
2228 translation.RotateZ(-fAlpha);
2229 translation.GetXYZ(vTrasl);
2231 //compute the new x,y,z of the track
2232 Double_t newX=fX-vTrasl[0];
2233 Double_t newY=fP[0]-vTrasl[1];
2234 Double_t newZ=fP[1]-vTrasl[2];
2236 //define the new parameters
2237 Double_t newParam[5];
2244 // recompute the covariance matrix:
2245 // 1. covV in the local system
2246 Double_t cosRot=TMath::Cos(fAlpha), sinRot=TMath::Sin(fAlpha);
2267 if(uUi.Determinant() <= 0.) {return kFALSE;}
2268 TMatrixD uUiQi(uUi,TMatrixD::kMult,qQi);
2269 TMatrixD m(qQi,TMatrixD::kTransposeMult,uUiQi);
2271 //2. compute the new covariance matrix of the track
2272 Double_t sigmaXX=m(0,0);
2273 Double_t sigmaXZ=m(2,0);
2274 Double_t sigmaXY=m(1,0);
2275 Double_t sigmaYY=GetSigmaY2()+m(1,1);
2276 Double_t sigmaYZ=fC[1]+m(1,2);
2277 Double_t sigmaZZ=fC[2]+m(2,2);
2278 Double_t covarianceYY=sigmaYY + (-1.)*((sigmaXY*sigmaXY)/sigmaXX);
2279 Double_t covarianceYZ=sigmaYZ-(sigmaXZ*sigmaXY/sigmaXX);
2280 Double_t covarianceZZ=sigmaZZ-((sigmaXZ*sigmaXZ)/sigmaXX);
2282 Double_t newCov[15];
2283 newCov[0]=covarianceYY;
2284 newCov[1]=covarianceYZ;
2285 newCov[2]=covarianceZZ;
2286 for(Int_t i=3;i<15;i++){
2290 // set the new parameters
2292 Set(newX,fAlpha,newParam,newCov);
2297 void AliExternalTrackParam::CheckCovariance() {
2299 // This function forces the diagonal elements of the covariance matrix to be positive.
2300 // In case the diagonal element is bigger than the maximal allowed value, it is set to
2301 // the limit and the off-diagonal elements that correspond to it are set to zero.
2302 fC[0] = TMath::Abs(fC[0]);
2304 double scl = TMath::Sqrt(kC0max/fC[0]);
2311 fC[2] = TMath::Abs(fC[2]);
2313 double scl = TMath::Sqrt(kC2max/fC[2]);
2320 fC[5] = TMath::Abs(fC[5]);
2322 double scl = TMath::Sqrt(kC5max/fC[5]);
2329 fC[9] = TMath::Abs(fC[9]);
2331 double scl = TMath::Sqrt(kC9max/fC[9]);
2338 fC[14] = TMath::Abs(fC[14]);
2339 if (fC[14]>kC14max) {
2340 double scl = TMath::Sqrt(kC14max/fC[14]);
2348 // The part below is used for tests and normally is commented out
2349 // TMatrixDSym m(5);
2353 // m(1,0)=fC[1]; m(1,1)=fC[2];
2354 // m(2,0)=fC[3]; m(2,1)=fC[4]; m(2,2)=fC[5];
2355 // m(3,0)=fC[6]; m(3,1)=fC[7]; m(3,2)=fC[8]; m(3,3)=fC[9];
2356 // 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];
2359 // m(0,2)=m(2,0); m(1,2)=m(2,1);
2360 // m(0,3)=m(3,0); m(1,3)=m(3,1); m(2,3)=m(3,2);
2361 // m(0,4)=m(4,0); m(1,4)=m(4,1); m(2,4)=m(4,2); m(3,4)=m(4,3);
2362 // m.EigenVectors(eig);
2364 // // assert(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0);
2365 // if (!(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0)) {
2366 // AliWarning("Negative eigenvalues of the covariance matrix!");
2372 Bool_t AliExternalTrackParam::ConstrainToVertex(const AliVVertex* vtx, Double_t b[3])
2374 // Constrain TPC inner params constrained
2379 Double_t dz[2], cov[3];
2380 if (!PropagateToDCABxByBz(vtx, b, 3, dz, cov))
2384 vtx->GetCovarianceMatrix(covar);
2386 Double_t p[2]= { fP[0] - dz[0], fP[1] - dz[1] };
2387 Double_t c[3]= { covar[2], 0., covar[5] };
2389 Double_t chi2C = GetPredictedChi2(p,c);
2399 //___________________________________________________________________________________________
2400 Bool_t AliExternalTrackParam::GetXatLabR(Double_t r,Double_t &x, Double_t bz, Int_t dir) const
2402 // Get local X of the track position estimated at the radius lab radius r.
2403 // The track curvature is accounted exactly
2405 // The flag "dir" can be used to remove the ambiguity of which intersection to take (out of 2 possible)
2406 // 0 - take the intersection closest to the current track position
2407 // >0 - go along the track (increasing fX)
2408 // <0 - go backward (decreasing fX)
2410 const Double_t &fy=fP[0], &sn = fP[2];
2412 double crv = GetC(bz);
2413 if (TMath::Abs(crv)<=kAlmost0) { // this is a straight track
2414 if (TMath::Abs(sn)>=kAlmost1) { // || to Y axis
2415 double det = (r-fX)*(r+fX);
2416 if (det<0) return kFALSE; // does not reach raduis r
2418 if (dir==0) return kTRUE;
2419 det = TMath::Sqrt(det);
2420 if (dir>0) { // along the track direction
2421 if (sn>0) {if (fy>det) return kFALSE;} // track is along Y axis and above the circle
2422 else {if (fy<-det) return kFALSE;} // track is against Y axis amd belo the circle
2424 else { // agains track direction
2425 if (sn>0) {if (fy<-det) return kFALSE;} // track is along Y axis
2426 else if (fy>det) return kFALSE; // track is against Y axis
2429 else if (TMath::Abs(sn)<=kAlmost0) { // || to X axis
2430 double det = (r-fy)*(r+fy);
2431 if (det<0) return kFALSE; // does not reach raduis r
2432 det = TMath::Sqrt(det);
2434 x = fX>0 ? det : -det; // choose the solution requiring the smalest step
2437 else if (dir>0) { // along the track direction
2438 if (fX > det) return kFALSE; // current point is in on the right from the circle
2439 else if (fX <-det) x = -det; // on the left
2440 else x = det; // within the circle
2442 else { // against the track direction
2443 if (fX <-det) return kFALSE;
2444 else if (fX > det) x = det;
2448 else { // general case of straight line
2449 double cs = TMath::Sqrt((1-sn)*(1+sn));
2450 double xsyc = fX*sn-fy*cs;
2451 double det = (r-xsyc)*(r+xsyc);
2452 if (det<0) return kFALSE; // does not reach raduis r
2453 det = TMath::Sqrt(det);
2454 double xcys = fX*cs+fy*sn;
2456 if (dir==0) t += t>0 ? -det:det; // chose the solution requiring the smalest step
2457 else if (dir>0) { // go in increasing fX direction. ( t+-det > 0)
2458 if (t>=-det) t += -det; // take minimal step giving t>0
2459 else return kFALSE; // both solutions have negative t
2461 else { // go in increasing fX direction. (t+-det < 0)
2462 if (t<det) t -= det; // take minimal step giving t<0
2463 else return kFALSE; // both solutions have positive t
2469 // get center of the track circle
2470 double tR = 1./crv; // track radius (for the moment signed)
2471 double cs = TMath::Sqrt((1-sn)*(1+sn));
2472 double x0 = fX - sn*tR;
2473 double y0 = fy + cs*tR;
2474 double r0 = TMath::Sqrt(x0*x0+y0*y0);
2475 // printf("Xc:%+e Yc:%+e\n",x0,y0);
2477 if (r0<=kAlmost0) return kFALSE; // the track is concentric to circle
2478 tR = TMath::Abs(tR);
2479 double tR2r0 = tR/r0;
2480 double g = 0.5*(r*r/(r0*tR) - tR2r0 - 1./tR2r0);
2481 double det = (1.-g)*(1.+g);
2482 if (det<0) return kFALSE; // does not reach raduis r
2483 det = TMath::Sqrt(det);
2485 // the intersection happens in 2 points: {x0+tR*C,y0+tR*S}
2486 // with C=f*c0+-|s0|*det and S=f*s0-+c0 sign(s0)*det
2487 // where s0 and c0 make direction for the circle center (=x0/r0 and y0/r0)
2489 double tmp = 1.+g*tR2r0;
2492 if (TMath::Abs(y0)>kAlmost0) { // when y0==0 the x,y is unique
2493 double dfx = tR2r0*TMath::Abs(y0)*det;
2494 double dfy = tR2r0*x0*TMath::Sign(det,y0);
2495 if (dir==0) { // chose the one which corresponds to smallest step
2496 double delta = (x-fX)*dfx-(y-fy)*dfy; // the choice of + in C will lead to smaller step if delta<0
2497 if (delta<0) x += dfx;
2500 else if (dir>0) { // along track direction: x must be > fX
2501 x -= dfx; // try the smallest step (dfx is positive)
2502 if (x<fX && (x+=dfx+dfx)<fX) return kFALSE;
2504 else { // backward: x must be < fX
2505 x += dfx; // try the smallest step (dfx is positive)
2506 if (x>fX && (x-=dfx+dfx)>fX) return kFALSE;
2509 else { // special case: track touching the circle just in 1 point
2510 if ( (dir>0&&x<fX) || (dir<0&&x>fX) ) return kFALSE;
2517 //_________________________________________________________
2518 Bool_t AliExternalTrackParam::GetXYZatR(Double_t xr,Double_t bz, Double_t *xyz, Double_t* alpSect) const
2520 // This method has 3 modes of behaviour
2521 // 1) xyz[3] array is provided but alpSect pointer is 0: calculate the position of track intersection
2522 // with circle of radius xr and fill it in xyz array
2523 // 2) alpSect pointer is provided: find alpha of the sector where the track reaches local coordinate xr
2524 // Note that in this case xr is NOT the radius but the local coordinate.
2525 // If the xyz array is provided, it will be filled by track lab coordinates at local X in this sector
2526 // 3) Neither alpSect nor xyz pointers are provided: just check if the track reaches radius xr
2529 double crv = GetC(bz);
2530 if ( (TMath::Abs(bz))<kAlmost0Field ) crv=0.;
2531 const double &fy = fP[0];
2532 const double &fz = fP[1];
2533 const double &sn = fP[2];
2534 const double &tgl = fP[3];
2536 // general circle parameterization:
2537 // x = (r0+tR)cos(phi0) - tR cos(t+phi0)
2538 // y = (r0+tR)sin(phi0) - tR sin(t+phi0)
2539 // where qb is the sign of the curvature, tR is the track's signed radius and r0
2540 // is the DCA of helix to origin
2542 double tR = 1./crv; // track radius signed
2543 double cs = TMath::Sqrt((1-sn)*(1+sn));
2544 double x0 = fX - sn*tR; // helix center coordinates
2545 double y0 = fy + cs*tR;
2546 double phi0 = TMath::ATan2(y0,x0); // angle of PCA wrt to the origin
2547 if (tR<0) phi0 += TMath::Pi();
2548 if (phi0 > TMath::Pi()) phi0 -= 2.*TMath::Pi();
2549 else if (phi0 <-TMath::Pi()) phi0 += 2.*TMath::Pi();
2550 double cs0 = TMath::Cos(phi0);
2551 double sn0 = TMath::Sin(phi0);
2552 double r0 = x0*cs0 + y0*sn0 - tR; // DCA to origin
2553 double r2R = 1.+r0/tR;
2556 if (r2R<kAlmost0) return kFALSE; // helix is centered at the origin, no specific intersection with other concetric circle
2557 if (!xyz && !alpSect) return kTRUE;
2558 double xr2R = xr/tR;
2559 double r2Ri = 1./r2R;
2560 // the intersection cos(t) = [1 + (r0/tR+1)^2 - (r0/tR)^2]/[2(1+r0/tR)]
2561 double cosT = 0.5*(r2R + (1-xr2R*xr2R)*r2Ri);
2562 if ( TMath::Abs(cosT)>kAlmost1 ) {
2563 // printf("Does not reach : %f %f\n",r0,tR);
2564 return kFALSE; // track does not reach the radius xr
2567 double t = TMath::ACos(cosT);
2569 // intersection point
2571 xyzi[0] = x0 - tR*TMath::Cos(t+phi0);
2572 xyzi[1] = y0 - tR*TMath::Sin(t+phi0);
2573 if (xyz) { // if postition is requested, then z is needed:
2574 double t0 = TMath::ATan2(cs,-sn) - phi0;
2575 double z0 = fz - t0*tR*tgl;
2576 xyzi[2] = z0 + tR*t*tgl;
2580 Local2GlobalPosition(xyzi,fAlpha);
2589 double &alp = *alpSect;
2590 // determine the sector of crossing
2591 double phiPos = TMath::Pi()+TMath::ATan2(-xyzi[1],-xyzi[0]);
2592 int sect = ((Int_t)(phiPos*TMath::RadToDeg()))/20;
2593 alp = TMath::DegToRad()*(20*sect+10);
2594 double x2r,f1,f2,r1,r2,dx,dy2dx,yloc=0, ylocMax = xr*TMath::Tan(TMath::Pi()/18); // min max Y within sector at given X
2597 Double_t ca=TMath::Cos(alp-fAlpha), sa=TMath::Sin(alp-fAlpha);
2598 if ((cs*ca+sn*sa)<0) {
2599 AliDebug(1,Form("Rotation to target sector impossible: local cos(phi) would become %.2f",cs*ca+sn*sa));
2604 if (TMath::Abs(f1) >= kAlmost1) {
2605 AliDebug(1,Form("Rotation to target sector impossible: local sin(phi) would become %.2f",f1));
2609 double tmpX = fX*ca + fy*sa;
2610 double tmpY = -fX*sa + fy*ca;
2612 // estimate Y at X=xr
2616 if (TMath::Abs(f2) >= kAlmost1) {
2617 AliDebug(1,Form("Propagation in target sector failed ! %.10e",f2));
2620 r1 = TMath::Sqrt((1.-f1)*(1.+f1));
2621 r2 = TMath::Sqrt((1.-f2)*(1.+f2));
2622 dy2dx = (f1+f2)/(r1+r2);
2623 yloc = tmpY + dx*dy2dx;
2624 if (yloc>ylocMax) {alp += 2*TMath::Pi()/18; sect++;}
2625 else if (yloc<-ylocMax) {alp -= 2*TMath::Pi()/18; sect--;}
2627 if (alp >= TMath::Pi()) alp -= 2*TMath::Pi();
2628 else if (alp < -TMath::Pi()) alp += 2*TMath::Pi();
2629 // if (sect>=18) sect = 0;
2630 // if (sect<=0) sect = 17;
2633 // if alpha was requested, then recalculate the position at intersection in sector
2637 if (TMath::Abs(x2r)<0.05) xyz[2] = fz + dx*(r2 + f2*dy2dx)*tgl;
2639 // for small dx/R the linear apporximation of the arc by the segment is OK,
2640 // but at large dx/R the error is very large and leads to incorrect Z propagation
2641 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
2642 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
2643 // Similarly, the rotation angle in linear in dx only for dx<<R
2644 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
2645 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
2646 xyz[2] = fz + rot/crv*tgl;
2648 Local2GlobalPosition(xyz,alp);