1 /**************************************************************************
2 * Copyright(c) 1998-1999, ALICE Experiment at CERN, All rights reserved. *
4 * Author: The ALICE Off-line Project. *
5 * Contributors are mentioned in the code where appropriate. *
7 * Permission to use, copy, modify and distribute this software and its *
8 * documentation strictly for non-commercial purposes is hereby granted *
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11 * appear in the supporting documentation. The authors make no claims *
12 * about the suitability of this software for any purpose. It is *
13 * provided "as is" without express or implied warranty. *
14 **************************************************************************/
18 //-----------------------------------------------------------------
19 // Implementation of the TPC track class
20 // This class is used by the AliTPCtracker class
21 // Origin: Iouri Belikov, CERN, Jouri.Belikov@cern.ch
22 //-----------------------------------------------------------------
24 #include <Riostream.h>
26 #include "AliTPCtrack.h"
27 #include "AliCluster.h"
28 #include "AliBarrelTrack.h"
29 #include "AliESDtrack.h"
33 //_________________________________________________________________________
34 AliTPCtrack::AliTPCtrack(): AliKalmanTrack()
36 //-------------------------------------------------
37 // default constructor
38 //-------------------------------------------------
39 fX = fP0 = fP1 = fP2 = fP3 = fP3 = fP4 = 0.0;
41 fNWrong = fNRotation = fNumber = 0; // [SR, 01.04.2003]
44 //_________________________________________________________________________
45 AliTPCtrack::AliTPCtrack(UInt_t index, const Double_t xx[5],
46 const Double_t cc[15], Double_t xref, Double_t alpha) : AliKalmanTrack() {
47 //-----------------------------------------------------------------
48 // This is the main track constructor.
49 //-----------------------------------------------------------------
52 if (fAlpha<-TMath::Pi()) fAlpha += 2*TMath::Pi();
53 if (fAlpha>=TMath::Pi()) fAlpha -= 2*TMath::Pi();
56 fP0=xx[0]; fP1=xx[1]; fP2=xx[2]; fP3=xx[3]; fP4=xx[4];
59 fC10=cc[1]; fC11=cc[2];
60 fC20=cc[3]; fC21=cc[4]; fC22=cc[5];
61 fC30=cc[6]; fC31=cc[7]; fC32=cc[8]; fC33=cc[9];
62 fC40=cc[10]; fC41=cc[11]; fC42=cc[12]; fC43=cc[13]; fC44=cc[14];
65 SetNumberOfClusters(1);
78 //_____________________________________________________________________________
79 AliTPCtrack::AliTPCtrack(const AliKalmanTrack& t,Double_t alpha) :
81 //-----------------------------------------------------------------
82 // Conversion AliKalmanTrack -> AliTPCtrack.
83 //-----------------------------------------------------------------
85 SetNumberOfClusters(0);
89 if (fAlpha < -TMath::Pi()) fAlpha += 2*TMath::Pi();
90 else if (fAlpha >= TMath::Pi()) fAlpha -= 2*TMath::Pi();
92 //Conversion of the track parameters
93 Double_t x,p[5]; t.GetExternalParameters(x,p);
94 fX=x; x=GetConvConst();
101 //Conversion of the covariance matrix
102 Double_t c[15]; t.GetExternalCovariance(c);
103 c[10]/=x; c[11]/=x; c[12]/=x; c[13]/=x; c[14]/=x*x;
105 Double_t c22=fX*fX*c[14] - 2*fX*c[12] + c[5];
106 Double_t c32=fX*c[13] - c[8];
107 Double_t c20=fX*c[10] - c[3], c21=fX*c[11] - c[4], c42=fX*c[14] - c[12];
110 fC10=c[1 ]; fC11=c[2 ];
111 fC20=c20; fC21=c21; fC22=c22;
112 fC30=c[6 ]; fC31=c[7 ]; fC32=c32; fC33=c[9 ];
113 fC40=c[10]; fC41=c[11]; fC42=c42; fC43=c[13]; fC44=c[14];
126 //_____________________________________________________________________________
127 AliTPCtrack::AliTPCtrack(const AliESDtrack& t) : AliKalmanTrack() {
128 //-----------------------------------------------------------------
129 // Conversion AliESDtrack -> AliTPCtrack.
130 //-----------------------------------------------------------------
131 SetNumberOfClusters(t.GetTPCclusters(fIndex));
132 SetLabel(t.GetLabel());
133 SetMass(t.GetMass());
135 fdEdx = t.GetTPCsignal();
136 fAlpha = t.GetAlpha();
137 if (fAlpha < -TMath::Pi()) fAlpha += 2*TMath::Pi();
138 else if (fAlpha >= TMath::Pi()) fAlpha -= 2*TMath::Pi();
140 //Conversion of the track parameters
141 Double_t x,p[5]; t.GetExternalParameters(x,p);
142 fX=x; x=GetConvConst();
149 //Conversion of the covariance matrix
150 Double_t c[15]; t.GetExternalCovariance(c);
151 c[10]/=x; c[11]/=x; c[12]/=x; c[13]/=x; c[14]/=x*x;
153 Double_t c22=fX*fX*c[14] - 2*fX*c[12] + c[5];
154 Double_t c32=fX*c[13] - c[8];
155 Double_t c20=fX*c[10] - c[3], c21=fX*c[11] - c[4], c42=fX*c[14] - c[12];
158 fC10=c[1 ]; fC11=c[2 ];
159 fC20=c20; fC21=c21; fC22=c22;
160 fC30=c[6 ]; fC31=c[7 ]; fC32=c32; fC33=c[9 ];
161 fC40=c[10]; fC41=c[11]; fC42=c42; fC43=c[13]; fC44=c[14];
163 if ((t.GetStatus()&AliESDtrack::kTIME) == 0) return;
165 Double_t times[10]; t.GetIntegratedTimes(times); SetIntegratedTimes(times);
166 SetIntegratedLength(t.GetIntegratedLength());
177 // SetFakeRatio(t.GetTPCFakeRatio());
180 //_____________________________________________________________________________
181 AliTPCtrack::AliTPCtrack(const AliTPCtrack& t) : AliKalmanTrack(t) {
182 //-----------------------------------------------------------------
183 // This is a track copy constructor.
184 //-----------------------------------------------------------------
189 fP0=t.fP0; fP1=t.fP1; fP2=t.fP2; fP3=t.fP3; fP4=t.fP4;
192 fC10=t.fC10; fC11=t.fC11;
193 fC20=t.fC20; fC21=t.fC21; fC22=t.fC22;
194 fC30=t.fC30; fC31=t.fC31; fC32=t.fC32; fC33=t.fC33;
195 fC40=t.fC40; fC41=t.fC41; fC42=t.fC42; fC43=t.fC43; fC44=t.fC44;
197 //Int_t n=GetNumberOfClusters();
198 for (Int_t i=0; i<kMaxRow; i++) fIndex[i]=t.fIndex[i];
202 fNFoundable = t.fNFoundable;
203 fBConstrain = t.fBConstrain;
204 fLastPoint = t.fLastPoint;
205 fFirstPoint = t.fFirstPoint;
206 fRemoval = t.fRemoval ;
207 fTrackType = t.fTrackType;
211 //_____________________________________________________________________________
213 void AliTPCtrack::GetBarrelTrack(AliBarrelTrack *track) const{
215 // Create a Barrel Track out of this track
216 // Current track is propagated to the reference plane
222 Double_t xr, vec[5], cov[15];
224 track->SetLabel(GetLabel());
225 track->SetX(fX, fAlpha);
226 track->SetNClusters(GetNumberOfClusters(), GetChi2());
228 GetIntegratedTimes(times);
229 track->SetTime(times, GetIntegratedLength());
231 track->SetMass(GetMass());
232 track->SetdEdX(GetdEdx());
234 track->SetNWrongClusters(fNWrong);
235 track->SetNRotate(fNRotation);
237 GetExternalParameters(xr, vec);
238 track->SetStateVector(vec);
240 GetExternalCovariance(cov);
241 track->SetCovarianceMatrix(cov);
244 //_____________________________________________________________________________
245 Int_t AliTPCtrack::Compare(const TObject *o) const {
246 //-----------------------------------------------------------------
247 // This function compares tracks according to the their curvature
248 //-----------------------------------------------------------------
249 AliTPCtrack *t=(AliTPCtrack*)o;
250 //Double_t co=TMath::Abs(t->Get1Pt());
251 //Double_t c =TMath::Abs(Get1Pt());
252 Double_t co=t->GetSigmaY2()*t->GetSigmaZ2();
253 Double_t c =GetSigmaY2()*GetSigmaZ2();
255 else if (c<co) return -1;
259 //_____________________________________________________________________________
260 void AliTPCtrack::GetExternalCovariance(Double_t cc[15]) const {
261 //-------------------------------------------------------------------------
262 // This function returns an external representation of the covriance matrix.
263 // (See comments in AliTPCtrack.h about external track representation)
264 //-------------------------------------------------------------------------
265 Double_t a=GetConvConst();
267 Double_t c22=fX*fX*fC44-2*fX*fC42+fC22;
268 Double_t c32=fX*fC43-fC32;
269 Double_t c20=fX*fC40-fC20, c21=fX*fC41-fC21, c42=fX*fC44-fC42;
272 cc[1 ]=fC10; cc[2 ]=fC11;
273 cc[3 ]=c20; cc[4 ]=c21; cc[5 ]=c22;
274 cc[6 ]=fC30; cc[7 ]=fC31; cc[8 ]=c32; cc[9 ]=fC33;
275 cc[10]=fC40*a; cc[11]=fC41*a; cc[12]=c42*a; cc[13]=fC43*a; cc[14]=fC44*a*a;
279 //_____________________________________________________________________________
280 Double_t AliTPCtrack::GetPredictedChi2(const AliCluster *c) const
282 //-----------------------------------------------------------------
283 // This function calculates a predicted chi2 increment.
284 //-----------------------------------------------------------------
285 Double_t r00=c->GetSigmaY2(), r01=0., r11=c->GetSigmaZ2();
286 r00+=fC00; r01+=fC10; r11+=fC11;
288 Double_t det=r00*r11 - r01*r01;
289 if (TMath::Abs(det) < 1.e-10) {
290 Int_t n=GetNumberOfClusters();
291 if (n>4) cerr<<n<<" AliKalmanTrack warning: Singular matrix !\n";
294 Double_t tmp=r00; r00=r11; r11=tmp; r01=-r01;
296 Double_t dy=c->GetY() - fP0, dz=c->GetZ() - fP1;
298 return (dy*r00*dy + 2*r01*dy*dz + dz*r11*dz)/det;
301 Double_t AliTPCtrack::GetYat(Double_t xk) const {
302 //-----------------------------------------------------------------
303 // This function calculates the Y-coordinate of a track at the plane x=xk.
304 //-----------------------------------------------------------------
305 Double_t c1=fP4*fX - fP2, r1=TMath::Sqrt(1.- c1*c1);
306 Double_t c2=fP4*xk - fP2;
308 Int_t n=GetNumberOfClusters();
309 if (n>4) cerr<<n<<"AliTPCtrack::GetYat: can't evaluate the y-coord !\n";
312 Double_t r2=TMath::Sqrt(1.- c2*c2);
313 return fP0 + (xk-fX)*(c1+c2)/(r1+r2);
316 //_____________________________________________________________________________
317 Int_t AliTPCtrack::PropagateTo(Double_t xk,Double_t /*x0*/,Double_t rho) {
318 //-----------------------------------------------------------------
319 // This function propagates a track to a reference plane x=xk.
320 //-----------------------------------------------------------------
321 if (TMath::Abs(fP4*xk - fP2) >= 0.9) {
322 // Int_t n=GetNumberOfClusters();
323 //if (n>4) cerr<<n<<" AliTPCtrack warning: Propagation failed !\n";
327 // old position for time [SR, GSI 17.02.2003]
333 Double_t x1=fX, x2=x1+(xk-x1), dx=x2-x1, y1=fP0, z1=fP1;
334 Double_t c1=fP4*x1 - fP2, r1=sqrt(1.- c1*c1);
335 Double_t c2=fP4*x2 - fP2, r2=sqrt(1.- c2*c2);
337 fP0 += dx*(c1+c2)/(r1+r2);
338 fP1 += dx*(c1+c2)/(c1*r2 + c2*r1)*fP3;
341 Double_t rr=r1+r2, cc=c1+c2, xx=x1+x2;
342 Double_t f02=-dx*(2*rr + cc*(c1/r1 + c2/r2))/(rr*rr);
343 Double_t f04= dx*(rr*xx + cc*(c1*x1/r1+c2*x2/r2))/(rr*rr);
344 Double_t cr=c1*r2+c2*r1;
345 Double_t f12=-dx*fP3*(2*cr + cc*(c2*c1/r1-r1 + c1*c2/r2-r2))/(cr*cr);
346 Double_t f13= dx*cc/cr;
347 Double_t f14=dx*fP3*(cr*xx-cc*(r1*x2-c2*c1*x1/r1+r2*x1-c1*c2*x2/r2))/(cr*cr);
350 Double_t b00=f02*fC20 + f04*fC40, b01=f12*fC20 + f14*fC40 + f13*fC30;
351 Double_t b10=f02*fC21 + f04*fC41, b11=f12*fC21 + f14*fC41 + f13*fC31;
352 Double_t b20=f02*fC22 + f04*fC42, b21=f12*fC22 + f14*fC42 + f13*fC32;
353 Double_t b30=f02*fC32 + f04*fC43, b31=f12*fC32 + f14*fC43 + f13*fC33;
354 Double_t b40=f02*fC42 + f04*fC44, b41=f12*fC42 + f14*fC44 + f13*fC43;
357 Double_t a00=f02*b20+f04*b40,a01=f02*b21+f04*b41,a11=f12*b21+f14*b41+f13*b31;
359 //F*C*Ft = C + (a + b + bt)
361 fC10 += a01 + b01 + b10;
372 //Multiple scattering******************
373 Double_t d=sqrt((x1-fX)*(x1-fX)+(y1-fP0)*(y1-fP0)+(z1-fP1)*(z1-fP1));
374 Double_t p2=(1.+ GetTgl()*GetTgl())/(Get1Pt()*Get1Pt());
375 Double_t beta2=p2/(p2 + GetMass()*GetMass());
376 beta2 = TMath::Min(beta2,0.99999999999);
377 //Double_t theta2=14.1*14.1/(beta2*p2*1e6)*d/x0*rho;
378 Double_t theta2=1.0259e-6*10*10/20/(beta2*p2)*d*rho;
380 Double_t ey=fP4*fX - fP2, ez=fP3;
381 Double_t xz=fP4*ez, zz1=ez*ez+1, xy=fP2+ey;
383 fC22 += (2*ey*ez*ez*fP2+1-ey*ey+ez*ez+fP2*fP2*ez*ez)*theta2;
384 fC32 += ez*zz1*xy*theta2;
385 fC33 += zz1*zz1*theta2;
386 fC42 += xz*ez*xy*theta2;
387 fC43 += xz*zz1*theta2;
388 fC44 += xz*xz*theta2;
392 Double_t dc22 = (1-ey*ey+xz*xz*fX*fX)*theta2;
393 Double_t dc32 = (xz*fX*zz1)*theta2;
394 Double_t dc33 = (zz1*zz1)*theta2;
395 Double_t dc42 = (xz*fX*xz)*theta2;
396 Double_t dc43 = (zz1*xz)*theta2;
397 Double_t dc44 = (xz*xz)*theta2;
405 //Energy losses************************
406 Double_t dE=0.153e-3/beta2*(log(5940*beta2/(1-beta2)) - beta2)*d*rho;
410 //Double_t E = sqrt(p2+GetMass()*GetMass());
411 //Double_t mifac = TMath::Sqrt(1.+dE*dE/p2+2*E*dE/p2)-1;
412 //Double_t belfac = E*dE/p2;
414 fP4*=(1.- sqrt(p2+GetMass()*GetMass())/p2*dE);
417 // Integrated Time [SR, GSI, 17.02.2003]
419 if (IsStartedTimeIntegral()) {
420 Double_t l2 = (fX-oldX)*(fX-oldX)+(fP0-oldY)*(fP0-oldY)+(fP1-oldZ)*(fP1-oldZ);
421 AddTimeStep(TMath::Sqrt(l2));
428 //_____________________________________________________________________________
429 Int_t AliTPCtrack::PropagateToVertex(Double_t x0,Double_t rho)
431 //-----------------------------------------------------------------
432 // This function propagates tracks to the "vertex".
433 //-----------------------------------------------------------------
434 Double_t c=fP4*fX - fP2;
435 Double_t tgf=-fP2/(fP4*fP0 + sqrt(1-c*c));
436 Double_t snf=tgf/sqrt(1.+ tgf*tgf);
437 Double_t xv=(fP2+snf)/fP4;
438 return PropagateTo(xv,x0,rho);
441 //_____________________________________________________________________________
442 Int_t AliTPCtrack::Update(const AliCluster *c, Double_t chisq, UInt_t index) {
443 //-----------------------------------------------------------------
444 // This function associates a cluster with this track.
445 //-----------------------------------------------------------------
447 // update the number of wrong SR[20.03.2003]
448 Int_t absLabel = TMath::Abs(GetLabel());
449 if ( (c->GetLabel(0) != absLabel) &&
450 (c->GetLabel(0) != absLabel) &&
451 (c->GetLabel(0) != absLabel)) fNWrong++;
454 Double_t r00=c->GetSigmaY2(), r01=0., r11=c->GetSigmaZ2();
455 r00+=fC00; r01+=fC10; r11+=fC11;
456 Double_t det=r00*r11 - r01*r01;
457 Double_t tmp=r00; r00=r11/det; r11=tmp/det; r01=-r01/det;
459 Double_t k00=fC00*r00+fC10*r01, k01=fC00*r01+fC10*r11;
460 Double_t k10=fC10*r00+fC11*r01, k11=fC10*r01+fC11*r11;
461 Double_t k20=fC20*r00+fC21*r01, k21=fC20*r01+fC21*r11;
462 Double_t k30=fC30*r00+fC31*r01, k31=fC30*r01+fC31*r11;
463 Double_t k40=fC40*r00+fC41*r01, k41=fC40*r01+fC41*r11;
465 Double_t dy=c->GetY() - fP0, dz=c->GetZ() - fP1;
466 Double_t cur=fP4 + k40*dy + k41*dz, eta=fP2 + k20*dy + k21*dz;
467 if (TMath::Abs(cur*fX-eta) >= 0.9) {
468 // Int_t n=GetNumberOfClusters();
469 //if (n>4) cerr<<n<<" AliTPCtrack warning: Filtering failed !\n";
473 fP0 += k00*dy + k01*dz;
474 fP1 += k10*dy + k11*dz;
476 fP3 += k30*dy + k31*dz;
479 Double_t c01=fC10, c02=fC20, c03=fC30, c04=fC40;
480 Double_t c12=fC21, c13=fC31, c14=fC41;
482 fC00-=k00*fC00+k01*fC10; fC10-=k00*c01+k01*fC11;
483 fC20-=k00*c02+k01*c12; fC30-=k00*c03+k01*c13;
484 fC40-=k00*c04+k01*c14;
486 fC11-=k10*c01+k11*fC11;
487 fC21-=k10*c02+k11*c12; fC31-=k10*c03+k11*c13;
488 fC41-=k10*c04+k11*c14;
490 fC22-=k20*c02+k21*c12; fC32-=k20*c03+k21*c13;
491 fC42-=k20*c04+k21*c14;
493 fC33-=k30*c03+k31*c13;
494 fC43-=k40*c03+k41*c13;
496 fC44-=k40*c04+k41*c14;
498 Int_t n=GetNumberOfClusters();
500 SetNumberOfClusters(n+1);
501 SetChi2(GetChi2()+chisq);
506 //_____________________________________________________________________________
507 Int_t AliTPCtrack::Rotate(Double_t alpha)
509 //-----------------------------------------------------------------
510 // This function rotates this track.
511 //-----------------------------------------------------------------
513 if (alpha != 0) fNRotation++; // [SR, 01.04.2003]
516 if (fAlpha<-TMath::Pi()) fAlpha += 2*TMath::Pi();
517 if (fAlpha>=TMath::Pi()) fAlpha -= 2*TMath::Pi();
519 Double_t x1=fX, y1=fP0;
520 Double_t ca=cos(alpha), sa=sin(alpha);
521 Double_t r1=fP4*fX - fP2;
525 fP2=fP2*ca + (fP4*y1 + sqrt(1.- r1*r1))*sa;
527 Double_t r2=fP4*fX - fP2;
528 if (TMath::Abs(r2) >= 0.99999) {
529 Int_t n=GetNumberOfClusters();
530 if (n>4) cerr<<n<<" AliTPCtrack warning: Rotation failed !\n";
534 Double_t y0=fP0 + sqrt(1.- r2*r2)/fP4;
535 if ((fP0-y0)*fP4 >= 0.) {
536 Int_t n=GetNumberOfClusters();
537 if (n>4) cerr<<n<<" AliTPCtrack warning: Rotation failed !!!\n";
542 Double_t f00=ca-1, f24=(y1 - r1*x1/sqrt(1.- r1*r1))*sa,
543 f20=fP4*sa, f22=(ca + sa*r1/sqrt(1.- r1*r1))-1;
546 Double_t b00=fC00*f00, b02=fC00*f20+fC40*f24+fC20*f22;
547 Double_t b10=fC10*f00, b12=fC10*f20+fC41*f24+fC21*f22;
548 Double_t b20=fC20*f00, b22=fC20*f20+fC42*f24+fC22*f22;
549 Double_t b30=fC30*f00, b32=fC30*f20+fC43*f24+fC32*f22;
550 Double_t b40=fC40*f00, b42=fC40*f20+fC44*f24+fC42*f22;
553 Double_t a00=f00*b00, a02=f00*b02, a22=f20*b02+f24*b42+f22*b22;
555 // *** Double_t dy2=fCyy;
557 //F*C*Ft = C + (a + b + bt)
568 // *** fCyy+=dy2*sa*sa*r1*r1/(1.- r1*r1);
569 // *** fCzz+=d2y*sa*sa*fT*fT/(1.- r1*r1);
574 void AliTPCtrack::ResetCovariance() {
575 //------------------------------------------------------------------
576 //This function makes a track forget its history :)
577 //------------------------------------------------------------------
581 fC20=0.; fC21=0.; fC22*=10.;
582 fC30=0.; fC31=0.; fC32=0.; fC33*=10.;
583 fC40=0.; fC41=0.; fC42=0.; fC43=0.; fC44*=10.;
587 ////////////////////////////////////////////////////////////////////////
588 Double_t AliTPCtrack::Phi() const {
592 Double_t phi = TMath::ASin(GetSnp()) + fAlpha;
593 if (phi<0) phi+=2*TMath::Pi();
594 if (phi>=2*TMath::Pi()) phi-=2*TMath::Pi();
597 ////////////////////////////////////////////////////////////////////////
601 ////////////////////////////////////////////////////////////////////////
604 Float_t AliTPCtrack::Density(Int_t row0, Int_t row1)
607 // calculate cluster density
610 //if (row0<fFirstPoint) row0 = fFirstPoint;
611 if (row1>fLastPoint) row1 = fLastPoint;
614 for (Int_t i=row0;i<=row1;i++){
615 // Int_t index = fClusterIndex[i];
616 Int_t index = fIndex[i];
617 if (index!=-1) good++;
618 if (index>0) found++;
621 if (good>0) density = Float_t(found)/Float_t(good);
626 Float_t AliTPCtrack::Density2(Int_t row0, Int_t row1)
629 // calculate cluster density
633 for (Int_t i=row0;i<=row1;i++){
634 Int_t index = fIndex[i];
635 if (index!=-1) good++;
636 if (index>0) found++;
639 if (good>0) density = Float_t(found)/Float_t(good);
644 Double_t AliTPCtrack::GetZat0() const
647 // return virtual z - supposing that x = 0
648 if (TMath::Abs(fP2)>1) return 0;
649 if (TMath::Abs(fX*fP4-fP2)>1) return 0;
650 Double_t vz = fP1+fP3/fP4*(asin(-fP2)-asin(fX*fP4-fP2));
655 Double_t AliTPCtrack::GetD(Double_t x, Double_t y) const {
656 //------------------------------------------------------------------
657 // This function calculates the transverse impact parameter
658 // with respect to a point with global coordinates (x,y)
659 //------------------------------------------------------------------
660 //Double_t xt=fX, yt=fP0;
662 Double_t sn=TMath::Sin(fAlpha), cs=TMath::Cos(fAlpha);
663 Double_t a = x*cs + y*sn;
664 y = -x*sn + y*cs; x=a;
666 Double_t r = TMath::Abs(1/fP4);
667 Double_t x0 = TMath::Abs(fP2*r);
669 y0= fP0+TMath::Sqrt(1-(fP4*fX-fP2)*(fP4*fX-fP2))/fP4;
671 Double_t delta = TMath::Sqrt((x-x0)*(x-x0)+(y-y0)*(y-y0));
672 // Double_t delta = TMath::Sqrt(TMath::Abs(x*x-2*x0*x+x0*x0+ y*y-2*y*y0+y0*y0));
673 delta -= TMath::Abs(r);