Helix Class Reference

Extension of the generalized circle also caching the perigee coordinates. More...

#include <Helix.h>

Public Member Functions
	Helix ()
	Default constructor for ROOT compatibility.
	Helix (const PerigeeCircle &circleXY, const SZLine &szLine)
	Constructor combining a two dimensional circle with the linear augment in the sz space.
	Helix (const HelixParameters &parameters)
	Constructor taking all stored parameters for internal use.
	Helix (double curvature, double phi0, double impact, double tanLambda, double z0)
	Constructor from all helix parameter.
	Helix (double curvature, const Vector2D &phi0Vec, double impact, double tanLambda, double z0)
	Constructor from all helix parameter, phi given as a unit vector.
void	invalidate ()
	Sets all circle parameters to zero.
bool	isInvalid () const
	Indicates if the stored parameter combination designates a valid helix.
void	reverse ()
	Flips the travel direction of the helix in place, pivot point is unchanged.
Helix	reversed () const
	Returns a copy of the helix with flips the travel direction, pivot point is the same.
double	arcLength2DToClosest (const Vector3D &point, bool firstPeriod=true) const
	Calculates the perpendicular travel distance at which the helix has the closest approach to the given point.
double	arcLength2DToXY (const Vector2D &point) const
	Calculates the two dimensional arc length that is closest to two dimensional point in the xy projection.
double	arcLength2DToCylindricalR (double cylindricalR) const
	Calculates the two dimensional arc length that first reaches a cylindrical radius on the helix Returns NAN if the radius cannot be reached.
Vector3D	closest (const Vector3D &point, bool firstPeriod=true) const
	Calculates the point on the helix with the smallest total distance.
Vector3D	closestXY (const Vector2D &pointXY) const
	Calculates the point on the helix with the smallest perpendicular (xy) distance.
double	distance (const Vector3D &point) const
	Calculates the distance of the point to the point of closest approach on the helix.
double	distanceXY (const Vector2D &point) const
	Calculates the distance of the line parallel to the z axes through the given point.
double	passiveMoveBy (const Vector3D &by)
	Moves the coordinates system by the given vector.
HelixJacobian	passiveMoveByJacobian (const Vector3D &by) const
	Computes the Jacobi matrix for a move of the coordinate system by the given vector.
double	shiftPeriod (int nPeriods)
	Adjust the arclength measure to start n periods later.
Vector3D	atArcLength2D (double s) const
	Calculates the point, which lies at the give perpendicular travel distance (counted from the perigee)
Vector3D	atZ (double z) const
	Calculates the point, which lies at the given z coordinate.
Vector2D	xyAtZ (double z) const
	Calculates the point, which lies at the given z coordinate.
double	minimalCylindricalR () const
	Gives the minimal cylindrical radius the circle reaches (unsigned)
double	maximalCylindricalR () const
	Gives the maximal cylindrical radius the circle reaches.
double	curvatureXY () const
	Getter for the signed curvature in the xy projection.
double	impactXY () const
	Getter for the signed distance to the z axes at the perigee point.
double	omega () const
	Getter for the omega parameter of the common helix parameterisation.
double	d0 () const
	Getter for the signed distance to the z axes at the perigee point.
Vector2D	perigeeXY () const
	Getter for the perigee point in the xy projection.
Vector3D	perigee () const
	Getter for the perigee point of the helix.
double	tanLambda () const
	Getter for the proportinality factor from arc length in xy space to z.
double	cotTheta () const
	Getter for the proportinality factor from arc length in xy space to z.
double	z0 () const
	Getter for z coordinate at the perigee point of the helix.
double	zPeriod () const
	Getter for the distance in z at which the two points on the helix coincide in the xy projection.
double	arcLength2DPeriod () const
	Getter for the arc length of one trip around the helix.
double	perimeterXY () const
	Getter for the perimeter of the circle in the xy projection.
double	radiusXY () const
	Getter for the radius of the circle in the xy projection.
Vector2D	centerXY () const
	Getter for the central point of the helix.
Vector3D	tangential () const
	Getter for the unit three dimensional tangential vector at the perigee point of the helix.
const Vector2D &	phi0Vec () const
	Getter for the direction vector in the xy projection at the perigee of the helix.
double	phi0 () const
	Getter for the azimuth angle of the direction of flight at the perigee.
HelixParameters	helixParameters () const
	Getter for the five helix parameters in the order defined by EHelixParameter.h.
const PerigeeCircle &	circleXY () const
	Getter for the projection into xy space.
const SZLine &	szLine () const
	Getter for the projection into xy space.

Private Attributes
PerigeeCircle	m_circleXY
	Memory of the projection of the helix in xy space.
SZLine	m_szLine
	Memory of the of the linear relation between perpendicular travel distance and the z position.

Detailed Description

Extension of the generalized circle also caching the perigee coordinates.

Definition at line 28 of file Helix.h.

Constructor & Destructor Documentation

Helix() [1/5]

Helix ( )

inline

Default constructor for ROOT compatibility.

Definition at line 32 of file Helix.h.

        : m_circleXY()
        , m_szLine()
      {
      }

Helix() [2/5]

Helix ( const PerigeeCircle &  circleXY, const SZLine &  szLine  )

inline

Constructor combining a two dimensional circle with the linear augment in the sz space.

Definition at line 39 of file Helix.h.

        : m_circleXY(circleXY)
        , m_szLine(szLine)
      {
      }

Helix() [3/5]

Helix ( const HelixParameters &  parameters )

inlineexplicit

Constructor taking all stored parameters for internal use.

Definition at line 46 of file Helix.h.

        : m_circleXY(HelixUtil::getPerigeeParameters(parameters))
        , m_szLine(HelixUtil::getSZParameters(parameters))
      {
      }

Helix() [4/5]

Helix ( double  curvature, double  phi0, double  impact, double  tanLambda, double  z0  )

inline

Constructor from all helix parameter.

Definition at line 53 of file Helix.h.

        : m_circleXY(curvature, phi0, impact)
        , m_szLine(tanLambda, z0)
      {
      }

Helix() [5/5]

Helix ( double  curvature, const Vector2D &  phi0Vec, double  impact, double  tanLambda, double  z0  )

inline

Constructor from all helix parameter, phi given as a unit vector.

Definition at line 60 of file Helix.h.

        : m_circleXY(curvature, phi0Vec, impact)
        , m_szLine(tanLambda, z0)
      {
      }

Member Function Documentation

arcLength2DPeriod()

double arcLength2DPeriod ( ) const

inline

Getter for the arc length of one trip around the helix.

Definition at line 265 of file Helix.h.

      {
        return circleXY().arcLengthPeriod();
      }

arcLength2DToClosest()

double arcLength2DToClosest	(	const Vector3D &	point,
		bool	firstPeriod = true
	)		const

Calculates the perpendicular travel distance at which the helix has the closest approach to the given point.

Definition at line 29 of file Helix.cc.

{
  // The point may happen to lie in the center of the helix.
  double d0 = circleXY().distance(point.xy());
  double denominator = 1 + curvatureXY() * d0;
  if (denominator == 0) {
    // When this happens we can optimise the z distance for the closest point
    double arcLength2D = (point.z() - z0()) / tanLambda();
    if (not std::isfinite(arcLength2D)) {
      return 0.0;
    } else {
      return arcLength2D;
    }
  }
 
  // First approximation optimising the xy distance.
  double arcLength2D = circleXY().arcLengthTo(point.xy());
  // In case the helix is a flat circle with no extend into z this is the actual solution
  if (tanLambda() == 0) {
    return arcLength2D;
  }
 
  double deltaZ = point.z() - szLine().map(arcLength2D);
  if (not firstPeriod) {
    if (fabs(deltaZ) > zPeriod() / 2) {
      double newDeltaZ = std::remainder(deltaZ, zPeriod());
      double nPeriodShift = (deltaZ - newDeltaZ) / zPeriod();
      arcLength2D += nPeriodShift * perimeterXY();
      deltaZ = newDeltaZ;
    }
  }
 
  using boost::math::tools::brent_find_minima;
 
  auto distance3D = [this, &point](const double & s) -> double {
    Vector3D pos = atArcLength2D(s);
    return pos.distance(point);
  };
 
  double searchWidth = std::fmin(perimeterXY(), 2 * deltaZ / tanLambda());
 
  double lowerS = arcLength2D - searchWidth;
  double upperS = arcLength2D + searchWidth;
 
  int bits = std::numeric_limits<double>::digits;
  boost::uintmax_t nMaxIter = 100;
 
  std::pair<double, double> sBounds = brent_find_minima(distance3D, lowerS, upperS, bits, nMaxIter);
 
  // Stopped before iterations were exhausted?
  // bool converged = nMaxIter > 0;
 
  double firstDistance = distance3D(sBounds.first);
  double secondDistance = distance3D(sBounds.second);
  if (firstDistance < secondDistance) {
    return sBounds.first;
  } else {
    return sBounds.second;
  }
}

arcLength2DToCylindricalR()

double arcLength2DToCylindricalR ( double  cylindricalR ) const

inline

Calculates the two dimensional arc length that first reaches a cylindrical radius on the helix Returns NAN if the radius cannot be reached.

Definition at line 112 of file Helix.h.

      {
        return circleXY().arcLengthToCylindricalR(cylindricalR);
      }

arcLength2DToXY()

double arcLength2DToXY ( const Vector2D &  point ) const

inline

Calculates the two dimensional arc length that is closest to two dimensional point in the xy projection.

Always gives a solution in the first half period in the positive or negative direction

Definition at line 102 of file Helix.h.

      {
        return circleXY().arcLengthTo(point);
      }

atArcLength2D()

Vector3D atArcLength2D ( double  s ) const

inline

Calculates the point, which lies at the give perpendicular travel distance (counted from the perigee)

Definition at line 174 of file Helix.h.

      {
        return Vector3D(circleXY().atArcLength(s), szLine().map(s));
      }

atZ()

Vector3D atZ ( double  z ) const

inline

Calculates the point, which lies at the given z coordinate.

Definition at line 180 of file Helix.h.

      {
        return Vector3D(xyAtZ(z), z);
      }

centerXY()

Vector2D centerXY ( ) const

inline

Getter for the central point of the helix.

Definition at line 283 of file Helix.h.

      {
        return circleXY().center();
      }

circleXY()

const PerigeeCircle & circleXY ( ) const

inline

Getter for the projection into xy space.

Definition at line 320 of file Helix.h.

      {
        return m_circleXY;
      }

closest()

Vector3D closest ( const Vector3D &  point, bool  firstPeriod = true  ) const

inline

Calculates the point on the helix with the smallest total distance.

Definition at line 118 of file Helix.h.

      {
        double arcLength2D = arcLength2DToClosest(point, firstPeriod);
        return atArcLength2D(arcLength2D);
      }

closestXY()

Vector3D closestXY ( const Vector2D &  pointXY ) const

inline

Calculates the point on the helix with the smallest perpendicular (xy) distance.

Definition at line 125 of file Helix.h.

      {
        double arcLength2D = arcLength2DToXY(pointXY);
        return atArcLength2D(arcLength2D);
      }

cotTheta()

double cotTheta ( ) const

inline

Getter for the proportinality factor from arc length in xy space to z.

Definition at line 246 of file Helix.h.

      {
        return tanLambda();
      }

curvatureXY()

double curvatureXY ( ) const

inline

Getter for the signed curvature in the xy projection.

Definition at line 204 of file Helix.h.

      {
        return circleXY().curvature();
      }

d0()

double d0 ( ) const

inline

Getter for the signed distance to the z axes at the perigee point.

Definition at line 222 of file Helix.h.

      {
        return circleXY().d0();
      }

distance()

double distance ( const Vector3D &  point ) const

inline

Calculates the distance of the point to the point of closest approach on the helix.

Definition at line 132 of file Helix.h.

      {
        return point.distance(closest(point));
      }

distanceXY()

double distanceXY ( const Vector2D &  point ) const

inline

Calculates the distance of the line parallel to the z axes through the given point.

Definition at line 138 of file Helix.h.

      {
        return m_circleXY.distance(point);
      }

helixParameters()

HelixParameters helixParameters ( ) const

inline

Getter for the five helix parameters in the order defined by EHelixParameter.h.

Definition at line 307 of file Helix.h.

      {
        HelixParameters result;
        using namespace NHelixParameterIndices;
        result(c_Curv) = curvatureXY();
        result(c_Phi0) = phi0();
        result(c_I) = impactXY();
        result(c_TanL) = tanLambda();
        result(c_Z0) = z0();
        return result;
      }

impactXY()

double impactXY ( ) const

inline

Getter for the signed distance to the z axes at the perigee point.

Definition at line 210 of file Helix.h.

      {
        return circleXY().impact();
      }

invalidate()

void invalidate ( )

inline

Sets all circle parameters to zero.

Definition at line 67 of file Helix.h.

      {
        m_circleXY.invalidate();
        m_szLine.invalidate();
      }

isInvalid()

bool isInvalid ( ) const

inline

Indicates if the stored parameter combination designates a valid helix.

Definition at line 74 of file Helix.h.

      {
        return circleXY().isInvalid() and szLine().isInvalid();
      }

maximalCylindricalR()

double maximalCylindricalR ( ) const

inline

Gives the maximal cylindrical radius the circle reaches.

Definition at line 198 of file Helix.h.

      {
        return circleXY().maximalCylindricalR();
      }

minimalCylindricalR()

double minimalCylindricalR ( ) const

inline

Gives the minimal cylindrical radius the circle reaches (unsigned)

Definition at line 192 of file Helix.h.

      {
        return circleXY().minimalCylindricalR();
      }

omega()

double omega ( ) const

inline

Getter for the omega parameter of the common helix parameterisation.

Definition at line 216 of file Helix.h.

      {
        return circleXY().omega();
      }

passiveMoveBy()

double passiveMoveBy ( const Vector3D &  by )

inline

Moves the coordinates system by the given vector.

Updates perigee point in place.

Returns

arcLength2D that has to be traversed to the new origin

Definition at line 147 of file Helix.h.

      {
        // First keep the necessary shift of the perpendicular travel distance to the new perigee
        // point.
        double byS = circleXY().arcLengthTo(by.xy());
        m_circleXY.passiveMoveBy(by.xy());
        Vector2D bySZ(byS, by.z());
        m_szLine.passiveMoveBy(bySZ);
        return byS;
      }

passiveMoveByJacobian()

HelixJacobian passiveMoveByJacobian

(

const Vector3D &

by

)

const

Computes the Jacobi matrix for a move of the coordinate system by the given vector.

Definition at line 90 of file Helix.cc.

{
  // Fills the upper left 3x3 corner.
  PerigeeJacobian perigeeJacobian = circleXY().passiveMoveByJacobian(by.xy());
  SZJacobian szJacobian = SZUtil::identity();
  HelixJacobian jacobian = HelixUtil::stackBlocks(perigeeJacobian, szJacobian);
 
  double curv = curvatureXY();
  double tanL = tanLambda();
  double sArc = circleXY().arcLengthTo(by.xy());
 
  using namespace NHelixParameterIndices;
  jacobian(c_Z0, c_Curv) = tanL * (jacobian(c_Phi0, c_Curv) - sArc) / curv;
  jacobian(c_Z0, c_Phi0) = tanL * (jacobian(c_Phi0, c_Phi0) - 1.) / curv;
  jacobian(c_Z0, c_I) = tanL * jacobian(c_Phi0, c_I) / curv;
  jacobian(c_Z0, c_TanL) = sArc;
 
  return jacobian;
}

perigee()

Vector3D perigee ( ) const

inline

Getter for the perigee point of the helix.

Definition at line 234 of file Helix.h.

      {
        return Vector3D(perigeeXY(), z0());
      }

perigeeXY()

Vector2D perigeeXY ( ) const

inline

Getter for the perigee point in the xy projection.

Definition at line 228 of file Helix.h.

      {
        return circleXY().perigee();
      }

perimeterXY()

double perimeterXY ( ) const

inline

Getter for the perimeter of the circle in the xy projection.

Definition at line 271 of file Helix.h.

      {
        return circleXY().perimeter();
      }

phi0()

double phi0 ( ) const

inline

Getter for the azimuth angle of the direction of flight at the perigee.

Definition at line 301 of file Helix.h.

      {
        return circleXY().phi0();
      }

phi0Vec()

const Vector2D & phi0Vec ( ) const

inline

Getter for the direction vector in the xy projection at the perigee of the helix.

Definition at line 295 of file Helix.h.

      {
        return circleXY().phi0Vec();
      }

radiusXY()

double radiusXY ( ) const

inline

Getter for the radius of the circle in the xy projection.

Definition at line 277 of file Helix.h.

      {
        return circleXY().radius();
      }

reverse()

void reverse ( )

inline

Flips the travel direction of the helix in place, pivot point is unchanged.

Definition at line 80 of file Helix.h.

      {
        m_circleXY.reverse();
        m_szLine.reverse();
      }

reversed()

Helix reversed ( ) const

inline

Returns a copy of the helix with flips the travel direction, pivot point is the same.

Definition at line 87 of file Helix.h.

      {
        return Helix(circleXY().reversed(), szLine().reversed());
      }

shiftPeriod()

double shiftPeriod ( int  nPeriods )

inline

Adjust the arclength measure to start n periods later.

Returns

The arc length needed to travel n periods.

Definition at line 165 of file Helix.h.

      {
        double arcLength2D = nPeriods * fabs(perimeterXY());
        m_szLine.passiveMoveBy(Vector2D(arcLength2D, 0.0));
        return arcLength2D;
      }

szLine()

const SZLine & szLine ( ) const

inline

Getter for the projection into xy space.

Definition at line 326 of file Helix.h.

      {
        return m_szLine;
      }

tangential()

Vector3D tangential ( ) const

inline

Getter for the unit three dimensional tangential vector at the perigee point of the helix.

Definition at line 289 of file Helix.h.

      {
        return Vector3D(phi0Vec(), tanLambda()).unit();
      }

tanLambda()

double tanLambda ( ) const

inline

Getter for the proportinality factor from arc length in xy space to z.

Definition at line 240 of file Helix.h.

      {
        return m_szLine.tanLambda();
      }

xyAtZ()

Vector2D xyAtZ ( double  z ) const

inline

Calculates the point, which lies at the given z coordinate.

Definition at line 186 of file Helix.h.

      {
        return Vector2D(circleXY().atArcLength(szLine().inverseMap(z)));
      }

z0()

double z0 ( ) const

inline

Getter for z coordinate at the perigee point of the helix.

Definition at line 252 of file Helix.h.

      {
        return m_szLine.z0();
      }

zPeriod()

double zPeriod ( ) const

inline

Getter for the distance in z at which the two points on the helix coincide in the xy projection.

Definition at line 259 of file Helix.h.

      {
        return tanLambda() * arcLength2DPeriod();
      }

Member Data Documentation

m_circleXY

PerigeeCircle m_circleXY

private

Memory of the projection of the helix in xy space.

Definition at line 333 of file Helix.h.

m_szLine

SZLine m_szLine

private

Memory of the of the linear relation between perpendicular travel distance and the z position.

Definition at line 337 of file Helix.h.

---

The documentation for this class was generated from the following files:

• tracking/trackFindingCDC/geometry/include/Helix.h
• framework/dataobjects/src/Helix.cc
• tracking/trackFindingCDC/geometry/src/Helix.cc