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 ///////////////////////////////////////////////////////////////////////////////
28 #include <TMatrixDSym.h>
29 #include <TPolyMarker3D.h>
33 #include "AliExternalTrackParam.h"
34 #include "AliVVertex.h"
37 ClassImp(AliExternalTrackParam)
39 Double32_t AliExternalTrackParam::fgMostProbablePt=kMostProbablePt;
41 //_____________________________________________________________________________
42 AliExternalTrackParam::AliExternalTrackParam() :
48 // default constructor
50 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
51 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
54 //_____________________________________________________________________________
55 AliExternalTrackParam::AliExternalTrackParam(const AliExternalTrackParam &track):
63 for (Int_t i = 0; i < 5; i++) fP[i] = track.fP[i];
64 for (Int_t i = 0; i < 15; i++) fC[i] = track.fC[i];
67 //_____________________________________________________________________________
68 AliExternalTrackParam& AliExternalTrackParam::operator=(const AliExternalTrackParam &trkPar)
71 // assignment operator
75 AliVTrack::operator=(trkPar);
77 fAlpha = trkPar.fAlpha;
79 for (Int_t i = 0; i < 5; i++) fP[i] = trkPar.fP[i];
80 for (Int_t i = 0; i < 15; i++) fC[i] = trkPar.fC[i];
86 //_____________________________________________________________________________
87 AliExternalTrackParam::AliExternalTrackParam(Double_t x, Double_t alpha,
88 const Double_t param[5],
89 const Double_t covar[15]) :
95 // create external track parameters from given arguments
97 for (Int_t i = 0; i < 5; i++) fP[i] = param[i];
98 for (Int_t i = 0; i < 15; i++) fC[i] = covar[i];
101 //_____________________________________________________________________________
102 AliExternalTrackParam::AliExternalTrackParam(const AliVTrack *vTrack) :
108 // Constructor from virtual track,
109 // This is not a copy contructor !
112 if (vTrack->InheritsFrom("AliExternalTrackParam")) {
113 AliError("This is not a copy constructor. Use AliExternalTrackParam(const AliExternalTrackParam &) !");
114 AliWarning("Calling the default constructor...");
115 AliExternalTrackParam();
119 Double_t xyz[3],pxpypz[3],cv[21];
121 pxpypz[0]=vTrack->Px();
122 pxpypz[1]=vTrack->Py();
123 pxpypz[2]=vTrack->Pz();
124 vTrack->GetCovarianceXYZPxPyPz(cv);
125 Short_t sign = (Short_t)vTrack->Charge();
127 Set(xyz,pxpypz,cv,sign);
130 //_____________________________________________________________________________
131 AliExternalTrackParam::AliExternalTrackParam(Double_t xyz[3],Double_t pxpypz[3],
132 Double_t cv[21],Short_t sign) :
138 // constructor from the global parameters
141 Set(xyz,pxpypz,cv,sign);
144 //_____________________________________________________________________________
145 void AliExternalTrackParam::Set(Double_t xyz[3],Double_t pxpypz[3],
146 Double_t cv[21],Short_t sign)
149 // create external track parameters from the global parameters
150 // x,y,z,px,py,pz and their 6x6 covariance matrix
151 // A.Dainese 10.10.08
153 // Calculate alpha: the rotation angle of the corresponding local system.
155 // For global radial position inside the beam pipe, alpha is the
156 // azimuthal angle of the momentum projected on (x,y).
158 // For global radial position outside the ITS, alpha is the
159 // azimuthal angle of the centre of the TPC sector in which the point
162 Double_t radPos2 = xyz[0]*xyz[0]+xyz[1]*xyz[1];
163 Double_t radMax = 45.; // approximately ITS outer radius
164 if (radPos2 < radMax*radMax) { // inside the ITS
166 fAlpha = TMath::ATan2(pxpypz[1],pxpypz[0]);
167 } else { // outside the ITS
168 Float_t phiPos = TMath::Pi()+TMath::ATan2(-xyz[1], -xyz[0]);
170 TMath::DegToRad()*(20*((((Int_t)(phiPos*TMath::RadToDeg()))/20))+10);
173 // Get the vertex of origin and the momentum
174 TVector3 ver(xyz[0],xyz[1],xyz[2]);
175 TVector3 mom(pxpypz[0],pxpypz[1],pxpypz[2]);
177 // Rotate to the local coordinate system
178 ver.RotateZ(-fAlpha);
179 mom.RotateZ(-fAlpha);
181 // x of the reference plane
184 Double_t charge = (Double_t)sign;
188 fP[2] = TMath::Sin(mom.Phi());
189 fP[3] = mom.Pz()/mom.Pt();
190 fP[4] = TMath::Sign(1/mom.Pt(),charge);
192 // Covariance matrix (formulas to be simplified)
194 Double_t pt=1./TMath::Abs(fP[4]);
195 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
196 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
198 Double_t m00=-sn;// m10=cs;
199 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
200 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
201 Double_t m35=pt, m45=-pt*pt*fP[3];
207 Double_t cv34 = TMath::Sqrt(cv[3 ]*cv[3 ]+cv[4 ]*cv[4 ]);
208 Double_t a1=cv[13]-cv[9]*(m23*m44+m43*m24)/m23/m43;
209 Double_t a2=m23*m24-m23*(m23*m44+m43*m24)/m43;
210 Double_t a3=m43*m44-m43*(m23*m44+m43*m24)/m23;
211 Double_t a4=cv[14]-2.*cv[9]*m24*m44/m23/m43;
212 Double_t a5=m24*m24-2.*m24*m44*m23/m43;
213 Double_t a6=m44*m44-2.*m24*m44*m43/m23;
215 fC[0 ] = cv[0 ]+cv[2 ];
216 fC[1 ] = TMath::Sign(cv34,cv[3 ]/m00);
218 fC[3 ] = (cv[10]/m44-cv[6]/m43)/(m24/m44-m23/m43)/m00;
219 fC[10] = (cv[6]/m00-fC[3 ]*m23)/m43;
220 fC[6 ] = (cv[15]/m00-fC[10]*m45)/m35;
221 fC[4 ] = (cv[12]-cv[8]*m44/m43)/(m24-m23*m44/m43);
222 fC[11] = (cv[8]-fC[4]*m23)/m43;
223 fC[7 ] = cv[17]/m35-fC[11]*m45/m35;
224 fC[5 ] = TMath::Abs((a4-a6*a1/a3)/(a5-a6*a2/a3));
225 fC[14] = TMath::Abs(a1/a3-a2*fC[5]/a3);
226 fC[12] = (cv[9]-fC[5]*m23*m23-fC[14]*m43*m43)/m23/m43;
227 Double_t b1=cv[18]-fC[12]*m23*m45-fC[14]*m43*m45;
230 Double_t b4=cv[19]-fC[12]*m24*m45-fC[14]*m44*m45;
233 fC[8 ] = (b4-b6*b1/b3)/(b5-b6*b2/b3);
234 fC[13] = b1/b3-b2*fC[8]/b3;
235 fC[9 ] = TMath::Abs((cv[20]-fC[14]*(m45*m45)-fC[13]*2.*m35*m45)/(m35*m35));
240 //_____________________________________________________________________________
241 void AliExternalTrackParam::Reset() {
243 // Resets all the parameters to 0
246 for (Int_t i = 0; i < 5; i++) fP[i] = 0;
247 for (Int_t i = 0; i < 15; i++) fC[i] = 0;
250 //_____________________________________________________________________________
251 void AliExternalTrackParam::AddCovariance(const Double_t c[15]) {
253 // Add "something" to the track covarince matrix.
254 // May be needed to account for unknown mis-calibration/mis-alignment
257 fC[1] +=c[1]; fC[2] +=c[2];
258 fC[3] +=c[3]; fC[4] +=c[4]; fC[5] +=c[5];
259 fC[6] +=c[6]; fC[7] +=c[7]; fC[8] +=c[8]; fC[9] +=c[9];
260 fC[10]+=c[10]; fC[11]+=c[11]; fC[12]+=c[12]; fC[13]+=c[13]; fC[14]+=c[14];
264 Double_t AliExternalTrackParam::GetP() const {
265 //---------------------------------------------------------------------
266 // This function returns the track momentum
267 // Results for (nearly) straight tracks are meaningless !
268 //---------------------------------------------------------------------
269 if (TMath::Abs(fP[4])<=kAlmost0) return kVeryBig;
270 return TMath::Sqrt(1.+ fP[3]*fP[3])/TMath::Abs(fP[4]);
273 Double_t AliExternalTrackParam::Get1P() const {
274 //---------------------------------------------------------------------
275 // This function returns the 1/(track momentum)
276 //---------------------------------------------------------------------
277 return TMath::Abs(fP[4])/TMath::Sqrt(1.+ fP[3]*fP[3]);
280 //_______________________________________________________________________
281 Double_t AliExternalTrackParam::GetD(Double_t x,Double_t y,Double_t b) const {
282 //------------------------------------------------------------------
283 // This function calculates the transverse impact parameter
284 // with respect to a point with global coordinates (x,y)
285 // in the magnetic field "b" (kG)
286 //------------------------------------------------------------------
287 if (TMath::Abs(b) < kAlmost0Field) return GetLinearD(x,y);
288 Double_t rp4=GetC(b);
290 Double_t xt=fX, yt=fP[0];
292 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
293 Double_t a = x*cs + y*sn;
294 y = -x*sn + y*cs; x=a;
297 sn=rp4*xt - fP[2]; cs=rp4*yt + TMath::Sqrt(1.- fP[2]*fP[2]);
298 a=2*(xt*fP[2] - yt*TMath::Sqrt(1.- fP[2]*fP[2]))-rp4*(xt*xt + yt*yt);
299 return -a/(1 + TMath::Sqrt(sn*sn + cs*cs));
302 //_______________________________________________________________________
303 void AliExternalTrackParam::
304 GetDZ(Double_t x, Double_t y, Double_t z, Double_t b, Float_t dz[2]) const {
305 //------------------------------------------------------------------
306 // This function calculates the transverse and longitudinal impact parameters
307 // with respect to a point with global coordinates (x,y)
308 // in the magnetic field "b" (kG)
309 //------------------------------------------------------------------
310 Double_t f1 = fP[2], r1 = TMath::Sqrt(1. - f1*f1);
311 Double_t xt=fX, yt=fP[0];
312 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
313 Double_t a = x*cs + y*sn;
314 y = -x*sn + y*cs; x=a;
317 Double_t rp4=GetC(b);
318 if ((TMath::Abs(b) < kAlmost0Field) || (TMath::Abs(rp4) < kAlmost0)) {
319 dz[0] = -(xt*f1 - yt*r1);
320 dz[1] = fP[1] + (dz[0]*f1 - xt)/r1*fP[3] - z;
324 sn=rp4*xt - f1; cs=rp4*yt + r1;
325 a=2*(xt*f1 - yt*r1)-rp4*(xt*xt + yt*yt);
326 Double_t rr=TMath::Sqrt(sn*sn + cs*cs);
328 Double_t f2 = -sn/rr, r2 = TMath::Sqrt(1. - f2*f2);
329 dz[1] = fP[1] + fP[3]/rp4*TMath::ASin(f2*r1 - f1*r2) - z;
332 //_______________________________________________________________________
333 Double_t AliExternalTrackParam::GetLinearD(Double_t xv,Double_t yv) const {
334 //------------------------------------------------------------------
335 // This function calculates the transverse impact parameter
336 // with respect to a point with global coordinates (xv,yv)
337 // neglecting the track curvature.
338 //------------------------------------------------------------------
339 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
340 Double_t x= xv*cs + yv*sn;
341 Double_t y=-xv*sn + yv*cs;
343 Double_t d = (fX-x)*fP[2] - (fP[0]-y)*TMath::Sqrt(1.- fP[2]*fP[2]);
348 Bool_t AliExternalTrackParam::CorrectForMeanMaterial
349 (Double_t xOverX0, Double_t xTimesRho, Double_t mass, Bool_t anglecorr,
350 Double_t (*Bethe)(Double_t)) {
351 //------------------------------------------------------------------
352 // This function corrects the track parameters for the crossed material.
353 // "xOverX0" - X/X0, the thickness in units of the radiation length.
354 // "xTimesRho" - is the product length*density (g/cm^2).
355 // "mass" - the mass of this particle (GeV/c^2).
356 //------------------------------------------------------------------
361 Double_t &fC22=fC[5];
362 Double_t &fC33=fC[9];
363 Double_t &fC43=fC[13];
364 Double_t &fC44=fC[14];
366 //Apply angle correction, if requested
368 Double_t angle=TMath::Sqrt((1.+ fP3*fP3)/(1.- fP2*fP2));
375 Double_t beta2=p2/(p2 + mass*mass);
377 //Calculating the multiple scattering corrections******************
383 Double_t theta2=14.1*14.1/(beta2*p2*1e6)*TMath::Abs(xOverX0);
384 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
385 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
386 cC22 = theta2*(1.- fP2*fP2)*(1. + fP3*fP3);
387 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
388 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
389 cC44 = theta2*fP3*fP4*fP3*fP4;
392 //Calculating the energy loss corrections************************
394 if ((xTimesRho != 0.) && (beta2 < 1.)) {
395 Double_t dE=Bethe(p/mass)*xTimesRho;
396 Double_t e=TMath::Sqrt(p2 + mass*mass);
397 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
399 if (TMath::Abs(fP4*cP4)>100.) return kFALSE; //Do not track below 10 MeV/c
402 // Approximate energy loss fluctuation (M.Ivanov)
403 const Double_t knst=0.07; // To be tuned.
404 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
405 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
409 //Applying the corrections*****************************
420 Bool_t AliExternalTrackParam::CorrectForMaterial
421 (Double_t d, Double_t x0, Double_t mass, Double_t (*Bethe)(Double_t)) {
422 //------------------------------------------------------------------
423 // Deprecated function !
424 // Better use CorrectForMeanMaterial instead of it.
426 // This function corrects the track parameters for the crossed material
427 // "d" - the thickness (fraction of the radiation length)
428 // "x0" - the radiation length (g/cm^2)
429 // "mass" - the mass of this particle (GeV/c^2)
430 //------------------------------------------------------------------
435 Double_t &fC22=fC[5];
436 Double_t &fC33=fC[9];
437 Double_t &fC43=fC[13];
438 Double_t &fC44=fC[14];
442 Double_t beta2=p2/(p2 + mass*mass);
443 d*=TMath::Sqrt((1.+ fP3*fP3)/(1.- fP2*fP2));
445 //Multiple scattering******************
451 Double_t theta2=14.1*14.1/(beta2*p2*1e6)*TMath::Abs(d);
452 //Double_t theta2=1.0259e-6*14*14/28/(beta2*p2)*TMath::Abs(d)*9.36*2.33;
453 if(theta2>TMath::Pi()*TMath::Pi()) return kFALSE;
454 cC22 = theta2*(1.- fP2*fP2)*(1. + fP3*fP3);
455 cC33 = theta2*(1. + fP3*fP3)*(1. + fP3*fP3);
456 cC43 = theta2*fP3*fP4*(1. + fP3*fP3);
457 cC44 = theta2*fP3*fP4*fP3*fP4;
460 //Energy losses************************
462 if (x0!=0. && beta2<1) {
464 Double_t dE=Bethe(p/mass)*d;
465 Double_t e=TMath::Sqrt(p2 + mass*mass);
466 if ( TMath::Abs(dE) > 0.3*e ) return kFALSE; //30% energy loss is too much!
469 // Approximate energy loss fluctuation (M.Ivanov)
470 const Double_t knst=0.07; // To be tuned.
471 Double_t sigmadE=knst*TMath::Sqrt(TMath::Abs(dE));
472 cC44 += ((sigmadE*e/p2*fP4)*(sigmadE*e/p2*fP4));
485 Double_t AliExternalTrackParam::BetheBlochAleph(Double_t bg,
492 // This is the empirical ALEPH parameterization of the Bethe-Bloch formula.
493 // It is normalized to 1 at the minimum.
497 // The default values for the kp* parameters are for ALICE TPC.
498 // The returned value is in MIP units
501 Double_t beta = bg/TMath::Sqrt(1.+ bg*bg);
503 Double_t aa = TMath::Power(beta,kp4);
504 Double_t bb = TMath::Power(1./bg,kp5);
506 bb=TMath::Log(kp3+bb);
508 return (kp2-aa-bb)*kp1/aa;
511 Double_t AliExternalTrackParam::BetheBlochGeant(Double_t bg,
518 // This is the parameterization of the Bethe-Bloch formula inspired by Geant.
521 // kp0 - density [g/cm^3]
522 // kp1 - density effect first junction point
523 // kp2 - density effect second junction point
524 // kp3 - mean excitation energy [GeV]
527 // The default values for the kp* parameters are for silicon.
528 // The returned value is in [GeV/(g/cm^2)].
531 const Double_t mK = 0.307075e-3; // [GeV*cm^2/g]
532 const Double_t me = 0.511e-3; // [GeV/c^2]
533 const Double_t rho = kp0;
534 const Double_t x0 = kp1*2.303;
535 const Double_t x1 = kp2*2.303;
536 const Double_t mI = kp3;
537 const Double_t mZA = kp4;
538 const Double_t bg2 = bg*bg;
539 const Double_t maxT= 2*me*bg2; // neglecting the electron mass
543 const Double_t x=TMath::Log(bg);
544 const Double_t lhwI=TMath::Log(28.816*1e-9*TMath::Sqrt(rho*mZA)/mI);
548 const Double_t r=(x1-x)/(x1-x0);
549 d2 = lhwI + x - 0.5 + (0.5 - lhwI - x0)*r*r*r;
552 return mK*mZA*(1+bg2)/bg2*
553 (0.5*TMath::Log(2*me*bg2*maxT/(mI*mI)) - bg2/(1+bg2) - d2);
556 Double_t AliExternalTrackParam::BetheBlochSolid(Double_t bg) {
557 //------------------------------------------------------------------
558 // This is an approximation of the Bethe-Bloch formula,
559 // reasonable for solid materials.
560 // All the parameters are, in fact, for Si.
561 // The returned value is in [GeV/(g/cm^2)]
562 //------------------------------------------------------------------
564 return BetheBlochGeant(bg);
567 Double_t AliExternalTrackParam::BetheBlochGas(Double_t bg) {
568 //------------------------------------------------------------------
569 // This is an approximation of the Bethe-Bloch formula,
570 // reasonable for gas materials.
571 // All the parameters are, in fact, for Ne.
572 // The returned value is in [GeV/(g/cm^2)]
573 //------------------------------------------------------------------
575 const Double_t rho = 0.9e-3;
576 const Double_t x0 = 2.;
577 const Double_t x1 = 4.;
578 const Double_t mI = 140.e-9;
579 const Double_t mZA = 0.49555;
581 return BetheBlochGeant(bg,rho,x0,x1,mI,mZA);
584 Bool_t AliExternalTrackParam::Rotate(Double_t alpha) {
585 //------------------------------------------------------------------
586 // Transform this track to the local coord. system rotated
587 // by angle "alpha" (rad) with respect to the global coord. system.
588 //------------------------------------------------------------------
589 if (TMath::Abs(fP[2]) >= kAlmost1) {
590 AliError(Form("Precondition is not satisfied: |sin(phi)|>1 ! %f",fP[2]));
594 if (alpha < -TMath::Pi()) alpha += 2*TMath::Pi();
595 else if (alpha >= TMath::Pi()) alpha -= 2*TMath::Pi();
599 Double_t &fC00=fC[0];
600 Double_t &fC10=fC[1];
601 Double_t &fC20=fC[3];
602 Double_t &fC21=fC[4];
603 Double_t &fC22=fC[5];
604 Double_t &fC30=fC[6];
605 Double_t &fC32=fC[8];
606 Double_t &fC40=fC[10];
607 Double_t &fC42=fC[12];
610 Double_t ca=TMath::Cos(alpha-fAlpha), sa=TMath::Sin(alpha-fAlpha);
611 Double_t sf=fP2, cf=TMath::Sqrt(1.- fP2*fP2);
613 Double_t tmp=sf*ca - cf*sa;
614 if (TMath::Abs(tmp) >= kAlmost1) {
615 AliError(Form("Rotation failed ! %.10e",tmp));
624 if (TMath::Abs(cf)<kAlmost0) {
625 AliError(Form("Too small cosine value %f",cf));
629 Double_t rr=(ca+sf/cf*sa);
644 Bool_t AliExternalTrackParam::PropagateTo(Double_t xk, Double_t b) {
645 //----------------------------------------------------------------
646 // Propagate this track to the plane X=xk (cm) in the field "b" (kG)
647 //----------------------------------------------------------------
649 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
651 Double_t crv=GetC(b);
652 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
654 Double_t f1=fP[2], f2=f1 + crv*dx;
655 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
656 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
658 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
661 &fC10=fC[1], &fC11=fC[2],
662 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
663 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
664 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
666 Double_t r1=TMath::Sqrt(1.- f1*f1), r2=TMath::Sqrt(1.- f2*f2);
669 fP0 += dx*(f1+f2)/(r1+r2);
670 fP1 += dx*(r2 + f2*(f1+f2)/(r1+r2))*fP3; // Many thanks to P.Hristov !
675 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
676 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
677 Double_t f12= dx*fP3*f1/(r1*r1*r1);
678 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
680 Double_t f24= dx; f24*=cc;
683 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
684 Double_t b02=f24*fC40;
685 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
686 Double_t b12=f24*fC41;
687 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
688 Double_t b22=f24*fC42;
689 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
690 Double_t b42=f24*fC44;
691 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
692 Double_t b32=f24*fC43;
695 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
696 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
697 Double_t a22=f24*b42;
699 //F*C*Ft = C + (b + bt + a)
700 fC00 += b00 + b00 + a00;
701 fC10 += b10 + b01 + a01;
702 fC20 += b20 + b02 + a02;
705 fC11 += b11 + b11 + a11;
706 fC21 += b21 + b12 + a12;
709 fC22 += b22 + b22 + a22;
717 AliExternalTrackParam::Propagate(Double_t alpha, Double_t x, Double_t b) {
718 //------------------------------------------------------------------
719 // Transform this track to the local coord. system rotated
720 // by angle "alpha" (rad) with respect to the global coord. system,
721 // and propagate this track to the plane X=xk (cm) in the field "b" (kG)
722 //------------------------------------------------------------------
724 //Save the parameters
727 Double_t ps[5], cs[15];
728 for (Int_t i=0; i<5; i++) ps[i]=fP[i];
729 for (Int_t i=0; i<15; i++) cs[i]=fC[i];
732 if (PropagateTo(x,b)) return kTRUE;
734 //Restore the parameters, if the operation failed
737 for (Int_t i=0; i<5; i++) fP[i]=ps[i];
738 for (Int_t i=0; i<15; i++) fC[i]=cs[i];
743 void AliExternalTrackParam::Propagate(Double_t len, Double_t x[3],
744 Double_t p[3], Double_t bz) const {
745 //+++++++++++++++++++++++++++++++++++++++++
746 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
747 // Extrapolate track along simple helix in magnetic field
748 // Arguments: len -distance alogn helix, [cm]
749 // bz - mag field, [kGaus]
750 // Returns: x and p contain extrapolated positon and momentum
751 // The momentum returned for straight-line tracks is meaningless !
752 //+++++++++++++++++++++++++++++++++++++++++
755 if (OneOverPt() < kAlmost0 || TMath::Abs(bz) < kAlmost0Field || GetC(bz) < kAlmost0){ //straight-line tracks
756 Double_t unit[3]; GetDirection(unit);
761 p[0]=unit[0]/kAlmost0;
762 p[1]=unit[1]/kAlmost0;
763 p[2]=unit[2]/kAlmost0;
767 Double_t a = -kB2C*bz*GetSign();
769 x[0] += p[0]*TMath::Sin(rho*len)/a - p[1]*(1-TMath::Cos(rho*len))/a;
770 x[1] += p[1]*TMath::Sin(rho*len)/a + p[0]*(1-TMath::Cos(rho*len))/a;
774 p[0] = p0 *TMath::Cos(rho*len) - p[1]*TMath::Sin(rho*len);
775 p[1] = p[1]*TMath::Cos(rho*len) + p0 *TMath::Sin(rho*len);
779 Bool_t AliExternalTrackParam::Intersect(Double_t pnt[3], Double_t norm[3],
781 //+++++++++++++++++++++++++++++++++++++++++
782 // Origin: K. Shileev (Kirill.Shileev@cern.ch)
783 // Finds point of intersection (if exists) of the helix with the plane.
784 // Stores result in fX and fP.
785 // Arguments: planePoint,planeNorm - the plane defined by any plane's point
786 // and vector, normal to the plane
787 // Returns: kTrue if helix intersects the plane, kFALSE otherwise.
788 //+++++++++++++++++++++++++++++++++++++++++
789 Double_t x0[3]; GetXYZ(x0); //get track position in MARS
791 //estimates initial helix length up to plane
793 (pnt[0]-x0[0])*norm[0] + (pnt[1]-x0[1])*norm[1] + (pnt[2]-x0[2])*norm[2];
794 Double_t dist=99999,distPrev=dist;
796 while(TMath::Abs(dist)>0.00001){
797 //calculates helix at the distance s from x0 ALONG the helix
800 //distance between current helix position and plane
801 dist=(x[0]-pnt[0])*norm[0]+(x[1]-pnt[1])*norm[1]+(x[2]-pnt[2])*norm[2];
803 if(TMath::Abs(dist) >= TMath::Abs(distPrev)) {return kFALSE;}
807 //on exit pnt is intersection point,norm is track vector at that point,
809 for (Int_t i=0; i<3; i++) {pnt[i]=x[i]; norm[i]=p[i];}
814 AliExternalTrackParam::GetPredictedChi2(Double_t p[2],Double_t cov[3]) const {
815 //----------------------------------------------------------------
816 // Estimate the chi2 of the space point "p" with the cov. matrix "cov"
817 //----------------------------------------------------------------
818 Double_t sdd = fC[0] + cov[0];
819 Double_t sdz = fC[1] + cov[1];
820 Double_t szz = fC[2] + cov[2];
821 Double_t det = sdd*szz - sdz*sdz;
823 if (TMath::Abs(det) < kAlmost0) return kVeryBig;
825 Double_t d = fP[0] - p[0];
826 Double_t z = fP[1] - p[1];
828 return (d*szz*d - 2*d*sdz*z + z*sdd*z)/det;
831 Double_t AliExternalTrackParam::
832 GetPredictedChi2(Double_t p[3],Double_t covyz[3],Double_t covxyz[3]) const {
833 //----------------------------------------------------------------
834 // Estimate the chi2 of the 3D space point "p" and
835 // the full covariance matrix "covyz" and "covxyz"
837 // Cov(x,x) ... : covxyz[0]
838 // Cov(y,x) ... : covxyz[1] covyz[0]
839 // Cov(z,x) ... : covxyz[2] covyz[1] covyz[2]
840 //----------------------------------------------------------------
849 if (TMath::Abs(f) >= kAlmost1) return kVeryBig;
850 Double_t r=TMath::Sqrt(1.- f*f);
851 Double_t a=f/r, b=GetTgl()/r;
853 Double_t s2=333.*333.; //something reasonably big (cm^2)
856 v(0,0)= s2; v(0,1)= a*s2; v(0,2)= b*s2;;
857 v(1,0)=a*s2; v(1,1)=a*a*s2 + GetSigmaY2(); v(1,2)=a*b*s2 + GetSigmaZY();
858 v(2,0)=b*s2; v(2,1)=a*b*s2 + GetSigmaZY(); v(2,2)=b*b*s2 + GetSigmaZ2();
860 v(0,0)+=covxyz[0]; v(0,1)+=covxyz[1]; v(0,2)+=covxyz[2];
861 v(1,0)+=covxyz[1]; v(1,1)+=covyz[0]; v(1,2)+=covyz[1];
862 v(2,0)+=covxyz[2]; v(2,1)+=covyz[1]; v(2,2)+=covyz[2];
865 if (!v.IsValid()) return kVeryBig;
868 for (Int_t i = 0; i < 3; i++)
869 for (Int_t j = 0; j < 3; j++) chi2 += res[i]*res[j]*v(i,j);
876 Bool_t AliExternalTrackParam::
877 PropagateTo(Double_t p[3],Double_t covyz[3],Double_t covxyz[3],Double_t bz) {
878 //----------------------------------------------------------------
879 // Propagate this track to the plane
880 // the 3D space point "p" (with the covariance matrix "covyz" and "covxyz")
882 // The magnetic field is "bz" (kG)
884 // The track curvature and the change of the covariance matrix
885 // of the track parameters are negleted !
886 // (So the "step" should be small compared with 1/curvature)
887 //----------------------------------------------------------------
890 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
891 Double_t r=TMath::Sqrt(1.- f*f);
892 Double_t a=f/r, b=GetTgl()/r;
894 Double_t s2=333.*333.; //something reasonably big (cm^2)
897 tV(0,0)= s2; tV(0,1)= a*s2; tV(0,2)= b*s2;
898 tV(1,0)=a*s2; tV(1,1)=a*a*s2; tV(1,2)=a*b*s2;
899 tV(2,0)=b*s2; tV(2,1)=a*b*s2; tV(2,2)=b*b*s2;
902 pV(0,0)=covxyz[0]; pV(0,1)=covxyz[1]; pV(0,2)=covxyz[2];
903 pV(1,0)=covxyz[1]; pV(1,1)=covyz[0]; pV(1,2)=covyz[1];
904 pV(2,0)=covxyz[2]; pV(2,1)=covyz[1]; pV(2,2)=covyz[2];
909 if (!tpV.IsValid()) return kFALSE;
911 TMatrixDSym pW(3),tW(3);
912 for (Int_t i=0; i<3; i++)
913 for (Int_t j=0; j<3; j++) {
915 for (Int_t k=0; k<3; k++) {
916 pW(i,j) += tV(i,k)*tpV(k,j);
917 tW(i,j) += pV(i,k)*tpV(k,j);
921 Double_t t[3] = {GetX(), GetY(), GetZ()};
924 for (Int_t i=0; i<3; i++) x += (tW(0,i)*t[i] + pW(0,i)*p[i]);
925 Double_t crv=GetC(bz);
926 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
928 if (TMath::Abs(f) >= kAlmost1) return kFALSE;
932 for (Int_t i=0; i<3; i++) fP[0] += (tW(1,i)*t[i] + pW(1,i)*p[i]);
934 for (Int_t i=0; i<3; i++) fP[1] += (tW(2,i)*t[i] + pW(2,i)*p[i]);
939 Double_t *AliExternalTrackParam::GetResiduals(
940 Double_t *p,Double_t *cov,Bool_t updated) const {
941 //------------------------------------------------------------------
942 // Returns the track residuals with the space point "p" having
943 // the covariance matrix "cov".
944 // If "updated" is kTRUE, the track parameters expected to be updated,
945 // otherwise they must be predicted.
946 //------------------------------------------------------------------
947 static Double_t res[2];
949 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
951 r00-=fC[0]; r01-=fC[1]; r11-=fC[2];
953 r00+=fC[0]; r01+=fC[1]; r11+=fC[2];
955 Double_t det=r00*r11 - r01*r01;
957 if (TMath::Abs(det) < kAlmost0) return 0;
959 Double_t tmp=r00; r00=r11/det; r11=tmp/det;
961 if (r00 < 0.) return 0;
962 if (r11 < 0.) return 0;
964 Double_t dy = fP[0] - p[0];
965 Double_t dz = fP[1] - p[1];
967 res[0]=dy*TMath::Sqrt(r00);
968 res[1]=dz*TMath::Sqrt(r11);
973 Bool_t AliExternalTrackParam::Update(Double_t p[2], Double_t cov[3]) {
974 //------------------------------------------------------------------
975 // Update the track parameters with the space point "p" having
976 // the covariance matrix "cov"
977 //------------------------------------------------------------------
978 Double_t &fP0=fP[0], &fP1=fP[1], &fP2=fP[2], &fP3=fP[3], &fP4=fP[4];
981 &fC10=fC[1], &fC11=fC[2],
982 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
983 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
984 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
986 Double_t r00=cov[0], r01=cov[1], r11=cov[2];
987 r00+=fC00; r01+=fC10; r11+=fC11;
988 Double_t det=r00*r11 - r01*r01;
990 if (TMath::Abs(det) < kAlmost0) return kFALSE;
993 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
995 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
996 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
997 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
998 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
999 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
1001 Double_t dy=p[0] - fP0, dz=p[1] - fP1;
1002 Double_t sf=fP2 + k20*dy + k21*dz;
1003 if (TMath::Abs(sf) > kAlmost1) return kFALSE;
1005 fP0 += k00*dy + k01*dz;
1006 fP1 += k10*dy + k11*dz;
1008 fP3 += k30*dy + k31*dz;
1009 fP4 += k40*dy + k41*dz;
1011 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
1012 Double_t c12=fC21, c13=fC31, c14=fC41;
1014 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
1015 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
1016 fC40-=k00*c04+k01*c14;
1018 fC11-=k10*c01+k11*fC11;
1019 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
1020 fC41-=k10*c04+k11*c14;
1022 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
1023 fC42-=k20*c04+k21*c14;
1025 fC33-=k30*c03+k31*c13;
1026 fC43-=k30*c04+k31*c14;
1028 fC44-=k40*c04+k41*c14;
1034 AliExternalTrackParam::GetHelixParameters(Double_t hlx[6], Double_t b) const {
1035 //--------------------------------------------------------------------
1036 // External track parameters -> helix parameters
1037 // "b" - magnetic field (kG)
1038 //--------------------------------------------------------------------
1039 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1041 hlx[0]=fP[0]; hlx[1]=fP[1]; hlx[2]=fP[2]; hlx[3]=fP[3];
1043 hlx[5]=fX*cs - hlx[0]*sn; // x0
1044 hlx[0]=fX*sn + hlx[0]*cs; // y0
1046 hlx[2]=TMath::ASin(hlx[2]) + fAlpha; // phi0
1048 hlx[4]=GetC(b); // C
1052 static void Evaluate(const Double_t *h, Double_t t,
1053 Double_t r[3], //radius vector
1054 Double_t g[3], //first defivatives
1055 Double_t gg[3]) //second derivatives
1057 //--------------------------------------------------------------------
1058 // Calculate position of a point on a track and some derivatives
1059 //--------------------------------------------------------------------
1060 Double_t phase=h[4]*t+h[2];
1061 Double_t sn=TMath::Sin(phase), cs=TMath::Cos(phase);
1065 if (TMath::Abs(h[4])>kAlmost0) {
1066 r[0] += (sn - h[6])/h[4];
1067 r[1] -= (cs - h[7])/h[4];
1069 r[2] = h[1] + h[3]*t;
1071 g[0] = cs; g[1]=sn; g[2]=h[3];
1073 gg[0]=-h[4]*sn; gg[1]=h[4]*cs; gg[2]=0.;
1076 Double_t AliExternalTrackParam::GetDCA(const AliExternalTrackParam *p,
1077 Double_t b, Double_t &xthis, Double_t &xp) const {
1078 //------------------------------------------------------------
1079 // Returns the (weighed !) distance of closest approach between
1080 // this track and the track "p".
1081 // Other returned values:
1082 // xthis, xt - coordinates of tracks' reference planes at the DCA
1083 //-----------------------------------------------------------
1084 Double_t dy2=GetSigmaY2() + p->GetSigmaY2();
1085 Double_t dz2=GetSigmaZ2() + p->GetSigmaZ2();
1088 Double_t p1[8]; GetHelixParameters(p1,b);
1089 p1[6]=TMath::Sin(p1[2]); p1[7]=TMath::Cos(p1[2]);
1090 Double_t p2[8]; p->GetHelixParameters(p2,b);
1091 p2[6]=TMath::Sin(p2[2]); p2[7]=TMath::Cos(p2[2]);
1094 Double_t r1[3],g1[3],gg1[3]; Double_t t1=0.;
1095 Evaluate(p1,t1,r1,g1,gg1);
1096 Double_t r2[3],g2[3],gg2[3]; Double_t t2=0.;
1097 Evaluate(p2,t2,r2,g2,gg2);
1099 Double_t dx=r2[0]-r1[0], dy=r2[1]-r1[1], dz=r2[2]-r1[2];
1100 Double_t dm=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1104 Double_t gt1=-(dx*g1[0]/dx2 + dy*g1[1]/dy2 + dz*g1[2]/dz2);
1105 Double_t gt2=+(dx*g2[0]/dx2 + dy*g2[1]/dy2 + dz*g2[2]/dz2);
1106 Double_t h11=(g1[0]*g1[0] - dx*gg1[0])/dx2 +
1107 (g1[1]*g1[1] - dy*gg1[1])/dy2 +
1108 (g1[2]*g1[2] - dz*gg1[2])/dz2;
1109 Double_t h22=(g2[0]*g2[0] + dx*gg2[0])/dx2 +
1110 (g2[1]*g2[1] + dy*gg2[1])/dy2 +
1111 (g2[2]*g2[2] + dz*gg2[2])/dz2;
1112 Double_t h12=-(g1[0]*g2[0]/dx2 + g1[1]*g2[1]/dy2 + g1[2]*g2[2]/dz2);
1114 Double_t det=h11*h22-h12*h12;
1117 if (TMath::Abs(det)<1.e-33) {
1118 //(quasi)singular Hessian
1121 dt1=-(gt1*h22 - gt2*h12)/det;
1122 dt2=-(h11*gt2 - h12*gt1)/det;
1125 if ((dt1*gt1+dt2*gt2)>0) {dt1=-dt1; dt2=-dt2;}
1127 //check delta(phase1) ?
1128 //check delta(phase2) ?
1130 if (TMath::Abs(dt1)/(TMath::Abs(t1)+1.e-3) < 1.e-4)
1131 if (TMath::Abs(dt2)/(TMath::Abs(t2)+1.e-3) < 1.e-4) {
1132 if ((gt1*gt1+gt2*gt2) > 1.e-4/dy2/dy2)
1133 AliDebug(1," stopped at not a stationary point !");
1134 Double_t lmb=h11+h22; lmb=lmb-TMath::Sqrt(lmb*lmb-4*det);
1136 AliDebug(1," stopped at not a minimum !");
1141 for (Int_t div=1 ; ; div*=2) {
1142 Evaluate(p1,t1+dt1,r1,g1,gg1);
1143 Evaluate(p2,t2+dt2,r2,g2,gg2);
1144 dx=r2[0]-r1[0]; dy=r2[1]-r1[1]; dz=r2[2]-r1[2];
1145 dd=dx*dx/dx2 + dy*dy/dy2 + dz*dz/dz2;
1149 AliDebug(1," overshoot !"); break;
1159 if (max<=0) AliDebug(1," too many iterations !");
1161 Double_t cs=TMath::Cos(GetAlpha());
1162 Double_t sn=TMath::Sin(GetAlpha());
1163 xthis=r1[0]*cs + r1[1]*sn;
1165 cs=TMath::Cos(p->GetAlpha());
1166 sn=TMath::Sin(p->GetAlpha());
1167 xp=r2[0]*cs + r2[1]*sn;
1169 return TMath::Sqrt(dm*TMath::Sqrt(dy2*dz2));
1172 Double_t AliExternalTrackParam::
1173 PropagateToDCA(AliExternalTrackParam *p, Double_t b) {
1174 //--------------------------------------------------------------
1175 // Propagates this track and the argument track to the position of the
1176 // distance of closest approach.
1177 // Returns the (weighed !) distance of closest approach.
1178 //--------------------------------------------------------------
1180 Double_t dca=GetDCA(p,b,xthis,xp);
1182 if (!PropagateTo(xthis,b)) {
1183 //AliWarning(" propagation failed !");
1187 if (!p->PropagateTo(xp,b)) {
1188 //AliWarning(" propagation failed !";
1196 Bool_t AliExternalTrackParam::PropagateToDCA(const AliVVertex *vtx,
1197 Double_t b, Double_t maxd, Double_t dz[2], Double_t covar[3]) {
1199 // Propagate this track to the DCA to vertex "vtx",
1200 // if the (rough) transverse impact parameter is not bigger then "maxd".
1201 // Magnetic field is "b" (kG).
1203 // a) The track gets extapolated to the DCA to the vertex.
1204 // b) The impact parameters and their covariance matrix are calculated.
1206 // In the case of success, the returned value is kTRUE
1207 // (otherwise, it's kFALSE)
1209 Double_t alpha=GetAlpha();
1210 Double_t sn=TMath::Sin(alpha), cs=TMath::Cos(alpha);
1211 Double_t x=GetX(), y=GetParameter()[0], snp=GetParameter()[2];
1212 Double_t xv= vtx->GetX()*cs + vtx->GetY()*sn;
1213 Double_t yv=-vtx->GetX()*sn + vtx->GetY()*cs, zv=vtx->GetZ();
1216 //Estimate the impact parameter neglecting the track curvature
1217 Double_t d=TMath::Abs(x*snp - y*TMath::Sqrt(1.- snp*snp));
1218 if (d > maxd) return kFALSE;
1220 //Propagate to the DCA
1221 Double_t crv=GetC(b);
1222 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1224 Double_t tgfv=-(crv*x - snp)/(crv*y + TMath::Sqrt(1.-snp*snp));
1225 sn=tgfv/TMath::Sqrt(1.+ tgfv*tgfv); cs=TMath::Sqrt(1.- sn*sn);
1226 if (TMath::Abs(tgfv)>0.) cs = sn/tgfv;
1230 yv=-xv*sn + yv*cs; xv=x;
1232 if (!Propagate(alpha+TMath::ASin(sn),xv,b)) return kFALSE;
1234 if (dz==0) return kTRUE;
1235 dz[0] = GetParameter()[0] - yv;
1236 dz[1] = GetParameter()[1] - zv;
1238 if (covar==0) return kTRUE;
1239 Double_t cov[6]; vtx->GetCovarianceMatrix(cov);
1241 //***** Improvements by A.Dainese
1242 alpha=GetAlpha(); sn=TMath::Sin(alpha); cs=TMath::Cos(alpha);
1243 Double_t s2ylocvtx = cov[0]*sn*sn + cov[2]*cs*cs - 2.*cov[1]*cs*sn;
1244 covar[0] = GetCovariance()[0] + s2ylocvtx; // neglecting correlations
1245 covar[1] = GetCovariance()[1]; // between (x,y) and z
1246 covar[2] = GetCovariance()[2] + cov[5]; // in vertex's covariance matrix
1253 void AliExternalTrackParam::GetDirection(Double_t d[3]) const {
1254 //----------------------------------------------------------------
1255 // This function returns a unit vector along the track direction
1256 // in the global coordinate system.
1257 //----------------------------------------------------------------
1258 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1260 Double_t csp =TMath::Sqrt((1.- snp)*(1.+snp));
1261 Double_t norm=TMath::Sqrt(1.+ fP[3]*fP[3]);
1262 d[0]=(csp*cs - snp*sn)/norm;
1263 d[1]=(snp*cs + csp*sn)/norm;
1267 Bool_t AliExternalTrackParam::GetPxPyPz(Double_t p[3]) const {
1268 //---------------------------------------------------------------------
1269 // This function returns the global track momentum components
1270 // Results for (nearly) straight tracks are meaningless !
1271 //---------------------------------------------------------------------
1272 p[0]=fP[4]; p[1]=fP[2]; p[2]=fP[3];
1273 return Local2GlobalMomentum(p,fAlpha);
1276 Double_t AliExternalTrackParam::Px() const {
1277 //---------------------------------------------------------------------
1278 // Returns x-component of momentum
1279 // Result for (nearly) straight tracks is meaningless !
1280 //---------------------------------------------------------------------
1282 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1288 Double_t AliExternalTrackParam::Py() const {
1289 //---------------------------------------------------------------------
1290 // Returns y-component of momentum
1291 // Result for (nearly) straight tracks is meaningless !
1292 //---------------------------------------------------------------------
1294 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1300 Double_t AliExternalTrackParam::Pz() const {
1301 //---------------------------------------------------------------------
1302 // Returns z-component of momentum
1303 // Result for (nearly) straight tracks is meaningless !
1304 //---------------------------------------------------------------------
1306 Double_t p[3]={kVeryBig,kVeryBig,kVeryBig};
1312 Double_t AliExternalTrackParam::Xv() const {
1313 //---------------------------------------------------------------------
1314 // Returns x-component of first track point
1315 //---------------------------------------------------------------------
1317 Double_t r[3]={0.,0.,0.};
1323 Double_t AliExternalTrackParam::Yv() const {
1324 //---------------------------------------------------------------------
1325 // Returns y-component of first track point
1326 //---------------------------------------------------------------------
1328 Double_t r[3]={0.,0.,0.};
1334 Double_t AliExternalTrackParam::Zv() const {
1335 //---------------------------------------------------------------------
1336 // Returns z-component of first track point
1337 //---------------------------------------------------------------------
1339 Double_t r[3]={0.,0.,0.};
1345 Double_t AliExternalTrackParam::Theta() const {
1346 // return theta angle of momentum
1348 return 0.5*TMath::Pi() - TMath::ATan(fP[3]);
1351 Double_t AliExternalTrackParam::Phi() const {
1352 //---------------------------------------------------------------------
1353 // Returns the azimuthal angle of momentum
1355 //---------------------------------------------------------------------
1357 Double_t phi=TMath::ASin(fP[2]) + fAlpha;
1358 if (phi<0.) phi+=2.*TMath::Pi();
1359 else if (phi>=2.*TMath::Pi()) phi-=2.*TMath::Pi();
1364 Double_t AliExternalTrackParam::M() const {
1365 // return particle mass
1367 // No mass information available so far.
1368 // Redifine in derived class!
1373 Double_t AliExternalTrackParam::E() const {
1374 // return particle energy
1376 // No PID information available so far.
1377 // Redifine in derived class!
1382 Double_t AliExternalTrackParam::Eta() const {
1383 // return pseudorapidity
1385 return -TMath::Log(TMath::Tan(0.5 * Theta()));
1388 Double_t AliExternalTrackParam::Y() const {
1391 // No PID information available so far.
1392 // Redifine in derived class!
1397 Bool_t AliExternalTrackParam::GetXYZ(Double_t *r) const {
1398 //---------------------------------------------------------------------
1399 // This function returns the global track position
1400 //---------------------------------------------------------------------
1401 r[0]=fX; r[1]=fP[0]; r[2]=fP[1];
1402 return Local2GlobalPosition(r,fAlpha);
1405 Bool_t AliExternalTrackParam::GetCovarianceXYZPxPyPz(Double_t cv[21]) const {
1406 //---------------------------------------------------------------------
1407 // This function returns the global covariance matrix of the track params
1409 // Cov(x,x) ... : cv[0]
1410 // Cov(y,x) ... : cv[1] cv[2]
1411 // Cov(z,x) ... : cv[3] cv[4] cv[5]
1412 // Cov(px,x)... : cv[6] cv[7] cv[8] cv[9]
1413 // Cov(py,x)... : cv[10] cv[11] cv[12] cv[13] cv[14]
1414 // Cov(pz,x)... : cv[15] cv[16] cv[17] cv[18] cv[19] cv[20]
1416 // Results for (nearly) straight tracks are meaningless !
1417 //---------------------------------------------------------------------
1418 if (TMath::Abs(fP[4])<=kAlmost0) {
1419 for (Int_t i=0; i<21; i++) cv[i]=0.;
1422 if (TMath::Abs(fP[2]) > kAlmost1) {
1423 for (Int_t i=0; i<21; i++) cv[i]=0.;
1426 Double_t pt=1./TMath::Abs(fP[4]);
1427 Double_t cs=TMath::Cos(fAlpha), sn=TMath::Sin(fAlpha);
1428 Double_t r=TMath::Sqrt((1.-fP[2])*(1.+fP[2]));
1430 Double_t m00=-sn, m10=cs;
1431 Double_t m23=-pt*(sn + fP[2]*cs/r), m43=-pt*pt*(r*cs - fP[2]*sn);
1432 Double_t m24= pt*(cs - fP[2]*sn/r), m44=-pt*pt*(r*sn + fP[2]*cs);
1433 Double_t m35=pt, m45=-pt*pt*fP[3];
1439 cv[0 ] = fC[0]*m00*m00;
1440 cv[1 ] = fC[0]*m00*m10;
1441 cv[2 ] = fC[0]*m10*m10;
1445 cv[6 ] = m00*(fC[3]*m23 + fC[10]*m43);
1446 cv[7 ] = m10*(fC[3]*m23 + fC[10]*m43);
1447 cv[8 ] = fC[4]*m23 + fC[11]*m43;
1448 cv[9 ] = m23*(fC[5]*m23 + fC[12]*m43) + m43*(fC[12]*m23 + fC[14]*m43);
1449 cv[10] = m00*(fC[3]*m24 + fC[10]*m44);
1450 cv[11] = m10*(fC[3]*m24 + fC[10]*m44);
1451 cv[12] = fC[4]*m24 + fC[11]*m44;
1452 cv[13] = m23*(fC[5]*m24 + fC[12]*m44) + m43*(fC[12]*m24 + fC[14]*m44);
1453 cv[14] = m24*(fC[5]*m24 + fC[12]*m44) + m44*(fC[12]*m24 + fC[14]*m44);
1454 cv[15] = m00*(fC[6]*m35 + fC[10]*m45);
1455 cv[16] = m10*(fC[6]*m35 + fC[10]*m45);
1456 cv[17] = fC[7]*m35 + fC[11]*m45;
1457 cv[18] = m23*(fC[8]*m35 + fC[12]*m45) + m43*(fC[13]*m35 + fC[14]*m45);
1458 cv[19] = m24*(fC[8]*m35 + fC[12]*m45) + m44*(fC[13]*m35 + fC[14]*m45);
1459 cv[20] = m35*(fC[9]*m35 + fC[13]*m45) + m45*(fC[13]*m35 + fC[14]*m45);
1466 AliExternalTrackParam::GetPxPyPzAt(Double_t x, Double_t b, Double_t *p) const {
1467 //---------------------------------------------------------------------
1468 // This function returns the global track momentum extrapolated to
1469 // the radial position "x" (cm) in the magnetic field "b" (kG)
1470 //---------------------------------------------------------------------
1472 p[1]=fP[2]+(x-fX)*GetC(b);
1474 return Local2GlobalMomentum(p,fAlpha);
1478 AliExternalTrackParam::GetYAt(Double_t x, Double_t b, Double_t &y) const {
1479 //---------------------------------------------------------------------
1480 // This function returns the local Y-coordinate of the intersection
1481 // point between this track and the reference plane "x" (cm).
1482 // Magnetic field "b" (kG)
1483 //---------------------------------------------------------------------
1485 if(TMath::Abs(dx)<=kAlmost0) {y=fP[0]; return kTRUE;}
1487 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1489 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1490 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1492 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1493 y = fP[0] + dx*(f1+f2)/(r1+r2);
1498 AliExternalTrackParam::GetZAt(Double_t x, Double_t b, Double_t &z) const {
1499 //---------------------------------------------------------------------
1500 // This function returns the local Z-coordinate of the intersection
1501 // point between this track and the reference plane "x" (cm).
1502 // Magnetic field "b" (kG)
1503 //---------------------------------------------------------------------
1505 if(TMath::Abs(dx)<=kAlmost0) {z=fP[1]; return kTRUE;}
1507 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1509 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1510 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1512 Double_t r1=sqrt((1.-f1)*(1.+f1)), r2=sqrt((1.-f2)*(1.+f2));
1513 z = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3]; // Many thanks to P.Hristov !
1518 AliExternalTrackParam::GetXYZAt(Double_t x, Double_t b, Double_t *r) const {
1519 //---------------------------------------------------------------------
1520 // This function returns the global track position extrapolated to
1521 // the radial position "x" (cm) in the magnetic field "b" (kG)
1522 //---------------------------------------------------------------------
1524 if(TMath::Abs(dx)<=kAlmost0) return GetXYZ(r);
1526 Double_t f1=fP[2], f2=f1 + dx*GetC(b);
1528 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1529 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1531 Double_t r1=TMath::Sqrt((1.-f1)*(1.+f1)), r2=TMath::Sqrt((1.-f2)*(1.+f2));
1533 r[1] = fP[0] + dx*(f1+f2)/(r1+r2);
1534 r[2] = fP[1] + dx*(r2 + f2*(f1+f2)/(r1+r2))*fP[3];//Thanks to Andrea & Peter
1536 return Local2GlobalPosition(r,fAlpha);
1539 //_____________________________________________________________________________
1540 void AliExternalTrackParam::Print(Option_t* /*option*/) const
1542 // print the parameters and the covariance matrix
1544 printf("AliExternalTrackParam: x = %-12g alpha = %-12g\n", fX, fAlpha);
1545 printf(" parameters: %12g %12g %12g %12g %12g\n",
1546 fP[0], fP[1], fP[2], fP[3], fP[4]);
1547 printf(" covariance: %12g\n", fC[0]);
1548 printf(" %12g %12g\n", fC[1], fC[2]);
1549 printf(" %12g %12g %12g\n", fC[3], fC[4], fC[5]);
1550 printf(" %12g %12g %12g %12g\n",
1551 fC[6], fC[7], fC[8], fC[9]);
1552 printf(" %12g %12g %12g %12g %12g\n",
1553 fC[10], fC[11], fC[12], fC[13], fC[14]);
1556 Double_t AliExternalTrackParam::GetSnpAt(Double_t x,Double_t b) const {
1558 // Get sinus at given x
1560 Double_t crv=GetC(b);
1561 if (TMath::Abs(b) < kAlmost0Field) crv=0.;
1563 Double_t res = fP[2]+dx*crv;
1567 Bool_t AliExternalTrackParam::GetDistance(AliExternalTrackParam *param2, Double_t x, Double_t dist[3], Double_t bz){
1568 //------------------------------------------------------------------------
1569 // Get the distance between two tracks at the local position x
1570 // working in the local frame of this track.
1571 // Origin : Marian.Ivanov@cern.ch
1572 //-----------------------------------------------------------------------
1576 if (!GetYAt(x,bz,xyz[1])) return kFALSE;
1577 if (!GetZAt(x,bz,xyz[2])) return kFALSE;
1580 if (TMath::Abs(GetAlpha()-param2->GetAlpha())<kAlmost0){
1582 if (!param2->GetYAt(x,bz,xyz2[1])) return kFALSE;
1583 if (!param2->GetZAt(x,bz,xyz2[2])) return kFALSE;
1587 Double_t dfi = param2->GetAlpha()-GetAlpha();
1588 Double_t ca = TMath::Cos(dfi), sa = TMath::Sin(dfi);
1589 xyz2[0] = xyz[0]*ca+xyz[1]*sa;
1590 xyz2[1] = -xyz[0]*sa+xyz[1]*ca;
1593 if (!param2->GetYAt(xyz2[0],bz,xyz1[1])) return kFALSE;
1594 if (!param2->GetZAt(xyz2[0],bz,xyz1[2])) return kFALSE;
1596 xyz2[0] = xyz1[0]*ca-xyz1[1]*sa;
1597 xyz2[1] = +xyz1[0]*sa+xyz1[1]*ca;
1600 dist[0] = xyz[0]-xyz2[0];
1601 dist[1] = xyz[1]-xyz2[1];
1602 dist[2] = xyz[2]-xyz2[2];
1609 // Draw functionality.
1610 // Origin: Marian Ivanov, Marian.Ivanov@cern.ch
1613 void AliExternalTrackParam::DrawTrack(Float_t magf, Float_t minR, Float_t maxR, Float_t stepR){
1617 if (minR>maxR) return ;
1618 if (stepR<=0) return ;
1619 Int_t npoints = TMath::Nint((maxR-minR)/stepR)+1;
1620 if (npoints<1) return;
1621 TPolyMarker3D *polymarker = new TPolyMarker3D(npoints);
1622 FillPolymarker(polymarker, magf,minR,maxR,stepR);
1627 void AliExternalTrackParam::FillPolymarker(TPolyMarker3D *pol, Float_t magF, Float_t minR, Float_t maxR, Float_t stepR){
1629 // Fill points in the polymarker
1632 for (Double_t r=minR; r<maxR; r+=stepR){
1634 GetXYZAt(r,magF,point);
1635 pol->SetPoint(counter,point[0],point[1], point[2]);
1636 printf("xyz\t%f\t%f\t%f\n",point[0], point[1],point[2]);
1641 Int_t AliExternalTrackParam::GetIndex(Int_t i, Int_t j) const {
1643 Int_t min = TMath::Min(i,j);
1644 Int_t max = TMath::Max(i,j);
1646 return min+(max+1)*max/2;
1650 void AliExternalTrackParam::g3helx3(Double_t qfield,
1653 /******************************************************************
1655 * GEANT3 tracking routine in a constant field oriented *
1657 * Tracking is performed with a conventional *
1658 * helix step method *
1660 * Authors R.Brun, M.Hansroul ********* *
1661 * Rewritten V.Perevoztchikov *
1663 * Rewritten in C++ by I.Belikov *
1665 * qfield (kG) - particle charge times magnetic field *
1666 * step (cm) - step length along the helix *
1667 * vect[7](cm,GeV/c) - input/output x, y, z, px/p, py/p ,pz/p, p *
1669 ******************************************************************/
1670 const Int_t ix=0, iy=1, iz=2, ipx=3, ipy=4, ipz=5, ipp=6;
1672 Double_t cosx=vect[ipx], cosy=vect[ipy], cosz=vect[ipz];
1674 Double_t rho = qfield*kB2C/vect[ipp];
1675 Double_t tet = rho*step;
1677 Double_t tsint, sintt, sint, cos1t;
1678 if (TMath::Abs(tet) > 0.15) {
1679 sint = TMath::Sin(tet);
1681 tsint = (tet - sint)/tet;
1682 Double_t t=TMath::Sin(0.5*tet);
1691 Double_t f1 = step*sintt;
1692 Double_t f2 = step*cos1t;
1693 Double_t f3 = step*tsint*cosz;
1694 Double_t f4 = -tet*cos1t;
1697 vect[ix] += f1*cosx - f2*cosy;
1698 vect[iy] += f1*cosy + f2*cosx;
1699 vect[iz] += f1*cosz + f3;
1701 vect[ipx] += f4*cosx - f5*cosy;
1702 vect[ipy] += f4*cosy + f5*cosx;
1706 Bool_t AliExternalTrackParam::PropagateToBxByBz(Double_t xk, const Double_t b[3]) {
1707 //----------------------------------------------------------------
1708 // Extrapolate this track to the plane X=xk in the field b[].
1710 // X [cm] is in the "tracking coordinate system" of this track.
1711 // b[]={Bx,By,Bz} [kG] is in the Global coordidate system.
1712 //----------------------------------------------------------------
1715 if (TMath::Abs(dx)<=kAlmost0) return kTRUE;
1717 Double_t crv=GetC(b[2]);
1718 if (TMath::Abs(b[2]) < kAlmost0Field) crv=0.;
1720 Double_t f1=fP[2], f2=f1 + crv*dx;
1721 if (TMath::Abs(f1) >= kAlmost1) return kFALSE;
1722 if (TMath::Abs(f2) >= kAlmost1) return kFALSE;
1725 // Estimate the covariance matrix
1726 Double_t &fP3=fP[3], &fP4=fP[4];
1729 &fC10=fC[1], &fC11=fC[2],
1730 &fC20=fC[3], &fC21=fC[4], &fC22=fC[5],
1731 &fC30=fC[6], &fC31=fC[7], &fC32=fC[8], &fC33=fC[9],
1732 &fC40=fC[10], &fC41=fC[11], &fC42=fC[12], &fC43=fC[13], &fC44=fC[14];
1734 Double_t r1=TMath::Sqrt(1.- f1*f1), r2=TMath::Sqrt(1.- f2*f2);
1737 Double_t f02= dx/(r1*r1*r1); Double_t cc=crv/fP4;
1738 Double_t f04=0.5*dx*dx/(r1*r1*r1); f04*=cc;
1739 Double_t f12= dx*fP3*f1/(r1*r1*r1);
1740 Double_t f14=0.5*dx*dx*fP3*f1/(r1*r1*r1); f14*=cc;
1741 Double_t f13= dx/r1;
1742 Double_t f24= dx; f24*=cc;
1745 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
1746 Double_t b02=f24*fC40;
1747 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
1748 Double_t b12=f24*fC41;
1749 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
1750 Double_t b22=f24*fC42;
1751 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
1752 Double_t b42=f24*fC44;
1753 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
1754 Double_t b32=f24*fC43;
1757 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a02=f02*b22+f04*b42;
1758 Double_t a11=f12*b21+f14*b41+f13*b31,a12=f12*b22+f14*b42+f13*b32;
1759 Double_t a22=f24*b42;
1761 //F*C*Ft = C + (b + bt + a)
1762 fC00 += b00 + b00 + a00;
1763 fC10 += b10 + b01 + a01;
1764 fC20 += b20 + b02 + a02;
1767 fC11 += b11 + b11 + a11;
1768 fC21 += b21 + b12 + a12;
1771 fC22 += b22 + b22 + a22;
1776 // Appoximate step length
1777 Double_t step=dx*TMath::Abs(r2 + f2*(f1+f2)/(r1+r2));
1778 step *= TMath::Sqrt(1.+ GetTgl()*GetTgl());
1781 // Get the track's (x,y,z) and (px,py,pz) in the Global System
1782 Double_t r[3]; GetXYZ(r);
1783 Double_t p[3]; GetPxPyPz(p);
1790 // Rotate to the system where Bx=By=0.
1791 Double_t bt=TMath::Sqrt(b[0]*b[0] + b[1]*b[1]);
1792 Double_t cosphi=1., sinphi=0.;
1793 if (bt > kAlmost0) {cosphi=b[0]/bt; sinphi=b[1]/bt;}
1794 Double_t bb=TMath::Sqrt(b[0]*b[0] + b[1]*b[1] + b[2]*b[2]);
1795 Double_t costet=1., sintet=0.;
1796 if (bb > kAlmost0) {costet=b[2]/bb; sintet=bt/bb;}
1799 vect[0] = costet*cosphi*r[0] + costet*sinphi*r[1] - sintet*r[2];
1800 vect[1] = -sinphi*r[0] + cosphi*r[1];
1801 vect[2] = sintet*cosphi*r[0] + sintet*sinphi*r[1] + costet*r[2];
1803 vect[3] = costet*cosphi*p[0] + costet*sinphi*p[1] - sintet*p[2];
1804 vect[4] = -sinphi*p[0] + cosphi*p[1];
1805 vect[5] = sintet*cosphi*p[0] + sintet*sinphi*p[1] + costet*p[2];
1810 // Do the helix step
1811 g3helx3(GetSign()*bb,step,vect);
1814 // Rotate back to the Global System
1815 r[0] = cosphi*costet*vect[0] - sinphi*vect[1] + cosphi*sintet*vect[2];
1816 r[1] = sinphi*costet*vect[0] + cosphi*vect[1] + sinphi*sintet*vect[2];
1817 r[2] = -sintet*vect[0] + costet*vect[2];
1819 p[0] = cosphi*costet*vect[3] - sinphi*vect[4] + cosphi*sintet*vect[5];
1820 p[1] = sinphi*costet*vect[3] + cosphi*vect[4] + sinphi*sintet*vect[5];
1821 p[2] = -sintet*vect[3] + costet*vect[5];
1824 // Rotate back to the Tracking System
1825 Double_t cosalp = TMath::Cos(fAlpha);
1826 Double_t sinalp =-TMath::Sin(fAlpha);
1829 t = cosalp*r[0] - sinalp*r[1];
1830 r[1] = sinalp*r[0] + cosalp*r[1];
1833 t = cosalp*p[0] - sinalp*p[1];
1834 p[1] = sinalp*p[0] + cosalp*p[1];
1838 // Do the final correcting step to the target plane (linear approximation)
1839 Double_t x=r[0], y=r[1], z=r[2];
1840 if (TMath::Abs(dx) > kAlmost0) {
1841 if (TMath::Abs(p[0]) < kAlmost0) return kFALSE;
1849 // Calculate the track parameters
1850 t=TMath::Sqrt(p[0]*p[0] + p[1]*p[1]);
1856 fP[4] = GetSign()/(t*pp);
1861 Bool_t AliExternalTrackParam::Translate(Double_t *vTrasl,Double_t *covV){
1863 //Translation: in the event mixing, the tracks can be shifted
1864 //of the difference among primary vertices (vTrasl) and
1865 //the covariance matrix is changed accordingly
1866 //(covV = covariance of the primary vertex).
1867 //Origin: "Romita, Rossella" <R.Romita@gsi.de>
1869 TVector3 translation;
1870 // vTrasl coordinates in the local system
1871 translation.SetXYZ(vTrasl[0],vTrasl[1],vTrasl[2]);
1872 translation.RotateZ(-fAlpha);
1873 translation.GetXYZ(vTrasl);
1875 //compute the new x,y,z of the track
1876 Double_t newX=fX-vTrasl[0];
1877 Double_t newY=fP[0]-vTrasl[1];
1878 Double_t newZ=fP[1]-vTrasl[2];
1880 //define the new parameters
1881 Double_t newParam[5];
1888 // recompute the covariance matrix:
1889 // 1. covV in the local system
1890 Double_t cosRot=TMath::Cos(fAlpha), sinRot=TMath::Sin(fAlpha);
1911 if(uUi.Determinant() <= 0.) {return kFALSE;}
1912 TMatrixD uUiQi(uUi,TMatrixD::kMult,qQi);
1913 TMatrixD m(qQi,TMatrixD::kTransposeMult,uUiQi);
1915 //2. compute the new covariance matrix of the track
1916 Double_t sigmaXX=m(0,0);
1917 Double_t sigmaXZ=m(2,0);
1918 Double_t sigmaXY=m(1,0);
1919 Double_t sigmaYY=GetSigmaY2()+m(1,1);
1920 Double_t sigmaYZ=fC[1]+m(1,2);
1921 Double_t sigmaZZ=fC[2]+m(2,2);
1922 Double_t covarianceYY=sigmaYY + (-1.)*((sigmaXY*sigmaXY)/sigmaXX);
1923 Double_t covarianceYZ=sigmaYZ-(sigmaXZ*sigmaXY/sigmaXX);
1924 Double_t covarianceZZ=sigmaZZ-((sigmaXZ*sigmaXZ)/sigmaXX);
1926 Double_t newCov[15];
1927 newCov[0]=covarianceYY;
1928 newCov[1]=covarianceYZ;
1929 newCov[2]=covarianceZZ;
1930 for(Int_t i=3;i<15;i++){
1934 // set the new parameters
1936 Set(newX,fAlpha,newParam,newCov);