TGeoHelix.cxx

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// @(#)root/geom:$Id$
// Author: Andrei Gheata   28/04/04
 
/*************************************************************************
 * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
 * All rights reserved.                                                  *
 *                                                                       *
 * For the licensing terms see $ROOTSYS/LICENSE.                         *
 * For the list of contributors see $ROOTSYS/README/CREDITS.             *
 *************************************************************************/
 
/** \class TGeoHelix
\ingroup Shapes_classes
 
Class representing a helix curve
 
 A helix is a curve defined by the following equations:
 
~~~ {.cpp}
    x = (1/c) * COS(q*phi)
    y = (1/c) * SIN(q*phi)
    z = s * alfa
~~~
 
where:
 
~~~ {.cpp}
    c = 1/Rxy  - curvature in XY plane
    phi        - phi angle
    S = 2*PI*s - vertical separation between helix loops
    q = +/- 1  - (+)=left-handed, (-)=right-handed
~~~
 
  In particular, a helix describes the trajectory of a charged particle in magnetic
field. In such case, the helix is right-handed for negative particle charge.
To define a helix, one must define:
  - the curvature - positive defined
  - the Z step made after one full turn of the helix
  - the particle charge sign
  - the initial particle position and direction (force normalization to unit)
  - the magnetic field direction
 
A helix provides:
  - propagation to a given Z position (in global frame)
  Double_t *point = TGeoHelix::PropagateToZ(Double_t z);
  - propagation to an arbitrary plane, returning also the new point
  - propagation in a geometry until the next crossed surface
  - computation of the total track length along a helix
*/
 
#include "TMath.h"
#include "TGeoShape.h"
#include "TGeoMatrix.h"
#include "TGeoHelix.h"
 
ClassImp(TGeoHelix);
 
////////////////////////////////////////////////////////////////////////////////
/// Dummy constructor
 
TGeoHelix::TGeoHelix()
{
   fC = 0.;
   fS = 0.;
   fStep = 0.;
   fPhi = 0.;
   fPointInit[0] = fPointInit[1] = fPointInit[2] = 0.;
   fDirInit[0] = fDirInit[1] = fDirInit[2] = 0.;
   fPoint[0] = fPoint[1] = fPoint[2] = 0.;
   fDir[0] = fDir[1] = fDir[2] = 0.;
   fB[0] = fB[1] = fB[2] = 0.;
   fQ = 0;
   fMatrix = nullptr;
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
   TObject::SetBit(kHelixStraight, kFALSE);
   TObject::SetBit(kHelixCircle, kFALSE);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Normal constructor
 
TGeoHelix::TGeoHelix(Double_t curvature, Double_t hstep, Int_t charge)
{
   SetXYcurvature(curvature);
   SetHelixStep(hstep);
   fQ = 0;
   SetCharge(charge);
   fStep = 0.;
   fPhi = 0.;
   fPointInit[0] = fPointInit[1] = fPointInit[2] = 0.;
   fDirInit[0] = fDirInit[1] = fDirInit[2] = 0.;
   fPoint[0] = fPoint[1] = fPoint[2] = 0.;
   fDir[0] = fDir[1] = fDir[2] = 0.;
   fB[0] = fB[1] = fB[2] = 0.;
   fMatrix = new TGeoHMatrix();
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
   TObject::SetBit(kHelixStraight, kFALSE);
   TObject::SetBit(kHelixCircle, kFALSE);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Destructor
 
TGeoHelix::~TGeoHelix()
{
   if (fMatrix)
      delete fMatrix;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute safe linear step that can be made such that the error
/// between linear-helix extrapolation is less than EPSIL.
 
Double_t TGeoHelix::ComputeSafeStep(Double_t epsil) const
{
   if (TestBit(kHelixStraight) || TMath::Abs(fC) < TGeoShape::Tolerance())
      return 1.E30;
   Double_t c = GetTotalCurvature();
   Double_t step = TMath::Sqrt(2. * epsil / c);
   return step;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize coordinates of a point on the helix
 
void TGeoHelix::InitPoint(Double_t x0, Double_t y0, Double_t z0)
{
   fPointInit[0] = x0;
   fPointInit[1] = y0;
   fPointInit[2] = z0;
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set initial point on the helix.
 
void TGeoHelix::InitPoint(Double_t *point)
{
   InitPoint(point[0], point[1], point[2]);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize particle direction (tangent on the helix in initial point)
 
void TGeoHelix::InitDirection(Double_t dirx, Double_t diry, Double_t dirz, Bool_t is_normalized)
{
   fDirInit[0] = dirx;
   fDirInit[1] = diry;
   fDirInit[2] = dirz;
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
   if (is_normalized)
      return;
   Double_t norm = 1. / TMath::Sqrt(dirx * dirx + diry * diry + dirz * dirz);
   for (Int_t i = 0; i < 3; i++)
      fDirInit[i] *= norm;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize particle direction (tangent on the helix in initial point)
 
void TGeoHelix::InitDirection(Double_t *dir, Bool_t is_normalized)
{
   InitDirection(dir[0], dir[1], dir[2], is_normalized);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Compute helix total curvature
 
Double_t TGeoHelix::GetTotalCurvature() const
{
   Double_t k = fC / (1. + fC * fC * fS * fS);
   return k;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set XY curvature: c = 1/Rxy
 
void TGeoHelix::SetXYcurvature(Double_t curvature)
{
   fC = curvature;
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
   if (fC < 0) {
      Error("SetXYcurvature", "Curvature %f not valid. Must be positive.", fC);
      return;
   }
   if (TMath::Abs(fC) < TGeoShape::Tolerance()) {
      Warning("SetXYcurvature", "Curvature is zero. Helix is a straight line.");
      TObject::SetBit(kHelixStraight, kTRUE);
   }
}
 
////////////////////////////////////////////////////////////////////////////////
/// Positive charge means left-handed helix.
 
void TGeoHelix::SetCharge(Int_t charge)
{
   if (charge == 0) {
      Error("ctor", "charge cannot be 0 - define it positive for a left-handed helix, negative otherwise");
      return;
   }
   Int_t q = TMath::Sign(1, charge);
   if (q == fQ)
      return;
   fQ = q;
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Initialize particle direction (tangent on the helix in initial point)
 
void TGeoHelix::SetField(Double_t bx, Double_t by, Double_t bz, Bool_t is_normalized)
{
   fB[0] = bx;
   fB[1] = by;
   fB[2] = bz;
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
   if (is_normalized)
      return;
   Double_t norm = 1. / TMath::Sqrt(bx * bx + by * by + bz * bz);
   for (Int_t i = 0; i < 3; i++)
      fB[i] *= norm;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Set Z step of the helix on a complete turn. Positive or null.
 
void TGeoHelix::SetHelixStep(Double_t step)
{
   if (step < 0) {
      Error("ctor", "Z step %f not valid. Must be positive.", step);
      return;
   }
   TObject::SetBit(kHelixNeedUpdate, kTRUE);
   fS = 0.5 * step / TMath::Pi();
   if (fS < TGeoShape::Tolerance())
      TObject::SetBit(kHelixCircle, kTRUE);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Reset current point/direction to initial values
 
void TGeoHelix::ResetStep()
{
   fStep = 0.;
   memcpy(fPoint, fPointInit, 3 * sizeof(Double_t));
   memcpy(fDir, fDirInit, 3 * sizeof(Double_t));
}
 
////////////////////////////////////////////////////////////////////////////////
/// Make a step from current point along the helix and compute new point, direction and angle
/// To reach a plane/ shape boundary, one has to:
///  1. Compute the safety to the plane/boundary
///  2. Define / update a helix according local field and particle state (position, direction, charge)
///  3. Compute the magnetic safety (maximum distance for which the field can be considered constant)
///  4. Call TGeoHelix::Step() having as argument the minimum between 1. and 3.
///  5. Repeat from 1. until the step to be made is small enough.
///  6. Add to the total step the distance along a straight line from the last point
///     to the plane/shape boundary
 
void TGeoHelix::Step(Double_t step)
{
   Int_t i;
   fStep += step;
   if (TObject::TestBit(kHelixStraight)) {
      for (i = 0; i < 3; i++) {
         fPoint[i] = fPointInit[i] + fStep * fDirInit[i];
         fDir[i] = fDirInit[i];
      }
      return;
   }
   if (TObject::TestBit(kHelixNeedUpdate))
      UpdateHelix();
   Double_t r = 1. / fC;
   fPhi = fStep / TMath::Sqrt(r * r + fS * fS);
   Double_t vect[3];
   vect[0] = r * TMath::Cos(fPhi);
   vect[1] = -fQ * r * TMath::Sin(fPhi);
   vect[2] = fS * fPhi;
   fMatrix->LocalToMaster(vect, fPoint);
 
   Double_t ddb = fDirInit[0] * fB[0] + fDirInit[1] * fB[1] + fDirInit[2] * fB[2];
   Double_t f = -TMath::Sqrt(1. - ddb * ddb);
   vect[0] = f * TMath::Sin(fPhi);
   vect[1] = fQ * f * TMath::Cos(fPhi);
   vect[2] = ddb;
   TMath::Normalize(vect);
   fMatrix->LocalToMasterVect(vect, fDir);
}
 
////////////////////////////////////////////////////////////////////////////////
/// Propagate initial point up to a given Z position in MARS.
 
Double_t TGeoHelix::StepToPlane(Double_t *point, Double_t *norm)
{
   Double_t step = 0.;
   Double_t snext = 1.E30;
   Double_t dx, dy, dz;
   Double_t ddn, pdn;
   if (TObject::TestBit(kHelixNeedUpdate))
      UpdateHelix();
   dx = point[0] - fPoint[0];
   dy = point[1] - fPoint[1];
   dz = point[2] - fPoint[2];
   pdn = dx * norm[0] + dy * norm[1] + dz * norm[2];
   ddn = fDir[0] * norm[0] + fDir[1] * norm[1] + fDir[2] * norm[2];
   if (TObject::TestBit(kHelixStraight)) {
      // propagate straight line to plane
      if ((pdn * ddn) <= 0)
         return snext;
      snext = pdn / ddn;
      Step(snext);
      return snext;
   }
 
   Double_t r = 1. / fC;
   Double_t dist;
   Double_t safety = TMath::Abs(pdn);
   Double_t safestep = ComputeSafeStep();
   snext = 1.E30;
   Bool_t approaching = (ddn * pdn > 0) ? kTRUE : kFALSE;
   if (approaching)
      snext = pdn / ddn;
   else if (safety > 2. * r)
      return snext;
   while (snext > safestep) {
      dist = TMath::Max(safety, safestep);
      Step(dist);
      step += dist;
      dx = point[0] - fPoint[0];
      dy = point[1] - fPoint[1];
      dz = point[2] - fPoint[2];
      pdn = dx * norm[0] + dy * norm[1] + dz * norm[2];
      ddn = fDir[0] * norm[0] + fDir[1] * norm[1] + fDir[2] * norm[2];
      safety = TMath::Abs(pdn);
      approaching = (ddn * pdn > 0) ? kTRUE : kFALSE;
      snext = 1.E30;
      if (approaching)
         snext = pdn / ddn;
      else if (safety > 2. * r) {
         ResetStep();
         return snext;
      }
   }
   step += snext;
   Step(snext);
   return step;
}
 
////////////////////////////////////////////////////////////////////////////////
/// Update the local helix matrix.
 
void TGeoHelix::UpdateHelix()
{
   TObject::SetBit(kHelixNeedUpdate, kFALSE);
   fStep = 0.;
   memcpy(fPoint, fPointInit, 3 * sizeof(Double_t));
   memcpy(fDir, fDirInit, 3 * sizeof(Double_t));
   Double_t rot[9];
   Double_t tr[3];
   Double_t ddb = fDirInit[0] * fB[0] + fDirInit[1] * fB[1] + fDirInit[2] * fB[2];
   if ((1. - TMath::Abs(ddb)) < TGeoShape::Tolerance() || TMath::Abs(fC) < TGeoShape::Tolerance()) {
      // helix is just a straight line
      TObject::SetBit(kHelixStraight, kTRUE);
      fMatrix->Clear();
      return;
   }
   rot[2] = fB[0];
   rot[5] = fB[1];
   rot[8] = fB[2];
   if (ddb < 0)
      fS = -TMath::Abs(fS);
   Double_t fy = -fQ * TMath::Sqrt(1. - ddb * ddb);
   fy = 1. / fy;
   rot[1] = fy * (fDirInit[0] - fB[0] * ddb);
   rot[4] = fy * (fDirInit[1] - fB[1] * ddb);
   rot[7] = fy * (fDirInit[2] - fB[2] * ddb);
 
   rot[0] = rot[4] * rot[8] - rot[7] * rot[5];
   rot[3] = rot[7] * rot[2] - rot[1] * rot[8];
   rot[6] = rot[1] * rot[5] - rot[4] * rot[2];
 
   tr[0] = fPointInit[0] - rot[0] / fC;
   tr[1] = fPointInit[1] - rot[3] / fC;
   tr[2] = fPointInit[2] - rot[6] / fC;
 
   fMatrix->SetTranslation(tr);
   fMatrix->SetRotation(rot);
}