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 ROOT::Math::XYVector &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 ROOT::Math::XYZVector &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 ROOT::Math::XYVector &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.

ROOT::Math::XYZVector	closest (const ROOT::Math::XYZVector &point, bool firstPeriod=true) const
	Calculates the point on the helix with the smallest total distance.

ROOT::Math::XYZVector	closestXY (const ROOT::Math::XYVector &pointXY) const
	Calculates the point on the helix with the smallest perpendicular (xy) distance.

double	distance (const ROOT::Math::XYZVector &point) const
	Calculates the distance of the point to the point of closest approach on the helix.

double	distanceXY (const ROOT::Math::XYVector &point) const
	Calculates the distance of the line parallel to the z axes through the given point.

double	passiveMoveBy (const ROOT::Math::XYZVector &by)
	Moves the coordinates system by the given vector.

HelixJacobian	passiveMoveByJacobian (const ROOT::Math::XYZVector &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.

ROOT::Math::XYZVector	atArcLength2D (double s) const
	Calculates the point, which lies at the give perpendicular travel distance (counted from the perigee)

ROOT::Math::XYZVector	atZ (double z) const
	Calculates the point, which lies at the given z coordinate.

ROOT::Math::XYVector	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.

ROOT::Math::XYVector	perigeeXY () const
	Getter for the perigee point in the xy projection.

ROOT::Math::XYZVector	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.

ROOT::Math::XYVector	centerXY () const
	Getter for the central point of the helix.

ROOT::Math::XYZVector	tangential () const
	Getter for the unit three dimensional tangential vector at the perigee point of the helix.

const ROOT::Math::XYVector &	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 30 of file Helix.h.

Constructor & Destructor Documentation

◆ Helix() [1/5]

Helix ( )

inline

Default constructor for ROOT compatibility.

Definition at line 34 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 41 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 48 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 55 of file Helix.h.

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

◆ Helix() [5/5]

Helix	(	double	curvature,
		const ROOT::Math::XYVector &	phi0Vec,
		double	impact,
		double	tanLambda,
		double	z0 )

inline

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

Definition at line 62 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 270 of file Helix.h.

      {
        return circleXY().arcLengthPeriod();
      }

◆ arcLength2DToClosest()

double arcLength2DToClosest	(	const ROOT::Math::XYZVector &	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 31 of file Helix.cc.

{
  // The point may happen to lie in the center of the helix.
  double loc_d0 = circleXY().distance(VectorUtil::getXYVector(point));
  double denominator = 1 + curvatureXY() * loc_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(VectorUtil::getXYVector(point));
  // 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 {
    ROOT::Math::XYZVector pos = atArcLength2D(s);
    return VectorUtil::Distance(pos, point);
  };
 
  double searchWidth = std::fmin(perimeterXY(), 2 * deltaZ / tanLambda());
 
  double lowerS = arcLength2D - searchWidth;
  double upperS = arcLength2D + searchWidth;
 
  int bits = std::numeric_limits<double>::digits;
  size_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 114 of file Helix.h.

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

◆ arcLength2DToXY()

double arcLength2DToXY ( const ROOT::Math::XYVector & 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 104 of file Helix.h.

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

◆ atArcLength2D()

ROOT::Math::XYZVector atArcLength2D ( double s ) const

inline

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

Definition at line 176 of file Helix.h.

      {
        const auto& tmp = circleXY().atArcLength(s);
        return ROOT::Math::XYZVector(tmp.X(), tmp.Y(), szLine().map(s));
      }

◆ atZ()

ROOT::Math::XYZVector atZ ( double z ) const

inline

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

Definition at line 183 of file Helix.h.

      {
        const auto& tmp = xyAtZ(z);
        return ROOT::Math::XYZVector(tmp.X(), tmp.Y(), z);
      }

◆ centerXY()

ROOT::Math::XYVector centerXY ( ) const

inline

Getter for the central point of the helix.

Definition at line 288 of file Helix.h.

      {
        return circleXY().center();
      }

◆ circleXY()

const PerigeeCircle & circleXY ( ) const

inline

Getter for the projection into xy space.

Definition at line 326 of file Helix.h.

      {
        return m_circleXY;
      }

◆ closest()

ROOT::Math::XYZVector closest	(	const ROOT::Math::XYZVector &	point,
		bool	firstPeriod = true ) const

inline

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

Definition at line 120 of file Helix.h.

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

◆ closestXY()

ROOT::Math::XYZVector closestXY ( const ROOT::Math::XYVector & pointXY ) const

inline

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

Definition at line 127 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 251 of file Helix.h.

      {
        return tanLambda();
      }

◆ curvatureXY()

double curvatureXY ( ) const

inline

Getter for the signed curvature in the xy projection.

Definition at line 208 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 226 of file Helix.h.

      {
        return circleXY().d0();
      }

◆ distance()

double distance ( const ROOT::Math::XYZVector & point ) const

inline

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

Definition at line 134 of file Helix.h.

      {
        return VectorUtil::Distance(point, closest(point));
      }

◆ distanceXY()

double distanceXY ( const ROOT::Math::XYVector & point ) const

inline

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

Definition at line 140 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 313 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 214 of file Helix.h.

      {
        return circleXY().impact();
      }

◆ invalidate()

void invalidate ( )

inline

Sets all circle parameters to zero.

Definition at line 69 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 76 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 202 of file Helix.h.

      {
        return circleXY().maximalCylindricalR();
      }

◆ minimalCylindricalR()

double minimalCylindricalR ( ) const

inline

Gives the minimal cylindrical radius the circle reaches (unsigned)

Definition at line 196 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 220 of file Helix.h.

      {
        return circleXY().omega();
      }

◆ passiveMoveBy()

double passiveMoveBy ( const ROOT::Math::XYZVector & 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 149 of file Helix.h.

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

◆ passiveMoveByJacobian()

HelixJacobian passiveMoveByJacobian ( const ROOT::Math::XYZVector & by ) const

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

Definition at line 92 of file Helix.cc.

{
  // Fills the upper left 3x3 corner.
  PerigeeJacobian perigeeJacobian = circleXY().passiveMoveByJacobian(VectorUtil::getXYVector(by));
  SZJacobian szJacobian = SZUtil::identity();
  HelixJacobian jacobian = HelixUtil::stackBlocks(perigeeJacobian, szJacobian);
 
  double curv = curvatureXY();
  double tanL = tanLambda();
  double sArc = circleXY().arcLengthTo(VectorUtil::getXYVector(by));
 
  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()

ROOT::Math::XYZVector perigee ( ) const

inline

Getter for the perigee point of the helix.

Definition at line 238 of file Helix.h.

      {
        const auto& tmp = perigeeXY();
        return ROOT::Math::XYZVector(tmp.X(), tmp.Y(), z0());
      }

◆ perigeeXY()

ROOT::Math::XYVector perigeeXY ( ) const

inline

Getter for the perigee point in the xy projection.

Definition at line 232 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 276 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 307 of file Helix.h.

      {
        return circleXY().phi0();
      }

◆ phi0Vec()

const ROOT::Math::XYVector & phi0Vec ( ) const

inline

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

Definition at line 301 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 282 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 82 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 89 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 167 of file Helix.h.

      {
        double arcLength2D = nPeriods * fabs(perimeterXY());
        m_szLine.passiveMoveBy(ROOT::Math::XYVector(arcLength2D, 0.0));
        return arcLength2D;
      }

◆ szLine()

const SZLine & szLine ( ) const

inline

Getter for the projection into xy space.

Definition at line 332 of file Helix.h.

      {
        return m_szLine;
      }

◆ tangential()

ROOT::Math::XYZVector tangential ( ) const

inline

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

Definition at line 294 of file Helix.h.

      {
        const auto& tmp = phi0Vec();
        return VectorUtil::unit(ROOT::Math::XYZVector(tmp.X(), tmp.Y(), tanLambda()));
      }

◆ tanLambda()

double tanLambda ( ) const

inline

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

Definition at line 245 of file Helix.h.

      {
        return m_szLine.tanLambda();
      }

◆ xyAtZ()

ROOT::Math::XYVector xyAtZ ( double z ) const

inline

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

Definition at line 190 of file Helix.h.

      {
        return ROOT::Math::XYVector(circleXY().atArcLength(szLine().inverseMap(z)));
      }

◆ z0()

double z0 ( ) const

inline

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

Definition at line 257 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 264 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 339 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 343 of file Helix.h.

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

tracking/trackingUtilities/geometry/include/Helix.h
framework/dataobjects/src/Helix.cc
tracking/trackingUtilities/geometry/src/Helix.cc

Public Member Functions

Private Attributes

Detailed Description

Constructor & Destructor Documentation

◆ Helix() [1/5]

◆ Helix() [2/5]

◆ Helix() [3/5]

◆ Helix() [4/5]

◆ Helix() [5/5]

Member Function Documentation

◆ arcLength2DPeriod()

◆ arcLength2DToClosest()

◆ arcLength2DToCylindricalR()

◆ arcLength2DToXY()

◆ atArcLength2D()

◆ atZ()

◆ centerXY()

◆ circleXY()

◆ closest()

◆ closestXY()

◆ cotTheta()

◆ curvatureXY()

◆ d0()

◆ distance()

◆ distanceXY()

◆ helixParameters()

◆ impactXY()

◆ invalidate()

◆ isInvalid()

◆ maximalCylindricalR()

◆ minimalCylindricalR()

◆ omega()

◆ passiveMoveBy()

◆ passiveMoveByJacobian()

◆ perigee()

◆ perigeeXY()

◆ perimeterXY()

◆ phi0()

◆ phi0Vec()

◆ radiusXY()

◆ reverse()

◆ reversed()

◆ shiftPeriod()

◆ szLine()

◆ tangential()

◆ tanLambda()

◆ xyAtZ()

◆ z0()

◆ zPeriod()

Member Data Documentation

◆ m_circleXY

◆ m_szLine