1 /**************************************************************************
2 * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
4 * Author: The ALICE Off-line Project. *
5 * Contributors are mentioned in the code where appropriate. *
7 * Permission to use, copy, modify and distribute this software and its *
8 * documentation strictly for non-commercial purposes is hereby granted *
9 * without fee, provided that the above copyright notice appears in all *
10 * copies and that both the copyright notice and this permission notice *
11 * appear in the supporting documentation. The authors make no claims *
12 * about the suitability of this software for any purpose. It is *
13 * provided "as is" without express or implied warranty. *
14 **************************************************************************/
18 ///////////////////////////////////////////////////////////////////////////////
20 // Implementation of the external track parameterisation class. //
22 // This parameterisation is used to exchange tracks between the detectors. //
23 // A set of functions returning the position and the momentum of tracks //
24 // in the global coordinate system as well as the track impact parameters //
26 // Origin: I.Belikov, CERN, Jouri.Belikov@cern.ch //
27 ///////////////////////////////////////////////////////////////////////////////
31 #include <TMatrixDSym.h>
32 #include <TPolyMarker3D.h>
36 #include "AliExternalTrackParam.h"
37 #include "AliVVertex.h"
40 ClassImp(AliExternalTrackParam)
42 Double32_t AliExternalTrackParam::fgMostProbablePt=kMostProbablePt;
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);
185 //_____________________________________________________________________________
186 void AliExternalTrackParam::Set(Double_t xyz[3],Double_t pxpypz[3],
187 Double_t cv[21],Short_t sign)
190 // create external track parameters from the global parameters
191 // x,y,z,px,py,pz and their 6x6 covariance matrix
192 // A.Dainese 10.10.08
194 // Calculate alpha: the rotation angle of the corresponding local system.
196 // For global radial position inside the beam pipe, alpha is the
197 // azimuthal angle of the momentum projected on (x,y).
199 // For global radial position outside the ITS, alpha is the
200 // azimuthal angle of the centre of the TPC sector in which the point
203 const double kSafe = 1e-5;
204 Double_t radPos2 = xyz[0]*xyz[0]+xyz[1]*xyz[1];
205 Double_t radMax = 45.; // approximately ITS outer radius
206 if (radPos2 < radMax*radMax) { // inside the ITS
207 fAlpha = TMath::ATan2(pxpypz[1],pxpypz[0]);
208 } else { // outside the ITS
209 Float_t phiPos = TMath::Pi()+TMath::ATan2(-xyz[1], -xyz[0]);
211 TMath::DegToRad()*(20*((((Int_t)(phiPos*TMath::RadToDeg()))/20))+10);
214 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
215 // protection: avoid alpha being too close to 0 or +-pi/2
216 if (TMath::Abs(sn)<2*kSafe) {
217 if (fAlpha>0) fAlpha += fAlpha< TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
218 else fAlpha += fAlpha>-TMath::Pi()/2. ? -2*kSafe : 2*kSafe;
219 cs=TMath::Cos(fAlpha);
220 sn=TMath::Sin(fAlpha);
222 else if (TMath::Abs(cs)<2*kSafe) {
223 if (fAlpha>0) fAlpha += fAlpha> TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
224 else fAlpha += fAlpha>-TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
225 cs=TMath::Cos(fAlpha);
226 sn=TMath::Sin(fAlpha);
228 // Get the vertex of origin and the momentum
229 TVector3 ver(xyz[0],xyz[1],xyz[2]);
230 TVector3 mom(pxpypz[0],pxpypz[1],pxpypz[2]);
232 // avoid momenta along axis
233 if (TMath::Abs(mom[0])<kSafe) mom[0] = TMath::Sign(kSafe*TMath::Abs(mom[1]), mom[0]);
234 if (TMath::Abs(mom[1])<kSafe) mom[1] = TMath::Sign(kSafe*TMath::Abs(mom[0]), mom[1]);
236 // Rotate to the local coordinate system
237 ver.RotateZ(-fAlpha);
238 mom.RotateZ(-fAlpha);
241 // x of the reference plane
244 Double_t charge = (Double_t)sign;
248 fP[2] = TMath::Sin(mom.Phi());
249 fP[3] = mom.Pz()/mom.Pt();
250 fP[4] = TMath::Sign(1/mom.Pt(),charge);
252 if (TMath::Abs( 1-fP[2]) < 3*kSafe) fP[2] = 1.- 3*kSafe; //Protection
253 else if (TMath::Abs(-1-fP[2]) < 3*kSafe) fP[2] =-1.+ 3*kSafe; //Protection
255 // Covariance matrix (formulas to be simplified)
256 Double_t pt=1./TMath::Abs(fP[4]);
257 // avoid alpha+phi being to close to +-pi/2 in the cov.matrix evaluation
259 Double_t r=TMath::Sqrt((1.-fp2)*(1.+fp2));
261 Double_t m00=-sn;// m10=cs;
262 Double_t m23=-pt*(sn + fp2*cs/r), m43=-pt*pt*(r*cs - fp2*sn);
263 Double_t m24= pt*(cs - fp2*sn/r), m44=-pt*pt*(r*sn + fp2*cs);
264 Double_t m35=pt, m45=-pt*pt*fP[3];
270 Double_t cv34 = TMath::Sqrt(cv[3 ]*cv[3 ]+cv[4 ]*cv[4 ]);
271 Double_t a1=cv[13]-cv[9]*(m23*m44+m43*m24)/m23/m43;
272 Double_t a2=m23*m24-m23*(m23*m44+m43*m24)/m43;
273 Double_t a3=m43*m44-m43*(m23*m44+m43*m24)/m23;
274 Double_t a4=cv[14]+2.*cv[9]; //cv[14]-2.*cv[9]*m24*m44/m23/m43;
275 Double_t a5=m24*m24-2.*m24*m44*m23/m43;
276 Double_t a6=m44*m44-2.*m24*m44*m43/m23;
278 fC[0 ] = cv[0 ]+cv[2 ];
279 fC[1 ] = TMath::Sign(cv34,cv[3 ]/m00);
281 fC[3 ] = (cv[10]*m43-cv[6]*m44)/(m24*m43-m23*m44)/m00;
282 fC[10] = (cv[6]/m00-fC[3 ]*m23)/m43;
283 fC[6 ] = (cv[15]/m00-fC[10]*m45)/m35;
284 fC[4 ] = (cv[12]*m43-cv[8]*m44)/(m24*m43-m23*m44);
285 fC[11] = (cv[8]-fC[4]*m23)/m43;
286 fC[7 ] = cv[17]/m35-fC[11]*m45/m35;
287 fC[5 ] = TMath::Abs((a4*a3-a6*a1)/(a5*a3-a6*a2));
288 fC[14] = TMath::Abs((a1-a2*fC[5])/a3);
289 fC[12] = (cv[9]-fC[5]*m23*m23-fC[14]*m43*m43)/m23/m43;
290 Double_t b1=cv[18]-fC[12]*m23*m45-fC[14]*m43*m45;
293 Double_t b4=cv[19]-fC[12]*m24*m45-fC[14]*m44*m45;
296 fC[8 ] = (b4-b6*b1/b3)/(b5-b6*b2/b3);
297 fC[13] = b1/b3-b2*fC[8]/b3;
298 fC[9 ] = TMath::Abs((cv[20]-fC[14]*(m45*m45)-fC[13]*2.*m35*m45)/(m35*m35));
306 //_____________________________________________________________________________
307 void AliExternalTrackParam::Set(Double_t xyz[3],Double_t pxpypz[3],
308 Double_t cv[21],Short_t sign)
311 // create external track parameters from the global parameters
312 // x,y,z,px,py,pz and their 6x6 covariance matrix
313 // A.Dainese 10.10.08
315 // Calculate alpha: the rotation angle of the corresponding local system.
317 // For global radial position inside the beam pipe, alpha is the
318 // azimuthal angle of the momentum projected on (x,y).
320 // For global radial position outside the ITS, alpha is the
321 // azimuthal angle of the centre of the TPC sector in which the point
324 const double kSafe = 1e-5;
325 Double_t radPos2 = xyz[0]*xyz[0]+xyz[1]*xyz[1];
326 Double_t radMax = 45.; // approximately ITS outer radius
327 if (radPos2 < radMax*radMax) { // inside the ITS
328 fAlpha = TMath::ATan2(pxpypz[1],pxpypz[0]);
329 } else { // outside the ITS
330 Float_t phiPos = TMath::Pi()+TMath::ATan2(-xyz[1], -xyz[0]);
332 TMath::DegToRad()*(20*((((Int_t)(phiPos*TMath::RadToDeg()))/20))+10);
335 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
336 // protection: avoid alpha being too close to 0 or +-pi/2
337 if (TMath::Abs(sn)<2*kSafe) {
338 if (fAlpha>0) fAlpha += fAlpha< TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
339 else fAlpha += fAlpha>-TMath::Pi()/2. ? -2*kSafe : 2*kSafe;
340 cs=TMath::Cos(fAlpha);
341 sn=TMath::Sin(fAlpha);
343 else if (TMath::Abs(cs)<2*kSafe) {
344 if (fAlpha>0) fAlpha += fAlpha> TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
345 else fAlpha += fAlpha>-TMath::Pi()/2. ? 2*kSafe : -2*kSafe;
346 cs=TMath::Cos(fAlpha);
347 sn=TMath::Sin(fAlpha);
349 // Get the vertex of origin and the momentum
350 TVector3 ver(xyz[0],xyz[1],xyz[2]);
351 TVector3 mom(pxpypz[0],pxpypz[1],pxpypz[2]);
353 // Rotate to the local coordinate system
354 ver.RotateZ(-fAlpha);
355 mom.RotateZ(-fAlpha);
358 // x of the reference plane
361 Double_t charge = (Double_t)sign;
365 fP[2] = TMath::Sin(mom.Phi());
366 fP[3] = mom.Pz()/mom.Pt();
367 fP[4] = TMath::Sign(1/mom.Pt(),charge);
369 if (TMath::Abs( 1-fP[2]) < kSafe) fP[2] = 1.- kSafe; //Protection
370 else if (TMath::Abs(-1-fP[2]) < kSafe) fP[2] =-1.+ kSafe; //Protection
372 // Covariance matrix (formulas to be simplified)
373 Double_t pt=1./TMath::Abs(fP[4]);
374 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
376 Double_t cv34 = TMath::Sqrt(cv[3 ]*cv[3 ]+cv[4 ]*cv[4 ]);
379 double sgcheck = r*sn + fP[2]*cs;
380 if (TMath::Abs(sgcheck)>=1-kSafe) { // special case: lab phi is +-pi/2
382 sgcheck = TMath::Sign(1.0,sgcheck);
384 else if (TMath::Abs(sgcheck)<kSafe) {
385 sgcheck = TMath::Sign(1.0,cs);
386 special = 2; // special case: lab phi is 0
389 fC[0 ] = cv[0 ]+cv[2 ];
390 fC[1 ] = TMath::Sign(cv34,-cv[3 ]*sn);
395 double pti2 = pti*pti;
398 fC[4 ] = -sgcheck*cv[8]*r*pti;
399 fC[5 ] = TMath::Abs(cv[9]*r*r*pti2);
400 fC[6 ] = (cv[10]*fP[3]-sgcheck*cv[15])*pti/r;
401 fC[7 ] = (cv[17]-sgcheck*cv[12]*fP[3])*pti;
402 fC[8 ] = (-sgcheck*cv[18]+cv[13]*fP[3])*r*pti2;
403 fC[9 ] = TMath::Abs( cv[20]-2*sgcheck*cv[19]*fP[3]+cv[14]*fP[3]*fP[3])*pti2;
404 fC[10] = cv[10]*pti2/r*q;
405 fC[11] = -sgcheck*cv[12]*pti2*q;
406 fC[12] = cv[13]*r*pti*pti2*q;
407 fC[13] = (-sgcheck*cv[19]+cv[14]*fP[3])*r*pti2*pti;
408 fC[14] = TMath::Abs(cv[14]*pti2*pti2);
409 } else if (special==2) {
411 double pti2 = pti*pti;
413 fC[3 ] = -cv[10]*pti*cs/sn;
414 fC[4 ] = cv[12]*cs*pti;
415 fC[5 ] = TMath::Abs(cv[14]*cs*cs*pti2);
416 fC[6 ] = (sgcheck*cv[6]*fP[3]-cv[15])*pti/sn;
417 fC[7 ] = (cv[17]-sgcheck*cv[8]*fP[3])*pti;
418 fC[8 ] = (cv[19]-sgcheck*cv[13]*fP[3])*cs*pti2;
419 fC[9 ] = TMath::Abs( cv[20]-2*sgcheck*cv[18]*fP[3]+cv[9]*fP[3]*fP[3])*pti2;
420 fC[10] = sgcheck*cv[6]*pti2/sn*q;
421 fC[11] = -sgcheck*cv[8]*pti2*q;
422 fC[12] = -sgcheck*cv[13]*cs*pti*pti2*q;
423 fC[13] = (-sgcheck*cv[18]+cv[9]*fP[3])*pti2*pti*q;
424 fC[14] = TMath::Abs(cv[9]*pti2*pti2);
427 Double_t m00=-sn;// m10=cs;
428 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
429 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
430 Double_t m35=pt, m45=-pt*pt*fP[3];
436 Double_t a1=cv[13]-cv[9]*(m23*m44+m43*m24)/m23/m43;
437 Double_t a2=m23*m24-m23*(m23*m44+m43*m24)/m43;
438 Double_t a3=m43*m44-m43*(m23*m44+m43*m24)/m23;
439 Double_t a4=cv[14]+2.*cv[9]; //cv[14]-2.*cv[9]*m24*m44/m23/m43;
440 Double_t a5=m24*m24-2.*m24*m44*m23/m43;
441 Double_t a6=m44*m44-2.*m24*m44*m43/m23;
443 fC[3 ] = (cv[10]*m43-cv[6]*m44)/(m24*m43-m23*m44)/m00;
444 fC[10] = (cv[6]/m00-fC[3 ]*m23)/m43;
445 fC[6 ] = (cv[15]/m00-fC[10]*m45)/m35;
446 fC[4 ] = (cv[12]*m43-cv[8]*m44)/(m24*m43-m23*m44);
447 fC[11] = (cv[8]-fC[4]*m23)/m43;
448 fC[7 ] = cv[17]/m35-fC[11]*m45/m35;
449 fC[5 ] = TMath::Abs((a4*a3-a6*a1)/(a5*a3-a6*a2));
450 fC[14] = TMath::Abs((a1-a2*fC[5])/a3);
451 fC[12] = (cv[9]-fC[5]*m23*m23-fC[14]*m43*m43)/m23/m43;
452 Double_t b1=cv[18]-fC[12]*m23*m45-fC[14]*m43*m45;
455 Double_t b4=cv[19]-fC[12]*m24*m45-fC[14]*m44*m45;
458 fC[8 ] = (b4-b6*b1/b3)/(b5-b6*b2/b3);
459 fC[13] = b1/b3-b2*fC[8]/b3;
460 fC[9 ] = TMath::Abs((cv[20]-fC[14]*(m45*m45)-fC[13]*2.*m35*m45)/(m35*m35));
467 //_____________________________________________________________________________
468 void AliExternalTrackParam::Reset() {
470 // Resets all the parameters to 0
473 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
474 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
477 //_____________________________________________________________________________
478 void AliExternalTrackParam::AddCovariance(const Double_t c[15]) {
480 // Add "something" to the track covarince matrix.
481 // May be needed to account for unknown mis-calibration/mis-alignment
484 fC[1] +=c[1]; fC[2] +=c[2];
485 fC[3] +=c[3]; fC[4] +=c[4]; fC[5] +=c[5];
486 fC[6] +=c[6]; fC[7] +=c[7]; fC[8] +=c[8]; fC[9] +=c[9];
487 fC[10]+=c[10]; fC[11]+=c[11]; fC[12]+=c[12]; fC[13]+=c[13]; fC[14]+=c[14];
492 Double_t AliExternalTrackParam::GetP() const {
493 //---------------------------------------------------------------------
494 // This function returns the track momentum
495 // Results for (nearly) straight tracks are meaningless !
496 //---------------------------------------------------------------------
497 if (TMath::Abs(fP[4])<=kAlmost0) return kVeryBig;
498 return TMath::Sqrt(1.+ fP[3]*fP[3])/TMath::Abs(fP[4]);
501 Double_t AliExternalTrackParam::Get1P() const {
502 //---------------------------------------------------------------------
503 // This function returns the 1/(track momentum)
504 //---------------------------------------------------------------------
505 return TMath::Abs(fP[4])/TMath::Sqrt(1.+ fP[3]*fP[3]);
508 //_______________________________________________________________________
509 Double_t AliExternalTrackParam::GetD(Double_t x,Double_t y,Double_t b) const {
510 //------------------------------------------------------------------
511 // This function calculates the transverse impact parameter
512 // with respect to a point with global coordinates (x,y)
513 // in the magnetic field "b" (kG)
514 //------------------------------------------------------------------
515 if (TMath::Abs(b) < kAlmost0Field) return GetLinearD(x,y);
516 Double_t rp4=GetC(b);
518 Double_t xt=fX, yt=fP[0];
520 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
521 Double_t a = x*cs + y*sn;
522 y = -x*sn + y*cs; x=a;
525 sn=rp4*xt - fP[2]; cs=rp4*yt + TMath::Sqrt((1.- fP[2])*(1.+fP[2]));
526 a=2*(xt*fP[2] - yt*TMath::Sqrt((1.-fP[2])*(1.+fP[2])))-rp4*(xt*xt + yt*yt);
527 return -a/(1 + TMath::Sqrt(sn*sn + cs*cs));
530 //_______________________________________________________________________
531 void AliExternalTrackParam::
532 GetDZ(Double_t x, Double_t y, Double_t z, Double_t b, Float_t dz[2]) const {
533 //------------------------------------------------------------------
534 // This function calculates the transverse and longitudinal impact parameters
535 // with respect to a point with global coordinates (x,y)
536 // in the magnetic field "b" (kG)
537 //------------------------------------------------------------------
538 Double_t f1 = fP[2], r1 = TMath::Sqrt((1.-f1)*(1.+f1));
539 Double_t xt=fX, yt=fP[0];
540 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
541 Double_t a = x*cs + y*sn;
542 y = -x*sn + y*cs; x=a;
545 Double_t rp4=GetC(b);
546 if ((TMath::Abs(b) < kAlmost0Field) || (TMath::Abs(rp4) < kAlmost0)) {
547 dz[0] = -(xt*f1 - yt*r1);
548 dz[1] = fP[1] + (dz[0]*f1 - xt)/r1*fP[3] - z;
552 sn=rp4*xt - f1; cs=rp4*yt + r1;
553 a=2*(xt*f1 - yt*r1)-rp4*(xt*xt + yt*yt);
554 Double_t rr=TMath::Sqrt(sn*sn + cs*cs);
556 Double_t f2 = -sn/rr, r2 = TMath::Sqrt((1.-f2)*(1.+f2));
557 dz[1] = fP[1] + fP[3]/rp4*TMath::ASin(f2*r1 - f1*r2) - z;
560 //_______________________________________________________________________
561 Double_t AliExternalTrackParam::GetLinearD(Double_t xv,Double_t yv) const {
562 //------------------------------------------------------------------
563 // This function calculates the transverse impact parameter
564 // with respect to a point with global coordinates (xv,yv)
565 // neglecting the track curvature.
566 //------------------------------------------------------------------
567 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
568 Double_t x= xv*cs + yv*sn;
569 Double_t y=-xv*sn + yv*cs;
571 Double_t d = (fX-x)*fP[2] - (fP[0]-y)*TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
576 Bool_t AliExternalTrackParam::CorrectForMeanMaterialdEdx
577 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
580 //------------------------------------------------------------------
581 // This function corrects the track parameters for the crossed material.
582 // "xOverX0" - X/X0, the thickness in units of the radiation length.
583 // "xTimesRho" - is the product length*density (g/cm^2).
584 // It should be passed as negative when propagating tracks
585 // from the intreaction point to the outside of the central barrel.
586 // "mass" - the mass of this particle (GeV/c^2). Negative mass means charge=2 particle
587 // "dEdx" - mean enery loss (GeV/(g/cm^2)
588 // "anglecorr" - switch for the angular correction
589 //------------------------------------------------------------------
594 Double_t &fC22=fC[5];
595 Double_t &fC33=fC[9];
596 Double_t &fC43=fC[13];
597 Double_t &fC44=fC[14];
599 //Apply angle correction, if requested
601 Double_t angle=TMath::Sqrt((1.+ fP3*fP3)/((1-fP2)*(1.+fP2)));
607 if (mass<0) p += p; // q=2 particle
609 Double_t beta2=p2/(p2 + mass*mass);
611 //Calculating the multiple scattering corrections******************
617 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
618 Double_t theta2=0.0136*0.0136/(beta2*p2)*TMath::Abs(xOverX0);
619 if (GetUseLogTermMS()) {
620 double lt = 1+0.038*TMath::Log(TMath::Abs(xOverX0));
621 if (lt>0) theta2 *= lt*lt;
623 if (mass<0) theta2 *= 4; // q=2 particle
624 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
625 cC22 = theta2*((1.-fP2)*(1.+fP2))*(1. + fP3*fP3);
626 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
627 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
628 cC44 = theta2*fP3*fP4*fP3*fP4;
631 //Calculating the energy loss corrections************************
633 if ((xTimesRho != 0.) && (beta2 < 1.)) {
634 Double_t dE=dEdx*xTimesRho;
635 Double_t e=TMath::Sqrt(p2 + mass*mass);
636 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
637 if ( (1.+ dE/p2*(dE + 2*e)) < 0. ) return kFALSE;
638 cP4 = 1./TMath::Sqrt(1.+ dE/p2*(dE + 2*e)); //A precise formula by Ruben !
639 if (TMath::Abs(fP4*cP4)>100.) return kFALSE; //Do not track below 10 MeV/c
642 // Approximate energy loss fluctuation (M.Ivanov)
643 const Double_t knst=0.07; // To be tuned.
644 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
645 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
649 //Applying the corrections*****************************
661 Bool_t AliExternalTrackParam::CorrectForMeanMaterial
662 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
664 Double_t (*Bethe)(Double_t)) {
665 //------------------------------------------------------------------
666 // This function corrects the track parameters for the crossed material.
667 // "xOverX0" - X/X0, the thickness in units of the radiation length.
668 // "xTimesRho" - is the product length*density (g/cm^2).
669 // It should be passed as negative when propagating tracks
670 // from the intreaction point to the outside of the central barrel.
671 // "mass" - the mass of this particle (GeV/c^2). mass<0 means charge=2
672 // "anglecorr" - switch for the angular correction
673 // "Bethe" - function calculating the energy loss (GeV/(g/cm^2))
674 //------------------------------------------------------------------
676 Double_t bg=GetP()/mass;
679 AliDebug(2,Form("Mass %f corresponds to unknown PID particle",mass));
684 Double_t dEdx=Bethe(bg);
685 if (mass<0) dEdx *= 4;
687 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
690 Bool_t AliExternalTrackParam::CorrectForMeanMaterialZA
691 (Double_t xOverX0, Double_t xTimesRho, Double_t mass,
698 //------------------------------------------------------------------
699 // This function corrects the track parameters for the crossed material
700 // using the full Geant-like Bethe-Bloch formula parameterization
701 // "xOverX0" - X/X0, the thickness in units of the radiation length.
702 // "xTimesRho" - is the product length*density (g/cm^2).
703 // It should be passed as negative when propagating tracks
704 // from the intreaction point to the outside of the central barrel.
705 // "mass" - the mass of this particle (GeV/c^2). mass<0 means charge=2 particle
706 // "density" - mean density (g/cm^3)
707 // "zOverA" - mean Z/A
708 // "exEnergy" - mean exitation energy (GeV)
709 // "jp1" - density effect first junction point
710 // "jp2" - density effect second junction point
711 // "anglecorr" - switch for the angular correction
713 // The default values of the parameters are for silicon
715 //------------------------------------------------------------------
717 Double_t bg=GetP()/mass;
720 AliDebug(2,Form("Mass %f corresponds to unknown PID particle",mass));
725 Double_t dEdx=BetheBlochGeant(bg,density,jp1,jp2,exEnergy,zOverA);
727 if (mass<0) dEdx *= 4;
728 return CorrectForMeanMaterialdEdx(xOverX0,xTimesRho,mass,dEdx,anglecorr);
733 Bool_t AliExternalTrackParam::CorrectForMaterial
734 (Double_t d, Double_t x0, Double_t mass, Double_t (*Bethe)(Double_t)) {
735 //------------------------------------------------------------------
736 // Deprecated function !
737 // Better use CorrectForMeanMaterial instead of it.
739 // This function corrects the track parameters for the crossed material
740 // "d" - the thickness (fraction of the radiation length)
741 // It should be passed as negative when propagating tracks
742 // from the intreaction point to the outside of the central barrel.
743 // "x0" - the radiation length (g/cm^2)
744 // "mass" - the mass of this particle (GeV/c^2)
745 //------------------------------------------------------------------
747 return CorrectForMeanMaterial(d,x0*d,mass,kTRUE,Bethe);
751 Double_t AliExternalTrackParam::BetheBlochAleph(Double_t bg,
758 // This is the empirical ALEPH parameterization of the Bethe-Bloch formula.
759 // It is normalized to 1 at the minimum.
763 // The default values for the kp* parameters are for ALICE TPC.
764 // The returned value is in MIP units
767 Double_t beta = bg/TMath::Sqrt(1.+ bg*bg);
769 Double_t aa = TMath::Power(beta,kp4);
770 Double_t bb = TMath::Power(1./bg,kp5);
772 bb=TMath::Log(kp3+bb);
774 return (kp2-aa-bb)*kp1/aa;
777 Double_t AliExternalTrackParam::BetheBlochGeant(Double_t bg,
784 // This is the parameterization of the Bethe-Bloch formula inspired by Geant.
787 // kp0 - density [g/cm^3]
788 // kp1 - density effect first junction point
789 // kp2 - density effect second junction point
790 // kp3 - mean excitation energy [GeV]
793 // The default values for the kp* parameters are for silicon.
794 // The returned value is in [GeV/(g/cm^2)].
797 const Double_t mK = 0.307075e-3; // [GeV*cm^2/g]
798 const Double_t me = 0.511e-3; // [GeV/c^2]
799 const Double_t rho = kp0;
800 const Double_t x0 = kp1*2.303;
801 const Double_t x1 = kp2*2.303;
802 const Double_t mI = kp3;
803 const Double_t mZA = kp4;
804 const Double_t bg2 = bg*bg;
805 const Double_t maxT= 2*me*bg2; // neglecting the electron mass
809 const Double_t x=TMath::Log(bg);
810 const Double_t lhwI=TMath::Log(28.816*1e-9*TMath::Sqrt(rho*mZA)/mI);
814 const Double_t r=(x1-x)/(x1-x0);
815 d2 = lhwI + x - 0.5 + (0.5 - lhwI - x0)*r*r*r;
818 return mK*mZA*(1+bg2)/bg2*
819 (0.5*TMath::Log(2*me*bg2*maxT/(mI*mI)) - bg2/(1+bg2) - d2);
822 Double_t AliExternalTrackParam::BetheBlochSolid(Double_t bg) {
823 //------------------------------------------------------------------
824 // This is an approximation of the Bethe-Bloch formula,
825 // reasonable for solid materials.
826 // All the parameters are, in fact, for Si.
827 // The returned value is in [GeV/(g/cm^2)]
828 //------------------------------------------------------------------
830 return BetheBlochGeant(bg);
833 Double_t AliExternalTrackParam::BetheBlochGas(Double_t bg) {
834 //------------------------------------------------------------------
835 // This is an approximation of the Bethe-Bloch formula,
836 // reasonable for gas materials.
837 // All the parameters are, in fact, for Ne.
838 // The returned value is in [GeV/(g/cm^2)]
839 //------------------------------------------------------------------
841 const Double_t rho = 0.9e-3;
842 const Double_t x0 = 2.;
843 const Double_t x1 = 4.;
844 const Double_t mI = 140.e-9;
845 const Double_t mZA = 0.49555;
847 return BetheBlochGeant(bg,rho,x0,x1,mI,mZA);
850 Bool_t AliExternalTrackParam::Rotate(Double_t alpha) {
851 //------------------------------------------------------------------
852 // Transform this track to the local coord. system rotated
853 // by angle "alpha" (rad) with respect to the global coord. system.
854 //------------------------------------------------------------------
855 if (TMath::Abs(fP[2]) >= kAlmost1) {
856 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
860 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
861 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
865 Double_t &fC00=fC[0];
866 Double_t &fC10=fC[1];
867 Double_t &fC20=fC[3];
868 Double_t &fC21=fC[4];
869 Double_t &fC22=fC[5];
870 Double_t &fC30=fC[6];
871 Double_t &fC32=fC[8];
872 Double_t &fC40=fC[10];
873 Double_t &fC42=fC[12];
876 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
877 Double_t sf=fP2, cf=TMath::Sqrt((1.- fP2)*(1.+fP2)); // Improve precision
878 // RS: check if rotation does no invalidate track model (cos(local_phi)>=0, i.e. particle
879 // direction in local frame is along the X axis
880 if ((cf*ca+sf*sa)<0) {
881 AliDebug(1,Form("Rotation failed: local cos(phi) would become %.2f",cf*ca+sf*sa));
885 Double_t tmp=sf*ca - cf*sa;
887 if (TMath::Abs(tmp) >= kAlmost1) {
888 if (TMath::Abs(tmp) > 1.+ Double_t(FLT_EPSILON))
889 AliWarning(Form("Rotation failed ! %.10e",tmp));
897 if (TMath::Abs(cf)<kAlmost0) {
898 AliError(Form("Too small cosine value %f",cf));
902 Double_t rr=(ca+sf/cf*sa);
919 //______________________________________________________
920 Bool_t AliExternalTrackParam::RotateParamOnly(Double_t alpha)
922 // rotate to new frame, ignore covariance
923 if (TMath::Abs(fP[2]) >= kAlmost1) {
924 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
928 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
929 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
935 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
936 Double_t sf=fP2, cf=TMath::Sqrt((1.- fP2)*(1.+fP2)); // Improve precision
937 // RS: check if rotation does no invalidate track model (cos(local_phi)>=0, i.e. particle
938 // direction in local frame is along the X axis
939 if ((cf*ca+sf*sa)<0) {
940 AliDebug(1,Form("Rotation failed: local cos(phi) would become %.2f",cf*ca+sf*sa));
944 Double_t tmp=sf*ca - cf*sa;
946 if (TMath::Abs(tmp) >= kAlmost1) {
947 if (TMath::Abs(tmp) > 1.+ Double_t(FLT_EPSILON))
948 AliWarning(Form("Rotation failed ! %.10e",tmp));
958 Bool_t AliExternalTrackParam::Invert() {
959 //------------------------------------------------------------------
960 // Transform this track to the local coord. system rotated by 180 deg.
961 //------------------------------------------------------------------
963 fAlpha += TMath::Pi();
964 while (fAlpha < -TMath::Pi()) fAlpha += 2*TMath::Pi();
965 while (fAlpha >= TMath::Pi()) fAlpha -= 2*TMath::Pi();
972 fC[1] = -fC[1]; // since the fP1 and fP2 are not inverted, their covariances with others change sign
982 Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
983 //----------------------------------------------------------------
984 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
985 //----------------------------------------------------------------
987 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
989 Double_t crv=GetC(b);
990 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
992 Double_t x2r = crv*dx;
993 Double_t f1=fP[2], f2=f1 + x2r;
994 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
995 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
996 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
998 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
1001 &fC10=fC[1], &fC11=fC[2],
1002 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1003 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1004 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1006 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1007 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
1008 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
1011 double dy2dx = (f1+f2)/(r1+r2);
1014 if (TMath::Abs(x2r)<0.05) fP1 += dx*(r2 + f2*dy2dx)*fP3; // Many thanks to P.Hristov !
1016 // for small dx/R the linear apporximation of the arc by the segment is OK,
1017 // but at large dx/R the error is very large and leads to incorrect Z propagation
1018 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
1019 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
1020 // double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
1021 // double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
1022 // fP1 += rot/crv*fP3;
1024 fP1 += fP3/crv*TMath::ASin(r1*f2 - r2*f1); // more economic version from Yura.
1029 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
1030 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
1031 Double_t f12= dx*fP3*f1/(r1*r1*r1);
1032 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
1033 Double_t f13= dx/r1;
1034 Double_t f24= dx; f24*=cc;
1036 Double_t rinv = 1./r1;
1037 Double_t r3inv = rinv*rinv*rinv;
1038 Double_t f24= x2r/fP4;
1039 Double_t f02= dx*r3inv;
1040 Double_t f04=0.5*f24*f02;
1041 Double_t f12= f02*fP3*f1;
1042 Double_t f14=0.5*f24*f02*fP3*f1;
1043 Double_t f13= dx*rinv;
1046 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
1047 Double_t b02=f24*fC40;
1048 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
1049 Double_t b12=f24*fC41;
1050 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
1051 Double_t b22=f24*fC42;
1052 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
1053 Double_t b42=f24*fC44;
1054 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
1055 Double_t b32=f24*fC43;
1058 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
1059 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
1060 Double_t a22=f24*b42;
1062 //F*C*Ft = C + (b + bt + a)
1063 fC00 += b00 + b00 + a00;
1064 fC10 += b10 + b01 + a01;
1065 fC20 += b20 + b02 + a02;
1068 fC11 += b11 + b11 + a11;
1069 fC21 += b21 + b12 + a12;
1072 fC22 += b22 + b22 + a22;
1081 Bool_t AliExternalTrackParam::PropagateParamOnlyTo(Double_t xk, Double_t b) {
1082 //----------------------------------------------------------------
1083 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
1084 // Only parameters are propagated, not the matrix. To be used for small
1085 // distances only (<mm, i.e. misalignment)
1086 //----------------------------------------------------------------
1088 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
1090 Double_t crv=GetC(b);
1091 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1093 Double_t x2r = crv*dx;
1094 Double_t f1=fP[2], f2=f1 + x2r;
1095 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1096 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1097 if (TMath::Abs(fP[4])< kAlmost0) return kFALSE;
1099 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1100 if (TMath::Abs(r1)<kAlmost0) return kFALSE;
1101 if (TMath::Abs(r2)<kAlmost0) return kFALSE;
1104 double dy2dx = (f1+f2)/(r1+r2);
1106 fP[1] += dx*(r2 + f2*dy2dx)*fP[3]; // Many thanks to P.Hristov !
1113 AliExternalTrackParam::Propagate(Double_t alpha, Double_t x, Double_t b) {
1114 //------------------------------------------------------------------
1115 // Transform this track to the local coord. system rotated
1116 // by angle "alpha" (rad) with respect to the global coord. system,
1117 // and propagate this track to the plane X=xk (cm) in the field "b" (kG)
1118 //------------------------------------------------------------------
1120 //Save the parameters
1123 Double_t ps[5], cs[15];
1124 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
1125 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
1128 if (PropagateTo(x,b)) return kTRUE;
1130 //Restore the parameters, if the operation failed
1133 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
1134 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
1138 Bool_t AliExternalTrackParam::PropagateBxByBz
1139 (Double_t alpha, Double_t x, Double_t b[3]) {
1140 //------------------------------------------------------------------
1141 // Transform this track to the local coord. system rotated
1142 // by angle "alpha" (rad) with respect to the global coord. system,
1143 // and propagate this track to the plane X=xk (cm),
1144 // taking into account all three components of the B field, "b[3]" (kG)
1145 //------------------------------------------------------------------
1147 //Save the parameters
1150 Double_t ps[5], cs[15];
1151 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
1152 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
1155 if (PropagateToBxByBz(x,b)) return kTRUE;
1157 //Restore the parameters, if the operation failed
1160 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
1161 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
1166 void AliExternalTrackParam::Propagate(Double_t len, Double_t x[3],
1167 Double_t p[3], Double_t bz) const {
1168 //+++++++++++++++++++++++++++++++++++++++++
1169 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
1170 // Extrapolate track along simple helix in magnetic field
1171 // Arguments: len -distance alogn helix, [cm]
1172 // bz - mag field, [kGaus]
1173 // Returns: x and p contain extrapolated positon and momentum
1174 // The momentum returned for straight-line tracks is meaningless !
1175 //+++++++++++++++++++++++++++++++++++++++++
1178 if (OneOverPt() < kAlmost0 || TMath::Abs(bz) < kAlmost0Field || GetC(bz) < kAlmost0){ //straight-line tracks
1179 Double_t unit[3]; GetDirection(unit);
1184 p[0]=unit[0]/kAlmost0;
1185 p[1]=unit[1]/kAlmost0;
1186 p[2]=unit[2]/kAlmost0;
1190 Double_t a = -kB2C*bz*GetSign();
1191 Double_t rho = a/pp;
1192 x[0] += p[0]*TMath::Sin(rho*len)/a - p[1]*(1-TMath::Cos(rho*len))/a;
1193 x[1] += p[1]*TMath::Sin(rho*len)/a + p[0]*(1-TMath::Cos(rho*len))/a;
1194 x[2] += p[2]*len/pp;
1197 p[0] = p0 *TMath::Cos(rho*len) - p[1]*TMath::Sin(rho*len);
1198 p[1] = p[1]*TMath::Cos(rho*len) + p0 *TMath::Sin(rho*len);
1202 Bool_t AliExternalTrackParam::Intersect(Double_t pnt[3], Double_t norm[3],
1203 Double_t bz) const {
1204 //+++++++++++++++++++++++++++++++++++++++++
1205 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
1206 // Finds point of intersection (if exists) of the helix with the plane.
1207 // Stores result in fX and fP.
1208 // Arguments: planePoint,planeNorm - the plane defined by any plane's point
1209 // and vector, normal to the plane
1210 // Returns: kTrue if helix intersects the plane, kFALSE otherwise.
1211 //+++++++++++++++++++++++++++++++++++++++++
1212 Double_t x0[3]; GetXYZ(x0); //get track position in MARS
1214 //estimates initial helix length up to plane
1216 (pnt[0]-x0[0])*norm[0] + (pnt[1]-x0[1])*norm[1] + (pnt[2]-x0[2])*norm[2];
1217 Double_t dist=99999,distPrev=dist;
1219 while(TMath::Abs(dist)>0.00001){
1220 //calculates helix at the distance s from x0 ALONG the helix
1221 Propagate(s,x,p,bz);
1223 //distance between current helix position and plane
1224 dist=(x[0]-pnt[0])*norm[0]+(x[1]-pnt[1])*norm[1]+(x[2]-pnt[2])*norm[2];
1226 if(TMath::Abs(dist) >= TMath::Abs(distPrev)) {return kFALSE;}
1230 //on exit pnt is intersection point,norm is track vector at that point,
1232 for (Int_t i=0; i<3; i++) {pnt[i]=x[i]; norm[i]=p[i];}
1237 AliExternalTrackParam::GetPredictedChi2(const Double_t p[2],const Double_t cov[3]) const {
1238 //----------------------------------------------------------------
1239 // Estimate the chi2 of the space point "p" with the cov. matrix "cov"
1240 //----------------------------------------------------------------
1241 Double_t sdd = fC[0] + cov[0];
1242 Double_t sdz = fC[1] + cov[1];
1243 Double_t szz = fC[2] + cov[2];
1244 Double_t det = sdd*szz - sdz*sdz;
1246 if (TMath::Abs(det) < kAlmost0) return kVeryBig;
1248 Double_t d = fP[0] - p[0];
1249 Double_t z = fP[1] - p[1];
1251 return (d*szz*d - 2*d*sdz*z + z*sdd*z)/det;
1254 Double_t AliExternalTrackParam::
1255 GetPredictedChi2(const Double_t p[3],const Double_t covyz[3],const Double_t covxyz[3]) const {
1256 //----------------------------------------------------------------
1257 // Estimate the chi2 of the 3D space point "p" and
1258 // the full covariance matrix "covyz" and "covxyz"
1260 // Cov(x,x) ... : covxyz[0]
1261 // Cov(y,x) ... : covxyz[1] covyz[0]
1262 // Cov(z,x) ... : covxyz[2] covyz[1] covyz[2]
1263 //----------------------------------------------------------------
1271 Double_t f=GetSnp();
1272 if (TMath::Abs(f) >= kAlmost1) return kVeryBig;
1273 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1274 Double_t a=f/r, b=GetTgl()/r;
1276 Double_t s2=333.*333.; //something reasonably big (cm^2)
1279 v(0,0)= s2; v(0,1)= a*s2; v(0,2)= b*s2;;
1280 v(1,0)=a*s2; v(1,1)=a*a*s2 + GetSigmaY2(); v(1,2)=a*b*s2 + GetSigmaZY();
1281 v(2,0)=b*s2; v(2,1)=a*b*s2 + GetSigmaZY(); v(2,2)=b*b*s2 + GetSigmaZ2();
1283 v(0,0)+=covxyz[0]; v(0,1)+=covxyz[1]; v(0,2)+=covxyz[2];
1284 v(1,0)+=covxyz[1]; v(1,1)+=covyz[0]; v(1,2)+=covyz[1];
1285 v(2,0)+=covxyz[2]; v(2,1)+=covyz[1]; v(2,2)+=covyz[2];
1288 if (!v.IsValid()) return kVeryBig;
1291 for (Int_t i = 0; i < 3; i++)
1292 for (Int_t j = 0; j < 3; j++) chi2 += res[i]*res[j]*v(i,j);
1297 Double_t AliExternalTrackParam::
1298 GetPredictedChi2(const AliExternalTrackParam *t) const {
1299 //----------------------------------------------------------------
1300 // Estimate the chi2 (5 dof) of this track with respect to the track
1301 // given by the argument.
1302 // The two tracks must be in the same reference system
1303 // and estimated at the same reference plane.
1304 //----------------------------------------------------------------
1306 if (TMath::Abs(t->GetAlpha()-GetAlpha()) > FLT_EPSILON) {
1307 AliError("The reference systems of the tracks differ !");
1310 if (TMath::Abs(t->GetX()-GetX()) > FLT_EPSILON) {
1311 AliError("The reference of the tracks planes differ !");
1316 c(0,0)=GetSigmaY2();
1317 c(1,0)=GetSigmaZY(); c(1,1)=GetSigmaZ2();
1318 c(2,0)=GetSigmaSnpY(); c(2,1)=GetSigmaSnpZ(); c(2,2)=GetSigmaSnp2();
1319 c(3,0)=GetSigmaTglY(); c(3,1)=GetSigmaTglZ(); c(3,2)=GetSigmaTglSnp(); c(3,3)=GetSigmaTgl2();
1320 c(4,0)=GetSigma1PtY(); c(4,1)=GetSigma1PtZ(); c(4,2)=GetSigma1PtSnp(); c(4,3)=GetSigma1PtTgl(); c(4,4)=GetSigma1Pt2();
1322 c(0,0)+=t->GetSigmaY2();
1323 c(1,0)+=t->GetSigmaZY(); c(1,1)+=t->GetSigmaZ2();
1324 c(2,0)+=t->GetSigmaSnpY();c(2,1)+=t->GetSigmaSnpZ();c(2,2)+=t->GetSigmaSnp2();
1325 c(3,0)+=t->GetSigmaTglY();c(3,1)+=t->GetSigmaTglZ();c(3,2)+=t->GetSigmaTglSnp();c(3,3)+=t->GetSigmaTgl2();
1326 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();
1328 c(0,2)=c(2,0); c(1,2)=c(2,1);
1329 c(0,3)=c(3,0); c(1,3)=c(3,1); c(2,3)=c(3,2);
1330 c(0,4)=c(4,0); c(1,4)=c(4,1); c(2,4)=c(4,2); c(3,4)=c(4,3);
1333 if (!c.IsValid()) return kVeryBig;
1339 GetSnp() - t->GetSnp(),
1340 GetTgl() - t->GetTgl(),
1341 GetSigned1Pt() - t->GetSigned1Pt()
1345 for (Int_t i = 0; i < 5; i++)
1346 for (Int_t j = 0; j < 5; j++) chi2 += res[i]*res[j]*c(i,j);
1351 Bool_t AliExternalTrackParam::
1352 PropagateTo(Double_t p[3],Double_t covyz[3],Double_t covxyz[3],Double_t bz) {
1353 //----------------------------------------------------------------
1354 // Propagate this track to the plane
1355 // the 3D space point "p" (with the covariance matrix "covyz" and "covxyz")
1357 // The magnetic field is "bz" (kG)
1359 // The track curvature and the change of the covariance matrix
1360 // of the track parameters are negleted !
1361 // (So the "step" should be small compared with 1/curvature)
1362 //----------------------------------------------------------------
1364 Double_t f=GetSnp();
1365 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1366 Double_t r=TMath::Sqrt((1.-f)*(1.+f));
1367 Double_t a=f/r, b=GetTgl()/r;
1369 Double_t s2=333.*333.; //something reasonably big (cm^2)
1372 tV(0,0)= s2; tV(0,1)= a*s2; tV(0,2)= b*s2;
1373 tV(1,0)=a*s2; tV(1,1)=a*a*s2; tV(1,2)=a*b*s2;
1374 tV(2,0)=b*s2; tV(2,1)=a*b*s2; tV(2,2)=b*b*s2;
1377 pV(0,0)=covxyz[0]; pV(0,1)=covxyz[1]; pV(0,2)=covxyz[2];
1378 pV(1,0)=covxyz[1]; pV(1,1)=covyz[0]; pV(1,2)=covyz[1];
1379 pV(2,0)=covxyz[2]; pV(2,1)=covyz[1]; pV(2,2)=covyz[2];
1381 TMatrixDSym tpV(tV);
1384 if (!tpV.IsValid()) return kFALSE;
1386 TMatrixDSym pW(3),tW(3);
1387 for (Int_t i=0; i<3; i++)
1388 for (Int_t j=0; j<3; j++) {
1390 for (Int_t k=0; k<3; k++) {
1391 pW(i,j) += tV(i,k)*tpV(k,j);
1392 tW(i,j) += pV(i,k)*tpV(k,j);
1396 Double_t t[3] = {GetX(), GetY(), GetZ()};
1399 for (Int_t i=0; i<3; i++) x += (tW(0,i)*t[i] + pW(0,i)*p[i]);
1400 Double_t crv=GetC(bz);
1401 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1403 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
1407 for (Int_t i=0; i<3; i++) fP[0] += (tW(1,i)*t[i] + pW(1,i)*p[i]);
1409 for (Int_t i=0; i<3; i++) fP[1] += (tW(2,i)*t[i] + pW(2,i)*p[i]);
1414 Double_t *AliExternalTrackParam::GetResiduals(
1415 Double_t *p,Double_t *cov,Bool_t updated) const {
1416 //------------------------------------------------------------------
1417 // Returns the track residuals with the space point "p" having
1418 // the covariance matrix "cov".
1419 // If "updated" is kTRUE, the track parameters expected to be updated,
1420 // otherwise they must be predicted.
1421 //------------------------------------------------------------------
1422 static Double_t res[2];
1424 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1426 r00-=fC[0]; r01-=fC[1]; r11-=fC[2];
1428 r00+=fC[0]; r01+=fC[1]; r11+=fC[2];
1430 Double_t det=r00*r11 - r01*r01;
1432 if (TMath::Abs(det) < kAlmost0) return 0;
1434 Double_t tmp=r00; r00=r11/det; r11=tmp/det;
1436 if (r00 < 0.) return 0;
1437 if (r11 < 0.) return 0;
1439 Double_t dy = fP[0] - p[0];
1440 Double_t dz = fP[1] - p[1];
1442 res[0]=dy*TMath::Sqrt(r00);
1443 res[1]=dz*TMath::Sqrt(r11);
1448 Bool_t AliExternalTrackParam::Update(const Double_t p[2], const Double_t cov[3]) {
1449 //------------------------------------------------------------------
1450 // Update the track parameters with the space point "p" having
1451 // the covariance matrix "cov"
1452 //------------------------------------------------------------------
1453 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
1456 &fC10=fC[1], &fC11=fC[2],
1457 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1458 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1459 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1461 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
1462 r00+=fC00; r01+=fC10; r11+=fC11;
1463 Double_t det=r00*r11 - r01*r01;
1465 if (TMath::Abs(det) < kAlmost0) return kFALSE;
1468 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
1470 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
1471 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
1472 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
1473 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
1474 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
1476 Double_t dy=p[0] - fP0, dz=p[1] - fP1;
1477 Double_t sf=fP2 + k20*dy + k21*dz;
1478 if (TMath::Abs(sf) > kAlmost1) return kFALSE;
1480 fP0 += k00*dy + k01*dz;
1481 fP1 += k10*dy + k11*dz;
1483 fP3 += k30*dy + k31*dz;
1484 fP4 += k40*dy + k41*dz;
1486 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
1487 Double_t c12=fC21, c13=fC31, c14=fC41;
1489 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
1490 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
1491 fC40-=k00*c04+k01*c14;
1493 fC11-=k10*c01+k11*fC11;
1494 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
1495 fC41-=k10*c04+k11*c14;
1497 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
1498 fC42-=k20*c04+k21*c14;
1500 fC33-=k30*c03+k31*c13;
1501 fC43-=k30*c04+k31*c14;
1503 fC44-=k40*c04+k41*c14;
1511 AliExternalTrackParam::GetHelixParameters(Double_t hlx[6], Double_t b) const {
1512 //--------------------------------------------------------------------
1513 // External track parameters -> helix parameters
1514 // "b" - magnetic field (kG)
1515 //--------------------------------------------------------------------
1516 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1518 hlx[0]=fP[0]; hlx[1]=fP[1]; hlx[2]=fP[2]; hlx[3]=fP[3];
1520 hlx[5]=fX*cs - hlx[0]*sn; // x0
1521 hlx[0]=fX*sn + hlx[0]*cs; // y0
1523 hlx[2]=TMath::ASin(hlx[2]) + fAlpha; // phi0
1525 hlx[4]=GetC(b); // C
1529 static void Evaluate(const Double_t *h, Double_t t,
1530 Double_t r[3], //radius vector
1531 Double_t g[3], //first defivatives
1532 Double_t gg[3]) //second derivatives
1534 //--------------------------------------------------------------------
1535 // Calculate position of a point on a track and some derivatives
1536 //--------------------------------------------------------------------
1537 Double_t phase=h[4]*t+h[2];
1538 Double_t sn=TMath::Sin(phase), cs=TMath::Cos(phase);
1542 if (TMath::Abs(h[4])>kAlmost0) {
1543 r[0] += (sn - h[6])/h[4];
1544 r[1] -= (cs - h[7])/h[4];
1546 r[2] = h[1] + h[3]*t;
1548 g[0] = cs; g[1]=sn; g[2]=h[3];
1550 gg[0]=-h[4]*sn; gg[1]=h[4]*cs; gg[2]=0.;
1553 Double_t AliExternalTrackParam::GetDCA(const AliExternalTrackParam *p,
1554 Double_t b, Double_t &xthis, Double_t &xp) const {
1555 //------------------------------------------------------------
1556 // Returns the (weighed !) distance of closest approach between
1557 // this track and the track "p".
1558 // Other returned values:
1559 // xthis, xt - coordinates of tracks' reference planes at the DCA
1560 //-----------------------------------------------------------
1561 Double_t dy2=GetSigmaY2() + p->GetSigmaY2();
1562 Double_t dz2=GetSigmaZ2() + p->GetSigmaZ2();
1565 Double_t p1[8]; GetHelixParameters(p1,b);
1566 p1[6]=TMath::Sin(p1[2]); p1[7]=TMath::Cos(p1[2]);
1567 Double_t p2[8]; p->GetHelixParameters(p2,b);
1568 p2[6]=TMath::Sin(p2[2]); p2[7]=TMath::Cos(p2[2]);
1571 Double_t r1[3],g1[3],gg1[3]; Double_t t1=0.;
1572 Evaluate(p1,t1,r1,g1,gg1);
1573 Double_t r2[3],g2[3],gg2[3]; Double_t t2=0.;
1574 Evaluate(p2,t2,r2,g2,gg2);
1576 Double_t dx=r2[0]-r1[0], dy=r2[1]-r1[1], dz=r2[2]-r1[2];
1577 Double_t dm=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1581 Double_t gt1=-(dx*g1[0]/dx2 + dy*g1[1]/dy2 + dz*g1[2]/dz2);
1582 Double_t gt2=+(dx*g2[0]/dx2 + dy*g2[1]/dy2 + dz*g2[2]/dz2);
1583 Double_t h11=(g1[0]*g1[0] - dx*gg1[0])/dx2 +
1584 (g1[1]*g1[1] - dy*gg1[1])/dy2 +
1585 (g1[2]*g1[2] - dz*gg1[2])/dz2;
1586 Double_t h22=(g2[0]*g2[0] + dx*gg2[0])/dx2 +
1587 (g2[1]*g2[1] + dy*gg2[1])/dy2 +
1588 (g2[2]*g2[2] + dz*gg2[2])/dz2;
1589 Double_t h12=-(g1[0]*g2[0]/dx2 + g1[1]*g2[1]/dy2 + g1[2]*g2[2]/dz2);
1591 Double_t det=h11*h22-h12*h12;
1594 if (TMath::Abs(det)<1.e-33) {
1595 //(quasi)singular Hessian
1598 dt1=-(gt1*h22 - gt2*h12)/det;
1599 dt2=-(h11*gt2 - h12*gt1)/det;
1602 if ((dt1*gt1+dt2*gt2)>0) {dt1=-dt1; dt2=-dt2;}
1604 //check delta(phase1) ?
1605 //check delta(phase2) ?
1607 if (TMath::Abs(dt1)/(TMath::Abs(t1)+1.e-3) < 1.e-4)
1608 if (TMath::Abs(dt2)/(TMath::Abs(t2)+1.e-3) < 1.e-4) {
1609 if ((gt1*gt1+gt2*gt2) > 1.e-4/dy2/dy2)
1610 AliDebug(1," stopped at not a stationary point !");
1611 Double_t lmb=h11+h22; lmb=lmb-TMath::Sqrt(lmb*lmb-4*det);
1613 AliDebug(1," stopped at not a minimum !");
1618 for (Int_t div=1 ; ; div*=2) {
1619 Evaluate(p1,t1+dt1,r1,g1,gg1);
1620 Evaluate(p2,t2+dt2,r2,g2,gg2);
1621 dx=r2[0]-r1[0]; dy=r2[1]-r1[1]; dz=r2[2]-r1[2];
1622 dd=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1626 AliDebug(1," overshoot !"); break;
1636 if (max<=0) AliDebug(1," too many iterations !");
1638 Double_t cs=TMath::Cos(GetAlpha());
1639 Double_t sn=TMath::Sin(GetAlpha());
1640 xthis=r1[0]*cs + r1[1]*sn;
1642 cs=TMath::Cos(p->GetAlpha());
1643 sn=TMath::Sin(p->GetAlpha());
1644 xp=r2[0]*cs + r2[1]*sn;
1646 return TMath::Sqrt(dm*TMath::Sqrt(dy2*dz2));
1649 Double_t AliExternalTrackParam::
1650 PropagateToDCA(AliExternalTrackParam *p, Double_t b) {
1651 //--------------------------------------------------------------
1652 // Propagates this track and the argument track to the position of the
1653 // distance of closest approach.
1654 // Returns the (weighed !) distance of closest approach.
1655 //--------------------------------------------------------------
1657 Double_t dca=GetDCA(p,b,xthis,xp);
1659 if (!PropagateTo(xthis,b)) {
1660 //AliWarning(" propagation failed !");
1664 if (!p->PropagateTo(xp,b)) {
1665 //AliWarning(" propagation failed !";
1673 Bool_t AliExternalTrackParam::PropagateToDCA(const AliVVertex *vtx,
1674 Double_t b, Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1676 // Propagate this track to the DCA to vertex "vtx",
1677 // if the (rough) transverse impact parameter is not bigger then "maxd".
1678 // Magnetic field is "b" (kG).
1680 // a) The track gets extapolated to the DCA to the vertex.
1681 // b) The impact parameters and their covariance matrix are calculated.
1683 // In the case of success, the returned value is kTRUE
1684 // (otherwise, it's kFALSE)
1686 Double_t alpha=GetAlpha();
1687 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1688 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1689 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1690 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1693 //Estimate the impact parameter neglecting the track curvature
1694 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1695 if (d > maxd) return kFALSE;
1697 //Propagate to the DCA
1698 Double_t crv=GetC(b);
1699 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1701 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1702 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1703 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1707 yv=-xv*sn + yv*cs; xv=x;
1709 if (!Propagate(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1711 if (dz==0) return kTRUE;
1712 dz[0] = GetParameter()[0] - yv;
1713 dz[1] = GetParameter()[1] - zv;
1715 if (covar==0) return kTRUE;
1716 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1718 //***** Improvements by A.Dainese
1719 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1720 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1721 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1722 covar[1] = GetCovariance()[1]; // between (x,y) and z
1723 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1729 Bool_t AliExternalTrackParam::PropagateToDCABxByBz(const AliVVertex *vtx,
1730 Double_t b[3], Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1732 // Propagate this track to the DCA to vertex "vtx",
1733 // if the (rough) transverse impact parameter is not bigger then "maxd".
1735 // This function takes into account all three components of the magnetic
1736 // field given by the b[3] arument (kG)
1738 // a) The track gets extapolated to the DCA to the vertex.
1739 // b) The impact parameters and their covariance matrix are calculated.
1741 // In the case of success, the returned value is kTRUE
1742 // (otherwise, it's kFALSE)
1744 Double_t alpha=GetAlpha();
1745 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1746 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1747 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1748 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1751 //Estimate the impact parameter neglecting the track curvature
1752 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt((1.-snp)*(1.+snp)));
1753 if (d > maxd) return kFALSE;
1755 //Propagate to the DCA
1756 Double_t crv=GetC(b[2]);
1757 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1759 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt((1.-snp)*(1.+snp)));
1760 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt((1.-sn)*(1.+sn));
1761 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1765 yv=-xv*sn + yv*cs; xv=x;
1767 if (!PropagateBxByBz(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1769 if (dz==0) return kTRUE;
1770 dz[0] = GetParameter()[0] - yv;
1771 dz[1] = GetParameter()[1] - zv;
1773 if (covar==0) return kTRUE;
1774 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1776 //***** Improvements by A.Dainese
1777 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1778 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1779 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1780 covar[1] = GetCovariance()[1]; // between (x,y) and z
1781 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1787 void AliExternalTrackParam::GetDirection(Double_t d[3]) const {
1788 //----------------------------------------------------------------
1789 // This function returns a unit vector along the track direction
1790 // in the global coordinate system.
1791 //----------------------------------------------------------------
1792 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1794 Double_t csp =TMath::Sqrt((1.-snp)*(1.+snp));
1795 Double_t norm=TMath::Sqrt(1.+ fP[3]*fP[3]);
1796 d[0]=(csp*cs - snp*sn)/norm;
1797 d[1]=(snp*cs + csp*sn)/norm;
1801 Bool_t AliExternalTrackParam::GetPxPyPz(Double_t p[3]) const {
1802 //---------------------------------------------------------------------
1803 // This function returns the global track momentum components
1804 // Results for (nearly) straight tracks are meaningless !
1805 //---------------------------------------------------------------------
1806 p[0]=fP[4]; p[1]=fP[2]; p[2]=fP[3];
1807 return Local2GlobalMomentum(p,fAlpha);
1810 Double_t AliExternalTrackParam::Px() const {
1811 //---------------------------------------------------------------------
1812 // Returns x-component of momentum
1813 // Result for (nearly) straight tracks is meaningless !
1814 //---------------------------------------------------------------------
1816 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1822 Double_t AliExternalTrackParam::Py() const {
1823 //---------------------------------------------------------------------
1824 // Returns y-component of momentum
1825 // Result for (nearly) straight tracks is meaningless !
1826 //---------------------------------------------------------------------
1828 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1834 Double_t AliExternalTrackParam::Xv() const {
1835 //---------------------------------------------------------------------
1836 // Returns x-component of first track point
1837 //---------------------------------------------------------------------
1839 Double_t r[3]={0.,0.,0.};
1845 Double_t AliExternalTrackParam::Yv() const {
1846 //---------------------------------------------------------------------
1847 // Returns y-component of first track point
1848 //---------------------------------------------------------------------
1850 Double_t r[3]={0.,0.,0.};
1856 Double_t AliExternalTrackParam::Theta() const {
1857 // return theta angle of momentum
1859 return 0.5*TMath::Pi() - TMath::ATan(fP[3]);
1862 Double_t AliExternalTrackParam::Phi() const {
1863 //---------------------------------------------------------------------
1864 // Returns the azimuthal angle of momentum
1866 //---------------------------------------------------------------------
1868 Double_t phi=TMath::ASin(fP[2]) + fAlpha;
1869 if (phi<0.) phi+=2.*TMath::Pi();
1870 else if (phi>=2.*TMath::Pi()) phi-=2.*TMath::Pi();
1875 Double_t AliExternalTrackParam::M() const {
1876 // return particle mass
1878 // No mass information available so far.
1879 // Redifine in derived class!
1884 Double_t AliExternalTrackParam::E() const {
1885 // return particle energy
1887 // No PID information available so far.
1888 // Redifine in derived class!
1893 Double_t AliExternalTrackParam::Eta() const {
1894 // return pseudorapidity
1896 return -TMath::Log(TMath::Tan(0.5 * Theta()));
1899 Double_t AliExternalTrackParam::Y() const {
1902 // No PID information available so far.
1903 // Redifine in derived class!
1908 Bool_t AliExternalTrackParam::GetXYZ(Double_t *r) const {
1909 //---------------------------------------------------------------------
1910 // This function returns the global track position
1911 //---------------------------------------------------------------------
1912 r[0]=fX; r[1]=fP[0]; r[2]=fP[1];
1913 return Local2GlobalPosition(r,fAlpha);
1916 Bool_t AliExternalTrackParam::GetCovarianceXYZPxPyPz(Double_t cv[21]) const {
1917 //---------------------------------------------------------------------
1918 // This function returns the global covariance matrix of the track params
1920 // Cov(x,x) ... : cv[0]
1921 // Cov(y,x) ... : cv[1] cv[2]
1922 // Cov(z,x) ... : cv[3] cv[4] cv[5]
1923 // Cov(px,x)... : cv[6] cv[7] cv[8] cv[9]
1924 // Cov(py,x)... : cv[10] cv[11] cv[12] cv[13] cv[14]
1925 // Cov(pz,x)... : cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
1927 // Results for (nearly) straight tracks are meaningless !
1928 //---------------------------------------------------------------------
1929 if (TMath::Abs(fP[4])<=kAlmost0) {
1930 for (Int_t i=0; i<21; i++) cv[i]=0.;
1933 if (TMath::Abs(fP[2]) > kAlmost1) {
1934 for (Int_t i=0; i<21; i++) cv[i]=0.;
1937 Double_t pt=1./TMath::Abs(fP[4]);
1938 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1939 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
1941 Double_t m00=-sn, m10=cs;
1942 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
1943 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
1944 Double_t m35=pt, m45=-pt*pt*fP[3];
1950 cv[0 ] = fC[0]*m00*m00;
1951 cv[1 ] = fC[0]*m00*m10;
1952 cv[2 ] = fC[0]*m10*m10;
1956 cv[6 ] = m00*(fC[3]*m23 + fC[10]*m43);
1957 cv[7 ] = m10*(fC[3]*m23 + fC[10]*m43);
1958 cv[8 ] = fC[4]*m23 + fC[11]*m43;
1959 cv[9 ] = m23*(fC[5]*m23 + fC[12]*m43) + m43*(fC[12]*m23 + fC[14]*m43);
1960 cv[10] = m00*(fC[3]*m24 + fC[10]*m44);
1961 cv[11] = m10*(fC[3]*m24 + fC[10]*m44);
1962 cv[12] = fC[4]*m24 + fC[11]*m44;
1963 cv[13] = m23*(fC[5]*m24 + fC[12]*m44) + m43*(fC[12]*m24 + fC[14]*m44);
1964 cv[14] = m24*(fC[5]*m24 + fC[12]*m44) + m44*(fC[12]*m24 + fC[14]*m44);
1965 cv[15] = m00*(fC[6]*m35 + fC[10]*m45);
1966 cv[16] = m10*(fC[6]*m35 + fC[10]*m45);
1967 cv[17] = fC[7]*m35 + fC[11]*m45;
1968 cv[18] = m23*(fC[8]*m35 + fC[12]*m45) + m43*(fC[13]*m35 + fC[14]*m45);
1969 cv[19] = m24*(fC[8]*m35 + fC[12]*m45) + m44*(fC[13]*m35 + fC[14]*m45);
1970 cv[20] = m35*(fC[9]*m35 + fC[13]*m45) + m45*(fC[13]*m35 + fC[14]*m45);
1977 AliExternalTrackParam::GetPxPyPzAt(Double_t x, Double_t b, Double_t *p) const {
1978 //---------------------------------------------------------------------
1979 // This function returns the global track momentum extrapolated to
1980 // the radial position "x" (cm) in the magnetic field "b" (kG)
1981 //---------------------------------------------------------------------
1983 p[1]=fP[2]+(x-fX)*GetC(b);
1985 return Local2GlobalMomentum(p,fAlpha);
1989 AliExternalTrackParam::GetYAt(Double_t x, Double_t b, Double_t &y) const {
1990 //---------------------------------------------------------------------
1991 // This function returns the local Y-coordinate of the intersection
1992 // point between this track and the reference plane "x" (cm).
1993 // Magnetic field "b" (kG)
1994 //---------------------------------------------------------------------
1996 if(TMath::Abs(dx)<=kAlmost0) {y=fP[0]; return kTRUE;}
1998 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
2000 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2001 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2003 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
2004 y = fP[0] + dx*(f1+f2)/(r1+r2);
2009 AliExternalTrackParam::GetZAt(Double_t x, Double_t b, Double_t &z) const {
2010 //---------------------------------------------------------------------
2011 // This function returns the local Z-coordinate of the intersection
2012 // point between this track and the reference plane "x" (cm).
2013 // Magnetic field "b" (kG)
2014 //---------------------------------------------------------------------
2016 if(TMath::Abs(dx)<=kAlmost0) {z=fP[1]; return kTRUE;}
2018 Double_t crv=GetC(b);
2019 Double_t x2r = crv*dx;
2020 Double_t f1=fP[2], f2=f1 + x2r;
2022 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2023 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2025 Double_t r1=sqrt((1.-f1)*(1.+f1)), r2=sqrt((1.-f2)*(1.+f2));
2026 double dy2dx = (f1+f2)/(r1+r2);
2027 if (TMath::Abs(x2r)<0.05) {
2028 z = fP[1] + dx*(r2 + f2*dy2dx)*fP[3]; // Many thanks to P.Hristov !
2031 // for small dx/R the linear apporximation of the arc by the segment is OK,
2032 // but at large dx/R the error is very large and leads to incorrect Z propagation
2033 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
2034 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
2035 // Similarly, the rotation angle in linear in dx only for dx<<R
2036 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
2037 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
2038 z = fP[1] + rot/crv*fP[3];
2044 AliExternalTrackParam::GetXYZAt(Double_t x, Double_t b, Double_t *r) const {
2045 //---------------------------------------------------------------------
2046 // This function returns the global track position extrapolated to
2047 // the radial position "x" (cm) in the magnetic field "b" (kG)
2048 //---------------------------------------------------------------------
2050 if(TMath::Abs(dx)<=kAlmost0) return GetXYZ(r);
2052 Double_t crv=GetC(b);
2053 Double_t x2r = crv*dx;
2054 Double_t f1=fP[2], f2=f1 + dx*crv;
2056 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2057 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2059 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
2060 double dy2dx = (f1+f2)/(r1+r2);
2062 r[1] = fP[0] + dx*dy2dx;
2063 if (TMath::Abs(x2r)<0.05) {
2064 r[2] = fP[1] + dx*(r2 + f2*dy2dx)*fP[3];//Thanks to Andrea & Peter
2067 // for small dx/R the linear apporximation of the arc by the segment is OK,
2068 // but at large dx/R the error is very large and leads to incorrect Z propagation
2069 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
2070 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
2071 // Similarly, the rotation angle in linear in dx only for dx<<R
2072 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
2073 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
2074 r[2] = fP[1] + rot/crv*fP[3];
2077 return Local2GlobalPosition(r,fAlpha);
2080 //_____________________________________________________________________________
2081 void AliExternalTrackParam::Print(Option_t* /*option*/) const
2083 // print the parameters and the covariance matrix
2085 printf("AliExternalTrackParam: x = %-12g alpha = %-12g\n", fX, fAlpha);
2086 printf(" parameters: %12g %12g %12g %12g %12g\n",
2087 fP[0], fP[1], fP[2], fP[3], fP[4]);
2088 printf(" covariance: %12g\n", fC[0]);
2089 printf(" %12g %12g\n", fC[1], fC[2]);
2090 printf(" %12g %12g %12g\n", fC[3], fC[4], fC[5]);
2091 printf(" %12g %12g %12g %12g\n",
2092 fC[6], fC[7], fC[8], fC[9]);
2093 printf(" %12g %12g %12g %12g %12g\n",
2094 fC[10], fC[11], fC[12], fC[13], fC[14]);
2097 Double_t AliExternalTrackParam::GetSnpAt(Double_t x,Double_t b) const {
2099 // Get sinus at given x
2101 Double_t crv=GetC(b);
2102 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
2104 Double_t res = fP[2]+dx*crv;
2108 Bool_t AliExternalTrackParam::GetDistance(AliExternalTrackParam *param2, Double_t x, Double_t dist[3], Double_t bz){
2109 //------------------------------------------------------------------------
2110 // Get the distance between two tracks at the local position x
2111 // working in the local frame of this track.
2112 // Origin : Marian.Ivanov@cern.ch
2113 //-----------------------------------------------------------------------
2117 if (!GetYAt(x,bz,xyz[1])) return kFALSE;
2118 if (!GetZAt(x,bz,xyz[2])) return kFALSE;
2121 if (TMath::Abs(GetAlpha()-param2->GetAlpha())<kAlmost0){
2123 if (!param2->GetYAt(x,bz,xyz2[1])) return kFALSE;
2124 if (!param2->GetZAt(x,bz,xyz2[2])) return kFALSE;
2128 Double_t dfi = param2->GetAlpha()-GetAlpha();
2129 Double_t ca = TMath::Cos(dfi), sa = TMath::Sin(dfi);
2130 xyz2[0] = xyz[0]*ca+xyz[1]*sa;
2131 xyz2[1] = -xyz[0]*sa+xyz[1]*ca;
2134 if (!param2->GetYAt(xyz2[0],bz,xyz1[1])) return kFALSE;
2135 if (!param2->GetZAt(xyz2[0],bz,xyz1[2])) return kFALSE;
2137 xyz2[0] = xyz1[0]*ca-xyz1[1]*sa;
2138 xyz2[1] = +xyz1[0]*sa+xyz1[1]*ca;
2141 dist[0] = xyz[0]-xyz2[0];
2142 dist[1] = xyz[1]-xyz2[1];
2143 dist[2] = xyz[2]-xyz2[2];
2150 // Draw functionality.
2151 // Origin: Marian Ivanov, Marian.Ivanov@cern.ch
2154 void AliExternalTrackParam::DrawTrack(Float_t magf, Float_t minR, Float_t maxR, Float_t stepR){
2158 if (minR>maxR) return ;
2159 if (stepR<=0) return ;
2160 Int_t npoints = TMath::Nint((maxR-minR)/stepR)+1;
2161 if (npoints<1) return;
2162 TPolyMarker3D *polymarker = new TPolyMarker3D(npoints);
2163 FillPolymarker(polymarker, magf,minR,maxR,stepR);
2168 void AliExternalTrackParam::FillPolymarker(TPolyMarker3D *pol, Float_t magF, Float_t minR, Float_t maxR, Float_t stepR){
2170 // Fill points in the polymarker
2173 for (Double_t r=minR; r<maxR; r+=stepR){
2175 GetXYZAt(r,magF,point);
2176 pol->SetPoint(counter,point[0],point[1], point[2]);
2177 // printf("xyz\t%f\t%f\t%f\n",point[0], point[1],point[2]);
2182 Int_t AliExternalTrackParam::GetIndex(Int_t i, Int_t j) const {
2184 Int_t min = TMath::Min(i,j);
2185 Int_t max = TMath::Max(i,j);
2187 return min+(max+1)*max/2;
2191 void AliExternalTrackParam::g3helx3(Double_t qfield,
2194 /******************************************************************
2196 * GEANT3 tracking routine in a constant field oriented *
2198 * Tracking is performed with a conventional *
2199 * helix step method *
2201 * Authors R.Brun, M.Hansroul ********* *
2202 * Rewritten V.Perevoztchikov *
2204 * Rewritten in C++ by I.Belikov *
2206 * qfield (kG) - particle charge times magnetic field *
2207 * step (cm) - step length along the helix *
2208 * vect[7](cm,GeV/c) - input/output x, y, z, px/p, py/p ,pz/p, p *
2210 ******************************************************************/
2211 const Int_t ix=0, iy=1, iz=2, ipx=3, ipy=4, ipz=5, ipp=6;
2212 const Double_t kOvSqSix=TMath::Sqrt(1./6.);
2214 Double_t cosx=vect[ipx], cosy=vect[ipy], cosz=vect[ipz];
2216 Double_t rho = qfield*kB2C/vect[ipp];
2217 Double_t tet = rho*step;
2219 Double_t tsint, sintt, sint, cos1t;
2220 if (TMath::Abs(tet) > 0.03) {
2221 sint = TMath::Sin(tet);
2223 tsint = (tet - sint)/tet;
2224 Double_t t=TMath::Sin(0.5*tet);
2228 sintt = (1.-tet*kOvSqSix)*(1.+tet*kOvSqSix); // 1.- tsint;
2233 Double_t f1 = step*sintt;
2234 Double_t f2 = step*cos1t;
2235 Double_t f3 = step*tsint*cosz;
2236 Double_t f4 = -tet*cos1t;
2239 vect[ix] += f1*cosx - f2*cosy;
2240 vect[iy] += f1*cosy + f2*cosx;
2241 vect[iz] += f1*cosz + f3;
2243 vect[ipx] += f4*cosx - f5*cosy;
2244 vect[ipy] += f4*cosy + f5*cosx;
2248 Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]) {
2249 //----------------------------------------------------------------
2250 // Extrapolate this track to the plane X=xk in the field b[].
2252 // X [cm] is in the "tracking coordinate system" of this track.
2253 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
2254 //----------------------------------------------------------------
2257 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
2258 if (TMath::Abs(fP[4])<=kAlmost0) return kFALSE;
2259 // Do not propagate tracks outside the ALICE detector
2260 if (TMath::Abs(dx)>1e5 ||
2261 TMath::Abs(GetY())>1e5 ||
2262 TMath::Abs(GetZ())>1e5) {
2263 AliWarning(Form("Anomalous track, target X:%f",xk));
2268 Double_t crv=GetC(b[2]);
2269 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
2271 Double_t x2r = crv*dx;
2272 Double_t f1=fP[2], f2=f1 + x2r;
2273 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2274 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2277 // Estimate the covariance matrix
2278 Double_t &fP3=fP[3], &fP4=fP[4];
2281 &fC10=fC[1], &fC11=fC[2],
2282 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
2283 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
2284 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
2286 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
2290 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
2291 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
2292 Double_t f12= dx*fP3*f1/(r1*r1*r1);
2293 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
2294 Double_t f13= dx/r1;
2295 Double_t f24= dx; f24*=cc;
2297 Double_t rinv = 1./r1;
2298 Double_t r3inv = rinv*rinv*rinv;
2299 Double_t f24= x2r/fP4;
2300 Double_t f02= dx*r3inv;
2301 Double_t f04=0.5*f24*f02;
2302 Double_t f12= f02*fP3*f1;
2303 Double_t f14=0.5*f24*f02*fP3*f1;
2304 Double_t f13= dx*rinv;
2307 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
2308 Double_t b02=f24*fC40;
2309 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
2310 Double_t b12=f24*fC41;
2311 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
2312 Double_t b22=f24*fC42;
2313 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
2314 Double_t b42=f24*fC44;
2315 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
2316 Double_t b32=f24*fC43;
2319 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
2320 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
2321 Double_t a22=f24*b42;
2323 //F*C*Ft = C + (b + bt + a)
2324 fC00 += b00 + b00 + a00;
2325 fC10 += b10 + b01 + a01;
2326 fC20 += b20 + b02 + a02;
2329 fC11 += b11 + b11 + a11;
2330 fC21 += b21 + b12 + a12;
2333 fC22 += b22 + b22 + a22;
2339 // Appoximate step length
2340 double dy2dx = (f1+f2)/(r1+r2);
2341 Double_t step = (TMath::Abs(x2r)<0.05) ? dx*TMath::Abs(r2 + f2*dy2dx) // chord
2342 : 2.*TMath::ASin(0.5*dx*TMath::Sqrt(1.+dy2dx*dy2dx)*crv)/crv; // arc
2343 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
2345 // Get the track's (x,y,z) and (px,py,pz) in the Global System
2346 Double_t r[3]; GetXYZ(r);
2347 Double_t p[3]; GetPxPyPz(p);
2354 // Rotate to the system where Bx=By=0.
2355 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
2356 Double_t cosphi=1., sinphi=0.;
2357 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
2358 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
2359 Double_t costet=1., sintet=0.;
2360 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
2363 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
2364 vect[1] = -sinphi*r[0] + cosphi*r[1];
2365 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
2367 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
2368 vect[4] = -sinphi*p[0] + cosphi*p[1];
2369 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
2374 // Do the helix step
2375 g3helx3(GetSign()*bb,step,vect);
2378 // Rotate back to the Global System
2379 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
2380 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
2381 r[2] = -sintet*vect[0] + costet*vect[2];
2383 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
2384 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
2385 p[2] = -sintet*vect[3] + costet*vect[5];
2388 // Rotate back to the Tracking System
2389 Double_t cosalp = TMath::Cos(fAlpha);
2390 Double_t sinalp =-TMath::Sin(fAlpha);
2393 t = cosalp*r[0] - sinalp*r[1];
2394 r[1] = sinalp*r[0] + cosalp*r[1];
2397 t = cosalp*p[0] - sinalp*p[1];
2398 p[1] = sinalp*p[0] + cosalp*p[1];
2402 // Do the final correcting step to the target plane (linear approximation)
2403 Double_t x=r[0], y=r[1], z=r[2];
2404 if (TMath::Abs(dx) > kAlmost0) {
2405 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
2413 // Calculate the track parameters
2414 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
2420 fP[4] = GetSign()/(t*pp);
2425 Bool_t AliExternalTrackParam::PropagateParamOnlyBxByBzTo(Double_t xk, const Double_t b[3]) {
2426 //----------------------------------------------------------------
2427 // Extrapolate this track params (w/o cov matrix) to the plane X=xk in the field b[].
2429 // X [cm] is in the "tracking coordinate system" of this track.
2430 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
2431 //----------------------------------------------------------------
2434 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
2435 if (TMath::Abs(fP[4])<=kAlmost0) return kFALSE;
2436 // Do not propagate tracks outside the ALICE detector
2437 if (TMath::Abs(dx)>1e5 ||
2438 TMath::Abs(GetY())>1e5 ||
2439 TMath::Abs(GetZ())>1e5) {
2440 AliWarning(Form("Anomalous track, target X:%f",xk));
2445 Double_t crv=GetC(b[2]);
2446 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
2448 Double_t x2r = crv*dx;
2449 Double_t f1=fP[2], f2=f1 + x2r;
2450 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
2451 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
2453 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
2455 // Appoximate step length
2456 double dy2dx = (f1+f2)/(r1+r2);
2457 Double_t step = (TMath::Abs(x2r)<0.05) ? dx*TMath::Abs(r2 + f2*dy2dx) // chord
2458 : 2.*TMath::ASin(0.5*dx*TMath::Sqrt(1.+dy2dx*dy2dx)*crv)/crv; // arc
2459 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
2461 // Get the track's (x,y,z) and (px,py,pz) in the Global System
2462 Double_t r[3]; GetXYZ(r);
2463 Double_t p[3]; GetPxPyPz(p);
2469 // Rotate to the system where Bx=By=0.
2470 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
2471 Double_t cosphi=1., sinphi=0.;
2472 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
2473 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
2474 Double_t costet=1., sintet=0.;
2475 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
2478 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
2479 vect[1] = -sinphi*r[0] + cosphi*r[1];
2480 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
2482 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
2483 vect[4] = -sinphi*p[0] + cosphi*p[1];
2484 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
2488 // Do the helix step
2489 g3helx3(GetSign()*bb,step,vect);
2491 // Rotate back to the Global System
2492 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
2493 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
2494 r[2] = -sintet*vect[0] + costet*vect[2];
2496 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
2497 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
2498 p[2] = -sintet*vect[3] + costet*vect[5];
2500 // Rotate back to the Tracking System
2501 Double_t cosalp = TMath::Cos(fAlpha);
2502 Double_t sinalp =-TMath::Sin(fAlpha);
2505 t = cosalp*r[0] - sinalp*r[1];
2506 r[1] = sinalp*r[0] + cosalp*r[1];
2509 t = cosalp*p[0] - sinalp*p[1];
2510 p[1] = sinalp*p[0] + cosalp*p[1];
2513 // Do the final correcting step to the target plane (linear approximation)
2514 Double_t x=r[0], y=r[1], z=r[2];
2515 if (TMath::Abs(dx) > kAlmost0) {
2516 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
2524 // Calculate the track parameters
2525 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
2531 fP[4] = GetSign()/(t*pp);
2537 Bool_t AliExternalTrackParam::Translate(Double_t *vTrasl,Double_t *covV){
2539 //Translation: in the event mixing, the tracks can be shifted
2540 //of the difference among primary vertices (vTrasl) and
2541 //the covariance matrix is changed accordingly
2542 //(covV = covariance of the primary vertex).
2543 //Origin: "Romita, Rossella" <R.Romita@gsi.de>
2545 TVector3 translation;
2546 // vTrasl coordinates in the local system
2547 translation.SetXYZ(vTrasl[0],vTrasl[1],vTrasl[2]);
2548 translation.RotateZ(-fAlpha);
2549 translation.GetXYZ(vTrasl);
2551 //compute the new x,y,z of the track
2552 Double_t newX=fX-vTrasl[0];
2553 Double_t newY=fP[0]-vTrasl[1];
2554 Double_t newZ=fP[1]-vTrasl[2];
2556 //define the new parameters
2557 Double_t newParam[5];
2564 // recompute the covariance matrix:
2565 // 1. covV in the local system
2566 Double_t cosRot=TMath::Cos(fAlpha), sinRot=TMath::Sin(fAlpha);
2587 if(uUi.Determinant() <= 0.) {return kFALSE;}
2588 TMatrixD uUiQi(uUi,TMatrixD::kMult,qQi);
2589 TMatrixD m(qQi,TMatrixD::kTransposeMult,uUiQi);
2591 //2. compute the new covariance matrix of the track
2592 Double_t sigmaXX=m(0,0);
2593 Double_t sigmaXZ=m(2,0);
2594 Double_t sigmaXY=m(1,0);
2595 Double_t sigmaYY=GetSigmaY2()+m(1,1);
2596 Double_t sigmaYZ=fC[1]+m(1,2);
2597 Double_t sigmaZZ=fC[2]+m(2,2);
2598 Double_t covarianceYY=sigmaYY + (-1.)*((sigmaXY*sigmaXY)/sigmaXX);
2599 Double_t covarianceYZ=sigmaYZ-(sigmaXZ*sigmaXY/sigmaXX);
2600 Double_t covarianceZZ=sigmaZZ-((sigmaXZ*sigmaXZ)/sigmaXX);
2602 Double_t newCov[15];
2603 newCov[0]=covarianceYY;
2604 newCov[1]=covarianceYZ;
2605 newCov[2]=covarianceZZ;
2606 for(Int_t i=3;i<15;i++){
2610 // set the new parameters
2612 Set(newX,fAlpha,newParam,newCov);
2617 void AliExternalTrackParam::CheckCovariance() {
2619 // This function forces the diagonal elements of the covariance matrix to be positive.
2620 // In case the diagonal element is bigger than the maximal allowed value, it is set to
2621 // the limit and the off-diagonal elements that correspond to it are set to zero.
2623 fC[0] = TMath::Abs(fC[0]);
2625 double scl = TMath::Sqrt(kC0max/fC[0]);
2632 fC[2] = TMath::Abs(fC[2]);
2634 double scl = TMath::Sqrt(kC2max/fC[2]);
2641 fC[5] = TMath::Abs(fC[5]);
2643 double scl = TMath::Sqrt(kC5max/fC[5]);
2650 fC[9] = TMath::Abs(fC[9]);
2652 double scl = TMath::Sqrt(kC9max/fC[9]);
2659 fC[14] = TMath::Abs(fC[14]);
2660 if (fC[14]>kC14max) {
2661 double scl = TMath::Sqrt(kC14max/fC[14]);
2669 // The part below is used for tests and normally is commented out
2670 // TMatrixDSym m(5);
2674 // m(1,0)=fC[1]; m(1,1)=fC[2];
2675 // m(2,0)=fC[3]; m(2,1)=fC[4]; m(2,2)=fC[5];
2676 // m(3,0)=fC[6]; m(3,1)=fC[7]; m(3,2)=fC[8]; m(3,3)=fC[9];
2677 // 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];
2680 // m(0,2)=m(2,0); m(1,2)=m(2,1);
2681 // m(0,3)=m(3,0); m(1,3)=m(3,1); m(2,3)=m(3,2);
2682 // m(0,4)=m(4,0); m(1,4)=m(4,1); m(2,4)=m(4,2); m(3,4)=m(4,3);
2683 // m.EigenVectors(eig);
2685 // // assert(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0);
2686 // if (!(eig(0)>=0 && eig(1)>=0 && eig(2)>=0 && eig(3)>=0 && eig(4)>=0)) {
2687 // AliWarning("Negative eigenvalues of the covariance matrix!");
2693 Bool_t AliExternalTrackParam::ConstrainToVertex(const AliVVertex* vtx, Double_t b[3])
2695 // Constrain TPC inner params constrained
2700 Double_t dz[2], cov[3];
2701 if (!PropagateToDCABxByBz(vtx, b, 3, dz, cov))
2705 vtx->GetCovarianceMatrix(covar);
2707 Double_t p[2]= { fP[0] - dz[0], fP[1] - dz[1] };
2708 Double_t c[3]= { covar[2], 0., covar[5] };
2710 Double_t chi2C = GetPredictedChi2(p,c);
2720 //___________________________________________________________________________________________
2721 Bool_t AliExternalTrackParam::GetXatLabR(Double_t r,Double_t &x, Double_t bz, Int_t dir) const
2723 // Get local X of the track position estimated at the radius lab radius r.
2724 // The track curvature is accounted exactly
2726 // The flag "dir" can be used to remove the ambiguity of which intersection to take (out of 2 possible)
2727 // 0 - take the intersection closest to the current track position
2728 // >0 - go along the track (increasing fX)
2729 // <0 - go backward (decreasing fX)
2731 const Double_t &fy=fP[0], &sn = fP[2];
2732 const double kEps = 1.e-6;
2734 double crv = GetC(bz);
2735 if (TMath::Abs(crv)>kAlmost0) { // helix
2736 // get center of the track circle
2737 double tR = 1./crv; // track radius (for the moment signed)
2738 double cs = TMath::Sqrt((1-sn)*(1+sn));
2739 double x0 = fX - sn*tR;
2740 double y0 = fy + cs*tR;
2741 double r0 = TMath::Sqrt(x0*x0+y0*y0);
2742 // printf("Xc:%+e Yc:%+e tR:%e r0:%e\n",x0,y0,tR,r0);
2744 if (r0<=kAlmost0) return kFALSE; // the track is concentric to circle
2745 tR = TMath::Abs(tR);
2746 double tR2r0=1.,g=0,tmp=0;
2747 if (TMath::Abs(tR-r0)>kEps) {
2749 g = 0.5*(r*r/(r0*tR) - tR2r0 - 1./tR2r0);
2754 g = 0.5*r*r/(r0*tR) - 1;
2755 tmp = 0.5*r*r/(r0*r0);
2757 double det = (1.-g)*(1.+g);
2758 if (det<0) return kFALSE; // does not reach raduis r
2759 det = TMath::Sqrt(det);
2761 // the intersection happens in 2 points: {x0+tR*C,y0+tR*S}
2762 // with C=f*c0+-|s0|*det and S=f*s0-+c0 sign(s0)*det
2763 // where s0 and c0 make direction for the circle center (=x0/r0 and y0/r0)
2767 if (TMath::Abs(y0)>kAlmost0) { // when y0==0 the x,y is unique
2768 double dfx = tR2r0*TMath::Abs(y0)*det;
2769 double dfy = tR2r0*x0*TMath::Sign(det,y0);
2770 if (dir==0) { // chose the one which corresponds to smallest step
2771 double delta = (x-fX)*dfx-(y-fy)*dfy; // the choice of + in C will lead to smaller step if delta<0
2772 if (delta<0) x += dfx;
2775 else if (dir>0) { // along track direction: x must be > fX
2776 x -= dfx; // try the smallest step (dfx is positive)
2777 double dfeps = fX-x; // handle special case of very small step
2778 if (dfeps<-kEps) return kTRUE;
2779 if (TMath::Abs(dfeps)<kEps && // are we already in right r?
2780 TMath::Abs(fX*fX+fy*fy - r*r)<kEps) return fX;
2782 if (x-fX>0) return kTRUE;
2783 if (x-fX<-kEps) return kFALSE;
2784 x = fX; // don't move
2786 else { // backward: x must be < fX
2787 x += dfx; // try the smallest step (dfx is positive)
2788 double dfeps = x-fX; // handle special case of very small step
2789 if (dfeps<-kEps) return kTRUE;
2790 if (TMath::Abs(dfeps)<kEps && // are we already in right r?
2791 TMath::Abs(fX*fX+fy*fy - r*r)<kEps) return fX;
2793 if (x-fX<0) return kTRUE;
2794 if (x-fX>kEps) return kFALSE;
2795 x = fX; // don't move
2798 else { // special case: track touching the circle just in 1 point
2799 if ( (dir>0&&x<fX) || (dir<0&&x>fX) ) return kFALSE;
2802 else { // this is a straight track
2803 if (TMath::Abs(sn)>=kAlmost1) { // || to Y axis
2804 double det = (r-fX)*(r+fX);
2805 if (det<0) return kFALSE; // does not reach raduis r
2807 if (dir==0) return kTRUE;
2808 det = TMath::Sqrt(det);
2809 if (dir>0) { // along the track direction
2810 if (sn>0) {if (fy>det) return kFALSE;} // track is along Y axis and above the circle
2811 else {if (fy<-det) return kFALSE;} // track is against Y axis amd belo the circle
2813 else if(dir>0) { // agains track direction
2814 if (sn>0) {if (fy<-det) return kFALSE;} // track is along Y axis
2815 else if (fy>det) return kFALSE; // track is against Y axis
2818 else if (TMath::Abs(sn)<=kAlmost0) { // || to X axis
2819 double det = (r-fy)*(r+fy);
2820 if (det<0) return kFALSE; // does not reach raduis r
2821 det = TMath::Sqrt(det);
2823 x = fX>0 ? det : -det; // choose the solution requiring the smalest step
2826 else if (dir>0) { // along the track direction
2827 if (fX > det) return kFALSE; // current point is in on the right from the circle
2828 else if (fX <-det) x = -det; // on the left
2829 else x = det; // within the circle
2831 else { // against the track direction
2832 if (fX <-det) return kFALSE;
2833 else if (fX > det) x = det;
2837 else { // general case of straight line
2838 double cs = TMath::Sqrt((1-sn)*(1+sn));
2839 double xsyc = fX*sn-fy*cs;
2840 double det = (r-xsyc)*(r+xsyc);
2841 if (det<0) return kFALSE; // does not reach raduis r
2842 det = TMath::Sqrt(det);
2843 double xcys = fX*cs+fy*sn;
2845 if (dir==0) t += t>0 ? -det:det; // chose the solution requiring the smalest step
2846 else if (dir>0) { // go in increasing fX direction. ( t+-det > 0)
2847 if (t>=-det) t += -det; // take minimal step giving t>0
2848 else return kFALSE; // both solutions have negative t
2850 else { // go in increasing fX direction. (t+-det < 0)
2851 if (t<det) t -= det; // take minimal step giving t<0
2852 else return kFALSE; // both solutions have positive t
2860 //_________________________________________________________
2861 Bool_t AliExternalTrackParam::GetXYZatR(Double_t xr,Double_t bz, Double_t *xyz, Double_t* alpSect) const
2863 // This method has 3 modes of behaviour
2864 // 1) xyz[3] array is provided but alpSect pointer is 0: calculate the position of track intersection
2865 // with circle of radius xr and fill it in xyz array
2866 // 2) alpSect pointer is provided: find alpha of the sector where the track reaches local coordinate xr
2867 // Note that in this case xr is NOT the radius but the local coordinate.
2868 // If the xyz array is provided, it will be filled by track lab coordinates at local X in this sector
2869 // 3) Neither alpSect nor xyz pointers are provided: just check if the track reaches radius xr
2872 double crv = GetC(bz);
2873 if ( (TMath::Abs(bz))<kAlmost0Field ) crv=0.;
2874 const double &fy = fP[0];
2875 const double &fz = fP[1];
2876 const double &sn = fP[2];
2877 const double &tgl = fP[3];
2879 // general circle parameterization:
2880 // x = (r0+tR)cos(phi0) - tR cos(t+phi0)
2881 // y = (r0+tR)sin(phi0) - tR sin(t+phi0)
2882 // where qb is the sign of the curvature, tR is the track's signed radius and r0
2883 // is the DCA of helix to origin
2885 double tR = 1./crv; // track radius signed
2886 double cs = TMath::Sqrt((1-sn)*(1+sn));
2887 double x0 = fX - sn*tR; // helix center coordinates
2888 double y0 = fy + cs*tR;
2889 double phi0 = TMath::ATan2(y0,x0); // angle of PCA wrt to the origin
2890 if (tR<0) phi0 += TMath::Pi();
2891 if (phi0 > TMath::Pi()) phi0 -= 2.*TMath::Pi();
2892 else if (phi0 <-TMath::Pi()) phi0 += 2.*TMath::Pi();
2893 double cs0 = TMath::Cos(phi0);
2894 double sn0 = TMath::Sin(phi0);
2895 double r0 = x0*cs0 + y0*sn0 - tR; // DCA to origin
2896 double r2R = 1.+r0/tR;
2899 if (r2R<kAlmost0) return kFALSE; // helix is centered at the origin, no specific intersection with other concetric circle
2900 if (!xyz && !alpSect) return kTRUE;
2901 double xr2R = xr/tR;
2902 double r2Ri = 1./r2R;
2903 // the intersection cos(t) = [1 + (r0/tR+1)^2 - (r0/tR)^2]/[2(1+r0/tR)]
2904 double cosT = 0.5*(r2R + (1-xr2R*xr2R)*r2Ri);
2905 if ( TMath::Abs(cosT)>kAlmost1 ) {
2906 // printf("Does not reach : %f %f\n",r0,tR);
2907 return kFALSE; // track does not reach the radius xr
2910 double t = TMath::ACos(cosT);
2912 // intersection point
2914 xyzi[0] = x0 - tR*TMath::Cos(t+phi0);
2915 xyzi[1] = y0 - tR*TMath::Sin(t+phi0);
2916 if (xyz) { // if postition is requested, then z is needed:
2917 double t0 = TMath::ATan2(cs,-sn) - phi0;
2918 double z0 = fz - t0*tR*tgl;
2919 xyzi[2] = z0 + tR*t*tgl;
2923 Local2GlobalPosition(xyzi,fAlpha);
2932 double &alp = *alpSect;
2933 // determine the sector of crossing
2934 double phiPos = TMath::Pi()+TMath::ATan2(-xyzi[1],-xyzi[0]);
2935 int sect = ((Int_t)(phiPos*TMath::RadToDeg()))/20;
2936 alp = TMath::DegToRad()*(20*sect+10);
2937 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
2940 Double_t ca=TMath::Cos(alp-fAlpha), sa=TMath::Sin(alp-fAlpha);
2941 if ((cs*ca+sn*sa)<0) {
2942 AliDebug(1,Form("Rotation to target sector impossible: local cos(phi) would become %.2f",cs*ca+sn*sa));
2947 if (TMath::Abs(f1) >= kAlmost1) {
2948 AliDebug(1,Form("Rotation to target sector impossible: local sin(phi) would become %.2f",f1));
2952 double tmpX = fX*ca + fy*sa;
2953 double tmpY = -fX*sa + fy*ca;
2955 // estimate Y at X=xr
2959 if (TMath::Abs(f2) >= kAlmost1) {
2960 AliDebug(1,Form("Propagation in target sector failed ! %.10e",f2));
2963 r1 = TMath::Sqrt((1.-f1)*(1.+f1));
2964 r2 = TMath::Sqrt((1.-f2)*(1.+f2));
2965 dy2dx = (f1+f2)/(r1+r2);
2966 yloc = tmpY + dx*dy2dx;
2967 if (yloc>ylocMax) {alp += 2*TMath::Pi()/18; sect++;}
2968 else if (yloc<-ylocMax) {alp -= 2*TMath::Pi()/18; sect--;}
2970 if (alp >= TMath::Pi()) alp -= 2*TMath::Pi();
2971 else if (alp < -TMath::Pi()) alp += 2*TMath::Pi();
2972 // if (sect>=18) sect = 0;
2973 // if (sect<=0) sect = 17;
2976 // if alpha was requested, then recalculate the position at intersection in sector
2980 if (TMath::Abs(x2r)<0.05) xyz[2] = fz + dx*(r2 + f2*dy2dx)*tgl;
2982 // for small dx/R the linear apporximation of the arc by the segment is OK,
2983 // but at large dx/R the error is very large and leads to incorrect Z propagation
2984 // angle traversed delta = 2*asin(dist_start_end / R / 2), hence the arc is: R*deltaPhi
2985 // The dist_start_end is obtained from sqrt(dx^2+dy^2) = x/(r1+r2)*sqrt(2+f1*f2+r1*r2)
2986 // Similarly, the rotation angle in linear in dx only for dx<<R
2987 double chord = dx*TMath::Sqrt(1+dy2dx*dy2dx); // distance from old position to new one
2988 double rot = 2*TMath::ASin(0.5*chord*crv); // angular difference seen from the circle center
2989 xyz[2] = fz + rot/crv*tgl;
2991 Local2GlobalPosition(xyz,alp);
2999 Double_t AliExternalTrackParam::GetParameterAtRadius(Double_t r, Double_t bz, Int_t parType) const
3002 // Get track parameters at the radius of interest.
3003 // Given function is aimed to be used to interactivelly (tree->Draw())
3004 // access track properties at different radii
3006 // TO BE USED WITH SPECICAL CARE -
3007 // it is aimed to be used for rough calculation as constant field and
3008 // no correction for material is used
3010 // r - radius of interest
3011 // bz - magentic field
3012 // retun values dependens on parType:
3022 // parType = 7 - global position phi
3023 // parType = 8 - global direction phi
3024 // parType = 9 - direction phi- positionphi
3032 Bool_t res = GetXatLabR(r,localX,bz,1);
3038 // position parameters
3040 GetXYZAt(localX,bz,xyz);
3042 return xyz[parType];
3045 if (parType==6) return TMath::Sqrt(xyz[0]*xyz[0]+xyz[1]*xyz[1]);
3046 if (parType==7) return TMath::ATan2(xyz[1],xyz[0]);
3048 // momenta parameters
3050 GetPxPyPzAt(localX,bz,pxyz);
3051 if (parType==8) return TMath::ATan2(pxyz[1],pxyz[0]);
3053 Double_t diff = TMath::ATan2(pxyz[1],pxyz[0])-TMath::ATan2(xyz[1],xyz[0]);
3054 if (diff>TMath::Pi()) diff-=TMath::TwoPi();
3055 if (diff<-TMath::Pi()) diff+=TMath::TwoPi();
3058 if (parType>=3&&parType<6) {
3059 return pxyz[parType%3];