G4IntersectingCone.cc

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//
// Implementation of G4IntersectingCone, a utility class which calculates
// the intersection of an arbitrary line with a fixed cone
//
// Author: David C. Williams (davidw@scipp.ucsc.edu)
// --------------------------------------------------------------------

#include "G4IntersectingCone.hh"
#include "G4GeometryTolerance.hh"

// Constructor
//
G4IntersectingCone::G4IntersectingCone( const G4double r[2],
                                        const G4double z[2] )
{
  const G4double halfCarTolerance
    = 0.5 * G4GeometryTolerance::GetInstance()->GetSurfaceTolerance();

  // What type of cone are we?
  //
  type1 = (std::abs(z[1]-z[0]) > std::abs(r[1]-r[0]));

  if (type1) // tube like
  {
    B = (r[1] - r[0]) / (z[1] - z[0]);
    A = (r[0]*z[1] - r[1]*z[0]) / (z[1] -z[0]);
  }
  else // disk like
  {
    B = (z[1] - z[0]) / (r[1] - r[0]);
    A = (z[0]*r[1] - z[1]*r[0]) / (r[1] - r[0]);
  }

  // Calculate extent
  //
  rLo = std::min(r[0], r[1]) - halfCarTolerance;
  rHi = std::max(r[0], r[1]) + halfCarTolerance;
  zLo = std::min(z[0], z[1]) - halfCarTolerance;
  zHi = std::max(z[0], z[1]) + halfCarTolerance;
}

// Fake default constructor - sets only member data and allocates memory
//                            for usage restricted to object persistency.
//
G4IntersectingCone::G4IntersectingCone( __void__& )
  : zLo(0.), zHi(0.), rLo(0.), rHi(0.), A(0.), B(0.)
{
}

// Destructor
//
G4IntersectingCone::~G4IntersectingCone()
{
}

// HitOn
//
// Check r or z extent, as appropriate, to see if the point is possibly
// on the cone.
//
G4bool G4IntersectingCone::HitOn( const G4double r,
                                  const G4double z )
{
  //
  // Be careful! The inequalities cannot be "<=" and ">=" here without
  // punching a tiny hole in our shape!
  //
  if (type1)
  {
    if (z < zLo || z > zHi) return false;
  }
  else
  {
    if (r < rLo || r > rHi) return false;
  }

  return true;
}

// LineHitsCone
//
// Calculate the intersection of a line with our conical surface, ignoring
// any phi division
//
G4int G4IntersectingCone::LineHitsCone( const G4ThreeVector& p,
                                        const G4ThreeVector& v,
                                              G4double* s1, G4double* s2 )
{
  if (type1)
  {
    return LineHitsCone1( p, v, s1, s2 );
  }
  else
  {
    return LineHitsCone2( p, v, s1, s2 );
  }
}

// LineHitsCone1
//
// Calculate the intersections of a line with a conical surface. Only
// suitable if zPlane[0] != zPlane[1].
//
// Equation of a line:
//
//       x = x0 + s*tx      y = y0 + s*ty      z = z0 + s*tz
//
// Equation of a conical surface:
//
//       x**2 + y**2 = (A + B*z)**2
//
// Solution is quadratic:
//
//  a*s**2 + b*s + c = 0
//
// where:
//
//  a = tx**2 + ty**2 - (B*tz)**2
//
//  b = 2*( px*vx + py*vy - B*(A + B*pz)*vz )
//
//  c = x0**2 + y0**2 - (A + B*z0)**2
//
// Notice, that if a < 0, this indicates that the two solutions (assuming
// they exist) are in opposite cones (that is, given z0 = -A/B, one z < z0
// and the other z > z0). For our shapes, the invalid solution is one
// which produces A + Bz < 0, or the one where Bz is smallest (most negative).
// Since Bz = B*s*tz, if B*tz > 0, we want the largest s, otherwise,
// the smaller.
//
// If there are two solutions on one side of the cone, we want to make
// sure that they are on the "correct" side, that is A + B*z0 + s*B*tz >= 0.
//
// If a = 0, we have a linear problem: s = c/b, which again gives one solution.
// This should be rare.
//
// For b*b - 4*a*c = 0, we also have one solution, which is almost always
// a line just grazing the surface of a the cone, which we want to ignore.
// However, there are two other, very rare, possibilities:
// a line intersecting the z axis and either:
//       1. At the same angle std::atan(B) to just miss one side of the cone, or
//       2. Intersecting the cone apex (0,0,-A/B)
// We *don't* want to miss these! How do we identify them? Well, since
// this case is rare, we can at least swallow a little more CPU than we would
// normally be comfortable with. Intersection with the z axis means
// x0*ty - y0*tx = 0. Case (1) means a==0, and we've already dealt with that
// above. Case (2) means a < 0.
//
// Now: x0*tx + y0*ty = 0 in terms of roundoff error. We can write:
//             Delta = x0*tx + y0*ty
//             b = 2*( Delta - B*(A + B*z0)*tz )
// For:
//             b*b - 4*a*c = epsilon
// where epsilon is small, then:
//             Delta = epsilon/2/B
//
G4int G4IntersectingCone::LineHitsCone1( const G4ThreeVector& p,
                                         const G4ThreeVector& v,
                                               G4double* s1, G4double* s2 )
{
  static const G4double EPS = DBL_EPSILON; // Precision constant,
                                           // originally it was 1E-6
  G4double x0 = p.x(), y0 = p.y(), z0 = p.z();
  G4double tx = v.x(), ty = v.y(), tz = v.z();

  // Value of radical can be inaccurate due to loss of precision
  // if to calculate the coefficiets a,b,c like the following:
  //     G4double a = tx*tx + ty*ty - sqr(B*tz);
  //     G4double b = 2*( x0*tx + y0*ty - B*(A + B*z0)*tz);
  //     G4double c = x0*x0 + y0*y0 - sqr(A + B*z0);
  //
  // For more accurate calculation of radical the coefficients
  // are splitted in two components, radial and along z-axis
  //
  G4double ar = tx*tx + ty*ty;
  G4double az = sqr(B*tz);
  G4double br = 2*(x0*tx + y0*ty);
  G4double bz = 2*B*(A + B*z0)*tz;
  G4double cr = x0*x0 + y0*y0;
  G4double cz = sqr(A + B*z0);

  // Instead radical = b*b - 4*a*c
  G4double arcz = 4*ar*cz;
  G4double azcr = 4*az*cr;
  G4double radical = (br*br - 4*ar*cr) + ((std::max(arcz,azcr) - 2*bz*br) + std::min(arcz,azcr));

  // Find the coefficients
  G4double a = ar - az;
  G4double b = br - bz;
  G4double c = cr - cz;

  if (radical < -EPS*std::fabs(b))  { return 0; } // No solution

  if (radical < EPS*std::fabs(b))
  {
    //
    // The radical is roughly zero: check for special, very rare, cases
    //
    if (std::fabs(a) > 1/kInfinity)
      {
      if(B==0.) { return 0; }
      if ( std::fabs(x0*ty - y0*tx) < std::fabs(EPS/B) )
      {
         *s1 = -0.5*b/a;
         return 1;
      }
      return 0;
    }
  }
  else
  {
    radical = std::sqrt(radical);
  }

  if (a > 1/kInfinity)
  {
    G4double sa, sb, q = -0.5*( b + (b < 0 ? -radical : +radical) );
    sa = q/a;
    sb = c/q;
    if (sa < sb) { *s1 = sa; *s2 = sb; } else { *s1 = sb; *s2 = sa; }
    if (A + B*(z0+(*s1)*tz) < 0)  { return 0; }
    return 2;
  }
  else if (a < -1/kInfinity)
  {
    G4double sa, sb, q = -0.5*( b + (b < 0 ? -radical : +radical) );
    sa = q/a;
    sb = c/q;
    *s1 = (B*tz > 0)^(sa > sb) ? sb : sa;
    return 1;
  }
  else if (std::fabs(b) < 1/kInfinity)
  {
    return 0;
  }
  else
  {
    *s1 = -c/b;
    if (A + B*(z0+(*s1)*tz) < 0)  { return 0; }
    return 1;
  }
}

// LineHitsCone2
//
// See comments under LineHitsCone1. In this routine, case2, we have:
//
//   Z = A + B*R
//
// The solution is still quadratic:
//
//  a = tz**2 - B*B*(tx**2 + ty**2)
//
//  b = 2*( (z0-A)*tz - B*B*(x0*tx+y0*ty) )
//
//  c = ( (z0-A)**2 - B*B*(x0**2 + y0**2) )
//
// The rest is much the same, except some details.
//
// a > 0 now means we intersect only once in the correct hemisphere.
//
// a > 0 ? We only want solution which produces R > 0.
// since R = (z0+s*tz-A)/B, for tz/B > 0, this is the largest s
//          for tz/B < 0, this is the smallest s
// thus, same as in case 1 ( since sign(tz/B) = sign(tz*B) )
//
G4int G4IntersectingCone::LineHitsCone2( const G4ThreeVector& p,
                                         const G4ThreeVector& v,
                                               G4double* s1, G4double* s2 )
{
  static const G4double EPS = DBL_EPSILON; // Precision constant,
                                           // originally it was 1E-6
  G4double x0 = p.x(), y0 = p.y(), z0 = p.z();
  G4double tx = v.x(), ty = v.y(), tz = v.z();

  // Special case which might not be so rare: B = 0 (precisely)
  //
  if (B==0)
  {
    if (std::fabs(tz) < 1/kInfinity)  { return 0; }

    *s1 = (A-z0)/tz;
    return 1;
  }

  // Value of radical can be inaccurate due to loss of precision
  // if to calculate the coefficiets a,b,c like the following:
  //   G4double a = tz*tz - B2*(tx*tx + ty*ty);
  //   G4double b = 2*( (z0-A)*tz - B2*(x0*tx + y0*ty) );
  //   G4double c = sqr(z0-A) - B2*( x0*x0 + y0*y0 );
  //
  // For more accurate calculation of radical the coefficients
  // are splitted in two components, radial and along z-axis
  //
  G4double B2 = B*B;

  G4double az = tz*tz;
  G4double ar = B2*(tx*tx + ty*ty);
  G4double bz = 2*(z0-A)*tz;
  G4double br = 2*B2*(x0*tx + y0*ty);
  G4double cz = sqr(z0-A);
  G4double cr = B2*(x0*x0 + y0*y0);

  // Instead radical = b*b - 4*a*c
  G4double arcz = 4*ar*cz;
  G4double azcr = 4*az*cr;
  G4double radical = (br*br - 4*ar*cr) + ((std::max(arcz,azcr) - 2*bz*br) + std::min(arcz,azcr));

  // Find the coefficients
  G4double a = az - ar;
  G4double b = bz - br;
  G4double c = cz - cr;

  if (radical < -EPS*std::fabs(b)) { return 0; } // No solution

  if (radical < EPS*std::fabs(b))
  {
    //
    // The radical is roughly zero: check for special, very rare, cases
    //
    if (std::fabs(a) > 1/kInfinity)
    {
      if ( std::fabs(x0*ty - y0*tx) < std::fabs(EPS/B) )
      {
        *s1 = -0.5*b/a;
        return 1;
      }
      return 0;
    }
  }
  else
  {
    radical = std::sqrt(radical);
  }

  if (a < -1/kInfinity)
  {
    G4double sa, sb, q = -0.5*( b + (b < 0 ? -radical : +radical) );
    sa = q/a;
    sb = c/q;
    if (sa < sb) { *s1 = sa; *s2 = sb; } else { *s1 = sb; *s2 = sa; }
    if ((z0 + (*s1)*tz  - A)/B < 0)  { return 0; }
    return 2;
  }
  else if (a > 1/kInfinity)
  {
    G4double sa, sb, q = -0.5*( b + (b < 0 ? -radical : +radical) );
    sa = q/a;
    sb = c/q;
    *s1 = (tz*B > 0)^(sa > sb) ? sb : sa;
    return 1;
  }
  else if (std::fabs(b) < 1/kInfinity)
  {
    return 0;
  }
  else
  {
    *s1 = -c/b;
    if ((z0 + (*s1)*tz  - A)/B < 0)  { return 0; }
    return 1;
  }
}