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 Revision 1.1 2005/12/15 08:55:33 decaro
19 New TOF geometry description (V5) -G. Cara Romeo and A. De Caro
21 Revision 0.1 2005/07/19 A. De Caro
22 Modify Global methods IsInsideThePad & DistanceToPad
23 according to the PPR TOF geometry
24 Implement Global methods GetPadDx & GetPadDy & GetPadDz
25 Modify Global methods GetDetID & GetPlate & GetSector &
26 GetStrip & GetPadX & GetPadZ
27 according to the PPR TOF geometry
28 Modify Global methods GetPos & GetX & GetY & GetZ
29 according to the PPR TOF geometry
33 #include <Riostream.h>
34 ///////////////////////////////////////////////////////////////////////////////
36 // TOF Geometry class (PPR version) //
38 ///////////////////////////////////////////////////////////////////////////////
43 #include "AliTOFGeometry.h"
44 #include "AliTOFGeometryV4.h"
46 ClassImp(AliTOFGeometryV4)
48 const Int_t AliTOFGeometryV4::kNStripC = 20; // number of strips in C type module
50 const Float_t AliTOFGeometryV4::fgkZlenA = 106.0; // length (cm) of the A module
51 const Float_t AliTOFGeometryV4::fgkZlenB = 141.0; // length (cm) of the B module
52 const Float_t AliTOFGeometryV4::fgkZlenC = 177.5; // length (cm) of the C module
53 const Float_t AliTOFGeometryV4::fgkMaxhZtof = 371.5; // Max half z-size of TOF (cm)
55 const Float_t AliTOFGeometryV4::fgkDeadBndX = 1.0; // Dead Boundaries of a Strip along X direction (length) (cm)
56 const Float_t AliTOFGeometryV4::fgkDeadBndZ = 1.5; // Dead Boundaries of a Strip along Z direction (width) (cm)
57 const Float_t AliTOFGeometryV4::fgkOverSpc = 15.3; // Space available for sensitive layers in radial direction (cm)
59 const Float_t AliTOFGeometryV4::fgkDprecMin = 0.0000075;//num.prec.tolerance on Thmin
60 const Float_t AliTOFGeometryV4::fgkDprecMax = 0.0000100;//num.prec.tolerance on Thma
61 const Float_t AliTOFGeometryV4::fgkDprecCen = 0.0000005;//num.prec.tolerance on <Theta>
63 const Float_t AliTOFGeometryV4::fgkxTOF = 371.; // Inner radius of the TOF for Reconstruction (cm)
64 const Float_t AliTOFGeometryV4::fgkRmin = 370.; // Inner radius of the TOF (cm)
65 const Float_t AliTOFGeometryV4::fgkRmax = 399.; // Outer radius of the TOF (cm)
67 //_____________________________________________________________________________
68 AliTOFGeometryV4::AliTOFGeometryV4()
72 // AliTOFGeometryV4 default constructor
75 AliTOFGeometry::kNStripC = kNStripC; // number of strips in C type module
77 AliTOFGeometry::kZlenA = fgkZlenA; // length (cm) of the A module
78 AliTOFGeometry::kZlenB = fgkZlenB; // length (cm) of the B module
79 AliTOFGeometry::kZlenC = fgkZlenC; // length (cm) of the C module
80 AliTOFGeometry::kMaxhZtof = fgkMaxhZtof; // Max half z-size of TOF (cm)
82 AliTOFGeometry::fgkxTOF = fgkxTOF; // Inner radius of the TOF for Reconstruction (cm)
83 AliTOFGeometry::fgkRmin = fgkRmin; // Inner radius of the TOF (cm)
84 AliTOFGeometry::fgkRmax = fgkRmax; // Outer radius of the TOF (cm)
90 //_____________________________________________________________________________
91 AliTOFGeometryV4::~AliTOFGeometryV4()
94 // AliTOFGeometryV4 destructor
98 //_____________________________________________________________________________
99 void AliTOFGeometryV4::Init()
102 // Initialize strip Tilt Angles and Heights
104 // Strips Tilt Angles
106 Float_t const kangles[kNPlates][kMaxNstrip] ={
108 {44.494, 43.725, 42.946, 42.156, 41.357, 40.548, 39.729, 38.899,
109 38.060, 37.211, 36.353, 35.484, 34.606, 33.719, 32.822, 31.916,
110 31.001, 30.077, 29.144, 28.202 },
112 {26.884, 25.922, 24.952, 23.975, 22.989, 22.320, 21.016, 20.309,
113 19.015, 18.270, 16.989, 16.205, 14.941, 14.117, 12.871, 12.008,
114 10.784, 9.8807, 8.681, 0.0 },
116 { 7.5835, 6.4124, 5.4058, 4.2809, 3.2448, 2.1424, 1.078, -0., -1.078,
117 -2.1424, -3.2448, -4.2809, -5.4058, -6.4124, -7.5835, 0.0, 0.0, 0.0,
120 {-8.681, -9.8807, -10.784, -12.008, -12.871, -14.117, -14.941, -16.205,
121 -16.989, -18.27, -19.015, -20.309, -21.016, -22.32, -22.989,
122 -23.975, -24.952, -25.922, -26.884, 0. },
124 {-28.202, -29.144, -30.077, -31.001, -31.916, -32.822, -33.719, -34.606,
125 -35.484, -36.353, -37.211, -38.06, -38.899, -39.729, -40.548,
126 -41.357, -42.156, -42.946, -43.725, -44.494 }};
131 Float_t const kheights[kNPlates][kMaxNstrip]= {
133 {-5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5,
134 -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5 },
136 {-6.3, -7.1, -7.9, -8.7, -9.5, -3, -9.5, -3, -9.5, -3,
137 -9.5, -3.0, -9.5, -3.0, -9.5, -3, -9.5, -3, -9 , 0.},
139 { -3, -9, -4.5, -9, -4.5, -9, -4.5, -9, -4.5, -9,
140 -4.5, -9, -4.5, -9, -3, 0.0, 0.0, 0.0, 0.0, 0.0 },
142 { -9, -3, -9.5, -3, -9.5, -3, -9.5, -3, -9.5, -3, -9.5,
143 -3, -9.5, -3, -9.5, -8.7, -7.9, -7.1, -6.3, 0. },
145 {-5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5,
146 -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5 }};
148 // Deposit in fAngles, fHeights
150 for (Int_t iplate = 0; iplate < kNPlates; iplate++) {
151 for (Int_t istrip = 0; istrip < kMaxNstrip; istrip++) {
152 AliTOFGeometry::fAngles[iplate][istrip] = kangles[iplate][istrip];
153 AliTOFGeometry::fHeights[iplate][istrip] = kheights[iplate][istrip];
159 //_____________________________________________________________________________
160 Float_t AliTOFGeometryV4::DistanceToPad(Int_t *det, Float_t *pos, Float_t *dist3d)
163 // Returns distance of space point with coor pos (x,y,z) (cm) wrt
164 // pad with Detector Indices idet (iSect,iPlate,iStrip,iPadX,iPadZ)
167 //Transform pos into Sector Frame
173 Float_t radius = TMath::Sqrt(x*x+y*y);
174 Float_t phi=TMath::ATan2(y,x);
175 if(phi<0) phi=2.*TMath::Pi()+phi;
176 // Get the local angle in the sector philoc
177 Float_t angle = phi*kRaddeg-( Int_t (kRaddeg*phi/fPhiSec) + 0.5)*fPhiSec;
178 Float_t xs = radius*TMath::Cos(angle/kRaddeg);
179 Float_t ys = radius*TMath::Sin(angle/kRaddeg);
182 // Do the same for the selected pad
187 Float_t padRadius = TMath::Sqrt(g[0]*g[0]+g[1]*g[1]);
188 Float_t padPhi=TMath::ATan2(g[1],g[0]);
189 if(padPhi<0) padPhi=2.*TMath::Pi()+padPhi;
190 // Get the local angle in the sector philoc
191 Float_t padAngle = padPhi*kRaddeg-( Int_t (padPhi*kRaddeg/fPhiSec)+ 0.5) * fPhiSec;
192 Float_t padxs = padRadius*TMath::Cos(padAngle/kRaddeg);
193 Float_t padys = padRadius*TMath::Sin(padAngle/kRaddeg);
194 Float_t padzs = g[2];
196 //Now move to local pad coordinate frame. Translate:
198 Float_t xt = xs-padxs;
199 Float_t yt = ys-padys;
200 Float_t zt = zs-padzs;
203 Float_t alpha = GetAngles(det[1],det[2]);
204 Float_t xr = xt*TMath::Cos(alpha/kRaddeg)+zt*TMath::Sin(alpha/kRaddeg);
206 Float_t zr = -xt*TMath::Sin(alpha/kRaddeg)+zt*TMath::Cos(alpha/kRaddeg);
208 Float_t dist = TMath::Sqrt(xr*xr+yr*yr+zr*zr);
220 //_____________________________________________________________________________
221 Bool_t AliTOFGeometryV4::IsInsideThePad(Int_t *det, Float_t *pos)
224 // Returns true if space point with coor pos (x,y,z) (cm) falls
225 // inside pad with Detector Indices idet (iSect,iPlate,iStrip,iPadX,iPadZ)
228 Bool_t isInside=false;
230 //Transform pos into Sector Frame
236 Float_t radius = TMath::Sqrt(x*x+y*y);
237 Float_t phi=TMath::ATan2(y,x);
238 if(phi<0) phi=2.*TMath::Pi()+phi;
239 // Get the local angle in the sector philoc
240 Float_t angle = phi*kRaddeg-( Int_t (kRaddeg*phi/fPhiSec) + 0.5) *fPhiSec;
241 Float_t xs = radius*TMath::Cos(angle/kRaddeg);
242 Float_t ys = radius*TMath::Sin(angle/kRaddeg);
245 // Do the same for the selected pad
250 Float_t padRadius = TMath::Sqrt(g[0]*g[0]+g[1]*g[1]);
251 Float_t padPhi=TMath::ATan2(g[1],g[0]);
252 if(padPhi<0) padPhi=2.*TMath::Pi()+padPhi;
253 // Get the local angle in the sector philoc
254 Float_t padAngle = padPhi*kRaddeg-( Int_t (padPhi*kRaddeg/fPhiSec)+ 0.5) * fPhiSec;
255 Float_t padxs = padRadius*TMath::Cos(padAngle/kRaddeg);
256 Float_t padys = padRadius*TMath::Sin(padAngle/kRaddeg);
257 Float_t padzs = g[2];
259 //Now move to local pad coordinate frame. Translate:
261 Float_t xt = xs-padxs;
262 Float_t yt = ys-padys;
263 Float_t zt = zs-padzs;
267 Float_t alpha = GetAngles(det[1],det[2]);
268 Float_t xr = xt*TMath::Cos(alpha/kRaddeg)+zt*TMath::Sin(alpha/kRaddeg);
270 Float_t zr = -xt*TMath::Sin(alpha/kRaddeg)+zt*TMath::Cos(alpha/kRaddeg);
272 if(TMath::Abs(xr)<=0.75 && TMath::Abs(yr)<= (fgkXPad*0.5) && TMath::Abs(zr)<= (fgkZPad*0.5))
278 //_____________________________________________________________________________
279 Float_t AliTOFGeometryV4::GetX(Int_t *det)
282 // Returns X coordinate (cm)
285 Int_t isector = det[0];
286 Int_t iplate = det[1];
287 Int_t istrip = det[2];
288 Int_t ipadz = det[3];
289 Int_t ipadx = det[4];
291 // Find out distance d on the plane wrt median phi:
292 Float_t d = (ipadx+0.5)*fgkXPad-(kNpadX*fgkXPad)*0.5;
294 // The radius r in xy plane:
295 Float_t r = (fgkRmin+fgkRmax)/2.+fHeights[iplate][istrip]+
296 (ipadz-0.5)*fgkZPad*TMath::Sin(fAngles[iplate][istrip]/kRaddeg)-0.25;
298 // local azimuthal angle in the sector philoc
299 Float_t philoc = TMath:: ATan(d/r);
301 // azimuthal angle in the global frame phi
302 Float_t phi = philoc*kRaddeg+(isector+0.5 )*fPhiSec;
304 Float_t xCoor = r/TMath::Cos(philoc)*TMath::Cos(phi/kRaddeg);
309 //_____________________________________________________________________________
310 Float_t AliTOFGeometryV4::GetY(Int_t *det)
313 // Returns Y coordinate (cm)
316 Int_t isector = det[0];
317 Int_t iplate = det[1];
318 Int_t istrip = det[2];
319 Int_t ipadz = det[3];
320 Int_t ipadx = det[4];
322 // Find out distance d on the plane wrt median phi:
323 Float_t d = (ipadx+0.5)*fgkXPad-(kNpadX*fgkXPad)*0.5;
325 // The radius r in xy plane:
326 Float_t r = (fgkRmin+fgkRmax)/2.+fHeights[iplate][istrip]+
327 (ipadz-0.5)*fgkZPad*TMath::Sin(fAngles[iplate][istrip]/kRaddeg)-0.25;
329 // local azimuthal angle in the sector philoc
330 Float_t philoc = TMath:: ATan(d/r);
332 // azimuthal angle in the global frame phi
333 Float_t phi = philoc*kRaddeg+(isector+0.5 )*fPhiSec;
335 Float_t yCoor = r/TMath::Cos(philoc)*TMath::Sin(phi/kRaddeg);
341 //_____________________________________________________________________________
342 Float_t AliTOFGeometryV4::GetZ(Int_t *det)
345 // Returns Z coordinate (cm)
348 Int_t iplate = det[1];
349 Int_t istrip = det[2];
350 Int_t ipadz = det[3];
352 // The radius r in xy plane:
353 Float_t r = (fgkRmin+fgkRmax)/2.+fHeights[iplate][istrip];
355 Float_t zCoor = r*TMath::Tan(0.5*TMath::Pi()-GetStripTheta(iplate,istrip))-
356 (ipadz-0.5)*fgkZPad*TMath::Cos(fAngles[iplate][istrip]/kRaddeg);
361 //_____________________________________________________________________________
362 Int_t AliTOFGeometryV4::GetSector(Float_t *pos)
365 // Returns the Sector index
373 Float_t phi = TMath::ATan2(y,x);
374 if(phi<0.) phi=2.*TMath::Pi()+phi;
375 iSect = (Int_t) (phi*kRaddeg/fPhiSec);
381 //_____________________________________________________________________________
382 Int_t AliTOFGeometryV4::GetPadX(Float_t *pos)
385 // Returns the Pad index along X
394 Int_t isector = GetSector(pos);
396 AliError("Detector Index could not be determined");
398 Int_t iplate = GetPlate(pos);
400 AliError("Detector Index could not be determined");
402 Int_t istrip = GetStrip(pos);
404 AliError("Detector Index could not be determined");
408 Float_t rho=TMath::Sqrt(x*x+y*y);
409 Float_t phi = TMath::ATan2(y,x);
410 if(phi<0.) phi=2.*TMath::Pi()+phi;
412 // Get the local angle in the sector philoc
413 Float_t philoc = phi*kRaddeg-(isector+0.5)*fPhiSec;
414 philoc*=TMath::Pi()/180.;
415 // theta projected on the median of the sector
416 Float_t theta = TMath::ATan2(rho*TMath::Cos(philoc),z);
417 // The radius r in xy plane:
418 Float_t r = (fgkRmin+fgkRmax)/2.+fHeights[iplate][istrip]+
419 (theta-GetStripTheta(iplate, istrip))/
420 (GetMaxStripTheta(iplate, istrip)-GetMinStripTheta(iplate, istrip))
421 * 2.*fgkZPad*TMath::Sin(fAngles[iplate][istrip]/kRaddeg)-0.25;
423 // Find out distance projected onto the strip plane
424 Float_t d = (r*TMath::Tan(philoc)+(kNpadX*fgkXPad)*0.5);
426 iPadX = (Int_t) ( d/fgkXPad);
430 //_____________________________________________________________________________
431 Int_t AliTOFGeometryV4::GetPlate(Float_t *pos)
434 // Returns the Plate index
438 Int_t isector = GetSector(pos);
440 AliError("Detector Index could not be determined");
447 Float_t rho=TMath::Sqrt(x*x+y*y);
448 Float_t phi=TMath::ATan2(y,x);
449 if(phi<0) phi=2.*TMath::Pi()+phi;
450 // Get the local angle in the sector philoc
451 Float_t philoc = phi*kRaddeg-(isector+0.5)*fPhiSec;
452 philoc*=TMath::Pi()/180.;
453 // theta projected on the median of the sector
454 Float_t theta=TMath::ATan2(rho*TMath::Cos(philoc),z);
456 for (Int_t i=0; i<kNPlates; i++){
457 if ( GetMaxPlateTheta(i) >= theta &&
458 GetMinPlateTheta(i) <= theta)iPlate=i;
465 //_____________________________________________________________________________
466 Int_t AliTOFGeometryV4::GetStrip(Float_t *pos)
469 // Returns the Strip index
475 Int_t isector = GetSector(pos);
477 AliError("Detector Index could not be determined");
479 Int_t iplate = GetPlate(pos);
481 AliError("Detector Index could not be determined");
490 if(iplate==0 || iplate == 4)nstrips=kNStripC;
491 if(iplate==1 || iplate == 3)nstrips=kNStripB;
492 if(iplate==2) nstrips=kNStripA;
494 Float_t rho=TMath::Sqrt(x*x+y*y);
495 Float_t phi=TMath::ATan2(y,x);
496 if(phi<0) phi=2.*TMath::Pi()+phi;
497 // Get the local angle in the sector philoc
498 Float_t philoc = phi*kRaddeg-(isector+0.5)*fPhiSec;
499 philoc*=TMath::Pi()/180.;
500 // theta projected on the median of the sector
501 Float_t theta=TMath::ATan2(rho*TMath::Cos(philoc),z);
503 for (Int_t istrip=0; istrip<nstrips; istrip++){
506 GetMaxStripTheta(iplate,istrip) >= theta
508 GetMinStripTheta(iplate,istrip) <= theta ) iStrip = istrip;
515 //_____________________________________________________________________________
516 Int_t AliTOFGeometryV4::GetPadZ(Float_t *pos)
519 // Returns the Pad index along Z
523 Int_t isector = GetSector(pos);
525 AliError("Detector Index could not be determined");
527 Int_t iplate = GetPlate(pos);
529 AliError("Detector Index could not be determined");
531 Int_t istrip = GetStrip(pos);
533 AliError("Detector Index could not be determined");
541 Float_t rho=TMath::Sqrt(x*x+y*y);
542 Float_t phi=TMath::ATan2(y,x);
543 if(phi<0) phi=2.*TMath::Pi()+phi;
544 Float_t philoc = phi*kRaddeg-(isector+0.5)*fPhiSec;
545 philoc*=TMath::Pi()/180.;
546 Float_t theta=TMath::ATan2(rho*TMath::Cos(philoc),z);
548 if (theta >= GetStripTheta(iplate, istrip))iPadZ=1;
554 //_____________________________________________________________________________
555 Float_t AliTOFGeometryV4::GetMinPlateTheta(Int_t iPlate)
558 // Returns the minimum theta angle of a given plate iPlate (rad)
565 if(iPlate==0)delta = -1. ;
566 if(iPlate==1)delta = -0.5;
567 if(iPlate==3)delta = +0.5;
568 if(iPlate==4)delta = +1. ;
570 Float_t z=(fgkRmin+2.)*TMath::Tan(fAngles[iPlate][index]/kRaddeg)+delta;
571 Float_t r=(fgkRmin+fgkRmax)/2.+fHeights[iPlate][index];
572 z =z+fgkZPad*TMath::Cos(fAngles[iPlate][index]/kRaddeg);
573 r =r-fgkZPad*TMath::Sin(fAngles[iPlate][index]/kRaddeg);
575 Float_t thmin = 0.5*TMath::Pi()-TMath::ATan(z/r)-fgkDprecMin;
579 //_____________________________________________________________________________
580 Float_t AliTOFGeometryV4::GetMaxPlateTheta(Int_t iPlate)
583 // Returns the maximum theta angle of a given plate iPlate (rad)
586 if(iPlate==0 ||iPlate == 4)index=kNStripC-1;
587 if(iPlate==1 ||iPlate == 3)index=kNStripB-1;
588 if(iPlate==2) index=kNStripA-1;
591 if(iPlate==0)delta = -1. ;
592 if(iPlate==1)delta = -0.5;
593 if(iPlate==3)delta = +0.5;
594 if(iPlate==4)delta = +1. ;
596 Float_t z=(fgkRmin+2.)*TMath::Tan(fAngles[iPlate][index]/kRaddeg)+delta;
597 Float_t r=(fgkRmin+fgkRmax)/2.+fHeights[iPlate][index];
598 z =z-fgkZPad*TMath::Cos(fAngles[iPlate][index]/kRaddeg);
599 r= r+fgkZPad*TMath::Sin(fAngles[iPlate][index]/kRaddeg);
601 Float_t thmax = 0.5*TMath::Pi()-TMath::ATan(z/r)+fgkDprecMax;
606 //_____________________________________________________________________________
607 Float_t AliTOFGeometryV4::GetMaxStripTheta(Int_t iPlate, Int_t iStrip)
610 // Returns the maximum theta angle of a given strip iStrip (rad)
615 if(iPlate==0)delta = -1. ;
616 if(iPlate==1)delta = -0.5;
617 if(iPlate==3)delta = +0.5;
618 if(iPlate==4)delta = +1. ;
620 Float_t r =(fgkRmin+fgkRmax)/2.+fHeights[iPlate][iStrip];
621 Float_t z =(fgkRmin+2.)*TMath::Tan(fAngles[iPlate][iStrip]/kRaddeg)+delta;
622 z = z-fgkZPad*TMath::Cos(fAngles[iPlate][iStrip]/kRaddeg);
623 r = r+fgkZPad*TMath::Sin(fAngles[iPlate][iStrip]/kRaddeg);
624 Float_t thmax =0.5*TMath::Pi()-TMath::ATan(z/r)+fgkDprecMax;
628 //_____________________________________________________________________________
629 Float_t AliTOFGeometryV4::GetMinStripTheta(Int_t iPlate, Int_t iStrip)
632 // Returns the minimum theta angle of a given Strip iStrip (rad)
637 if(iPlate==0)delta = -1. ;
638 if(iPlate==1)delta = -0.5;
639 if(iPlate==3)delta = +0.5;
640 if(iPlate==4)delta = +1. ;
643 Float_t r =(fgkRmin+fgkRmax)/2.+fHeights[iPlate][iStrip];
644 Float_t z =(fgkRmin+2.)*TMath::Tan(fAngles[iPlate][iStrip]/kRaddeg)+delta;
645 z =z+fgkZPad*TMath::Cos(fAngles[iPlate][iStrip]/kRaddeg);
646 r =r-fgkZPad*TMath::Sin(fAngles[iPlate][iStrip]/kRaddeg);
647 Float_t thmin =0.5*TMath::Pi()-TMath::ATan(z/r)-fgkDprecMin;
652 //_____________________________________________________________________________
653 Float_t AliTOFGeometryV4::GetStripTheta(Int_t iPlate, Int_t iStrip)
656 // returns the median theta angle of a given strip iStrip (rad)
661 if(iPlate==0)delta = -1. ;
662 if(iPlate==1)delta = -0.5;
663 if(iPlate==3)delta = +0.5;
664 if(iPlate==4)delta = +1. ;
666 Float_t r =(fgkRmin+fgkRmax)/2.+fHeights[iPlate][iStrip];
667 Float_t z =(fgkRmin+2.)*TMath::Tan(fAngles[iPlate][iStrip]/kRaddeg)+delta;
668 Float_t theta =0.5*TMath::Pi()-TMath::ATan(z/r);
670 if(theta > 0.5*TMath::Pi() )theta+=fgkDprecCen;
671 if(theta < 0.5*TMath::Pi() )theta-=fgkDprecCen;
676 //_____________________________________________________________________________