GeneralizedCircle Class Reference

A generalized circle. More...

#include <GeneralizedCircle.h>

Public Member Functions
	GeneralizedCircle ()
	Default constructor for ROOT compatibility.
	GeneralizedCircle (double n0, double n1, double n2, double n3=0)
	Constructor with the four parameters of the generalized circle.
	GeneralizedCircle (double n0, const Vector2D &n12, double n3=0)
	Constructor with the four parameters of the generalized circle.
	GeneralizedCircle (const Line2D &n012)
	Constructor from a two dimensional line.
	GeneralizedCircle (const Circle2D &circle)
	Constructor from a two dimensional circle.
void	setCenterAndRadius (const Vector2D &center, double absRadius, ERotation orientation=ERotation::c_CounterClockwise)
	Setter for the circle center and radius.
void	setPerigeeParameters (double curvature, const Vector2D &tangential, double impact)
	Setter for the perigee parameters.
void	setPerigeeParameters (const double curvature, const double tangentialPhi, const double impact)
	Setter for the perigee parameters.
void	setN (const double n0, const double n1, const double n2, const double n3=0.0)
	Setter for all four circle parameters.
void	setN (const double n0, const Vector2D &n12, const double n3=0.0)
	Setter for all four circle parameters.
void	setN (const Line2D &n012)
	Setter for all four circle parameters from another circle.
void	setN (const GeneralizedCircle &n0123)
	Setter for all four circle parameters from another circle.
void	invalidate ()
	Sets all circle parameters to zero.
void	reverse ()
	Flips the orientation of the circle in place.
void	conformalTransform ()
	Transforms the generalized circle to conformal space inplace Applies the conformal map in the self-inverse form X = x / (x^2 + y^2) and Y = y / (x^2 +y^2) inplace It works most easily by the exchange of the circle parameters n0 <-> n3.
void	passiveMoveBy (const Vector2D &by)
	Moves the coordinate system by the given vector.
double	n0 () const
	Getter for the first circle parameter.
double	n1 () const
	Getter for the second circle parameter.
double	n2 () const
	Getter for the third circle parameter.
const Vector2D &	n12 () const
	Getter for the second and third circle parameter which natuarally from a vector.
double	n3 () const
	Getter for the fourth circle parameter.
bool	isInvalid () const
	Indicates if all circle parameters are zero.
bool	isValid () const
	Indicates if the combination of the circle parameters makes up a valid circle.
double	normalizationSquared () const
	Calculates the generalized circle specific squared norm.
GeneralizedCircle	reversed () const
	Returns a copy of the circle with opposite orientation.
GeneralizedCircle	conformalTransformed () const
	Returns a copy of the circle in conformal space.
Vector2D	gradient (const Vector2D &point) const
	Gradient of the distance field Gives the gradient of the approximated distance field for the given point.
Vector2D	normal (const Vector2D &point) const
	Normal vector to the circle near the given position.
Vector2D	tangential (const Vector2D &point) const
	Tangential vector to the circle near the given position.
Vector2D	closest (const Vector2D &point) const
	Closest approach on the circle to the point.
Vector2D	perigee () const
	Calculates the closest approach to the two dimensional origin.
Vector2D	apogee () const
	Calculates the point on the circle that is furthest away from the origin.
EForwardBackward	isForwardOrBackwardOf (const Vector2D &from, const Vector2D &to) const
	Calculates if the to vector is closer to the from vector following the along orientation of the circle or against.
Vector2D	chooseNextForwardOf (const Vector2D &start, const Vector2D &end1, const Vector2D &end2) const
	Returns the end point which is first reached if one follows the forward direction of the circle starting from the start point.
std::pair< Belle2::TrackFindingCDC::Vector2D, Belle2::TrackFindingCDC::Vector2D >	atCylindricalR (double cylindricalR) const
	Calculates the two points with the given cylindrical radius on the generalised circle.
Vector2D	atCylindricalRForwardOf (const Vector2D &startPoint, double cylindricalR) const
	Approach on the circle with the given cylindrical radius that lies in the forward direction of a start point.
double	fastDistance (const Vector2D &point) const
	Approximate distance.
double	fastImpact () const
	Approximate distance to the origin.
double	distance (const Vector2D &point) const
	Gives the proper distance of the point to the circle line retaining the sign of the fast distance.
double	distance (double fastDistance) const
	Helper function to translate the fast linearized distance measure of the generalized circle to the proper distance from the circle retaining the sign of the distance.
double	fastDistance (const double distance) const
	Helper function to translate the proper distance to the linearized distance measure of the generalized circle retaining the sign of the distance.
double	impact () const
	Gives the signed distance of the origin to the circle.
double	d0 () const
	Getter for the absolute distance to the z axes at the support point.
Vector2D	tangential () const
	Gives the tangential vector at the closest approach to the origin / at the perigee.
double	tangentialPhi () const
	Gives to azimuth angle phi of the direction of flight at the perigee.
double	minimalCylindricalR () const
	Gives the minimal cylindrical radius the circle reaches (unsigned)
double	maximalCylindricalR () const
	Gives the maximal cylindrical radius the circle reaches.
double	absDistance (const Vector2D &point) const
	Gives the proper absolute distance of the point to the circle line.
ERightLeft	isRightOrLeft (const Vector2D &point) const
	Indicates if the point is on the right or left side of the circle.
bool	isLeft (const Vector2D &rhs) const
	Return if the point given is left of the line.
bool	isRight (const Vector2D &rhs) const
	Return if the point given is right of the line.
bool	isLine () const
	Indicates if the generalized circle is actually a line.
bool	isCircle () const
	Indicates if the generalized circle is actually a circle.
double	radius () const
	Gives the signed radius of the circle. If it was a line this will be infinity.
double	absRadius () const
	Gives the signed radius of the circle. If it was a line this will be infinity.
double	curvature () const
	Gives the signed curvature of the generalized circle.
double	omega () const
	Gives the omega parameter as used by the framework helix.
Vector2D	center () const
	Gives the center of the circle. If it was a line both components will be infinity.
double	perimeter () const
	Gives the perimeter of the circle.
ERotation	orientation () const
	Gives the orientation of the circle.
double	arcLengthPeriod () const
	Getter for the arc length for a full round of the circle.
double	arcLengthBetween (const Vector2D &from, const Vector2D &to) const
	Calculates the arc length between two points of closest approach on the circle.
double	arcLengthTo (const Vector2D &to) const
	Calculates the arc length between the perigee and the given point.
double	arcLengthToCylindricalR (double cylindricalR) const
	Calculates the two dimensional arc length till the cylindrical radius is reached If the radius can not be reached return NAN.
double	arcLengthFactor (const double directDistance) const
	Helper function the calculate the factor between the length of a secant line and the length on the arc.
std::pair< Vector2D, Vector2D >	intersections (const GeneralizedCircle &generalizedCircle) const
	Calculates the two points common to both circles.
Vector2D	atArcLength (double arcLength) const
	Calculates the point, which lies at the give perpendicular travel distance (counted from the perigee)

Static Public Member Functions
static GeneralizedCircle	fromCenterAndRadius (const Vector2D &center, double absRadius, ERotation orientation=ERotation::c_CounterClockwise)
	Constructor from center, radius and a optional orientation.
static GeneralizedCircle	fromPerigeeParameters (double curvature, const Vector2D &tangential, double impact)
	Constructor of a generalized circle from perigee parameters.
static GeneralizedCircle	fromPerigeeParameters (double curvature, double tangentialPhi, double impact)
	Constructor of a generalized circle from perigee parameters.
static double	arcLengthFactor (double directDistance, double curvature)
	Helper function the calculate the factor between the length of a secant line and the length on the arc.

Protected Member Functions
void	setN0 (const double n0)
	Setter for first circle parameter.
void	setN1 (const double n1)
	Setter for second circle parameter.
void	setN2 (const double n2)
	Setter for third circle parameter.
void	setN12 (const double n1, const double n2)
	Setter for second and third circle parameter.
void	setN12 (const Vector2D &n12)
	Setter for second and third circle parameter.
void	setN3 (const double n3)
	Setter for fourth circle parameter.
void	normalize ()
	Normalizes the circle parameters.

Private Member Functions
void	scaleN (const double factor)
	Scales the circle parameters by a common factor.

Private Attributes
double	m_n3
	Memory for the fourth parameter.
Vector2D	m_n12
	Memory for the second and third parameter.
double	m_n0
	Memory for the first parameter.

Detailed Description

A generalized circle.

Makes a smooth generalization from a two dimensional normal line ( like Line2D ) to a circle.

The parameterisation is best suited for low curvature circle. The representation takes four parameters. They correspond to the normal circle parameters like
n0 = (m_x*m_x + m_y*m_y - r*r)/2r
n1 = -m_x/r
n2 = -m_y/r
n3 = 1/2r
where the normalization condition is n1*n1 + n2*n2 - 4 * n0 * n3 = 1. The overall sign is fixed in the following way: If the last parameter is positive the circle is assumed to be orientated counterclockwise else the circle is assumed to be orientated clockwise. The parameters n1 and n2 are indeed a vector in two dimensions and we keep them stored as Vector2D. Additionally we can represent a line with same parameters by setting n3 = 0. Compare Line2D.

Definition at line 51 of file GeneralizedCircle.h.

Constructor & Destructor Documentation

GeneralizedCircle() [1/5]

GeneralizedCircle

(

)

Default constructor for ROOT compatibility.

Definition at line 26 of file GeneralizedCircle.cc.

  : m_n3(0.0)
  , m_n12(0.0, 0.0)
  , m_n0(0.0)
{
}

GeneralizedCircle() [2/5]

GeneralizedCircle	(	double	n0,
		double	n1,
		double	n2,
		double	n3 = 0
	)

Constructor with the four parameters of the generalized circle.

Definition at line 33 of file GeneralizedCircle.cc.

  : m_n3(n3)
  , m_n12(n1, n2)
  , m_n0(n0)
{
  normalize();
}

GeneralizedCircle() [3/5]

GeneralizedCircle	(	double	n0,
		const Vector2D &	n12,
		double	n3 = 0
	)

Constructor with the four parameters of the generalized circle.

Definition at line 44 of file GeneralizedCircle.cc.

  : m_n3(n3)
  , m_n12(n12)
  , m_n0(n0)
{
  normalize();
}

GeneralizedCircle() [4/5]

GeneralizedCircle ( const Line2D &  n012 )

explicit

Constructor from a two dimensional line.

Definition at line 52 of file GeneralizedCircle.cc.

  : m_n3(0.0)
  , m_n12(n012.n12())
  , m_n0(n012.n0())
{
  normalize();
}

GeneralizedCircle() [5/5]

GeneralizedCircle ( const Circle2D &  circle )

explicit

Constructor from a two dimensional circle.

Definition at line 60 of file GeneralizedCircle.cc.

  : m_n3(1.0 / 2.0 / circle.radius())
  , m_n12(-circle.center().x() * (m_n3 * 2.0), -circle.center().y() * (m_n3 * 2.0))
  , m_n0((circle.center().normSquared() - circle.radiusSquared()) * m_n3)
{
}

Member Function Documentation

absDistance()

double absDistance ( const Vector2D &  point ) const

inline

Gives the proper absolute distance of the point to the circle line.

Definition at line 529 of file GeneralizedCircle.h.

      {
        return fabs(distance(point));
      }

absRadius()

double absRadius ( ) const

inline

Gives the signed radius of the circle. If it was a line this will be infinity.

Definition at line 574 of file GeneralizedCircle.h.

      {
        return fabs(radius());
      }

apogee()

Vector2D apogee

(

)

const

Calculates the point on the circle that is furthest away from the origin.

This results in Vector2D(NAN, NAN) in the straight line case.

Definition at line 168 of file GeneralizedCircle.cc.

{
  Vector2D result(n12());
  result.setCylindricalR(-impact() - 2 * radius());
  return result;
}

arcLengthBetween()

double arcLengthBetween	(	const Vector2D &	from,
		const Vector2D &	to
	)		const

Calculates the arc length between two points of closest approach on the circle.

The arc length is signed positive for travel in orientation direction. In the circle case the arc length is between -pi*radius and pi*radius, hence the discontinuity is on the far side of the circle relative to the given from point. The points are essentially first taken to their closest approach before we take the length on the curve. For the line case the length is the distance component parallel to the line.

Definition at line 244 of file GeneralizedCircle.cc.

{
  EForwardBackward lengthSign = isForwardOrBackwardOf(from, to);
  if (not NForwardBackward::isValid(lengthSign)) return NAN;
 
  // Handling the rare case that from and to correspond to opposing points on the circle
  if (lengthSign == EForwardBackward::c_Unknown) lengthSign = EForwardBackward::c_Forward;
 
  Vector2D closestAtFrom = closest(from);
  Vector2D closestAtTo = closest(to);
  double directDistance = closestAtFrom.distance(closestAtTo);
 
  return static_cast<double>(lengthSign) * arcLengthFactor(directDistance) * directDistance;
}

arcLengthFactor() [1/2]

double arcLengthFactor ( const double  directDistance ) const

inline

Helper function the calculate the factor between the length of a secant line and the length on the arc.

Smooth function expressing the relation between arc length and direct length only using the curvature of the circle as additional information. It enables better handling of the line limit compared to the former implementation which used the opening angle of the arc.

Definition at line 653 of file GeneralizedCircle.h.

      {
        return arcLengthFactor(directDistance, curvature());
      }

arcLengthFactor() [2/2]

double arcLengthFactor ( double  directDistance, double  curvature  )

static

Helper function the calculate the factor between the length of a secant line and the length on the arc.

Smooth function expressing the relation between arc length and direct length only using the curvature of the circle as additional information. It enables better handling of the line limit compared to the former implementation which used the opening angle of the arc.

Definition at line 290 of file GeneralizedCircle.cc.

{
  double x = directDistance * curvature / 2.0;
  return asinc(x);
}

arcLengthPeriod()

double arcLengthPeriod ( ) const

inline

Getter for the arc length for a full round of the circle.

Definition at line 618 of file GeneralizedCircle.h.

      {
        return std::fabs(perimeter());
      }

arcLengthTo()

double arcLengthTo

(

const Vector2D &

to

)

const

Calculates the arc length between the perigee and the given point.

Definition at line 259 of file GeneralizedCircle.cc.

{
  const Vector2D from = perigee();
 
  EForwardBackward lengthSign = isForwardOrBackwardOf(from, to);
  if (not NForwardBackward::isValid(lengthSign)) return NAN;
 
  // Handling the rare case that from and to correspond to opposing points on the circle
  if (lengthSign == EForwardBackward::c_Unknown) lengthSign = EForwardBackward::c_Forward;
 
  const Vector2D& closestAtFrom = from;
  Vector2D closestAtTo = closest(to);
  double directDistance = closestAtFrom.distance(closestAtTo);
 
  return static_cast<double>(lengthSign) * arcLengthFactor(directDistance) * directDistance;
}

arcLengthToCylindricalR()

double arcLengthToCylindricalR

(

double

cylindricalR

)

const

Calculates the two dimensional arc length till the cylindrical radius is reached If the radius can not be reached return NAN.

Note that there are two solutions which have equivalent arc lengths with different sign Always return the positive solution.

Definition at line 276 of file GeneralizedCircle.cc.

{
  // Slight trick here
  // Since the sought point is on the helix we treat it as the perigee
  // and the origin as the point to extrapolate to.
  // We know the distance of the origin to the circle, which is just d0
  // The direct distance from the origin to the imaginary perigee is just the given cylindricalR.
  const double dr = d0();
  const double directDistance =
    sqrt((cylindricalR + dr) * (cylindricalR - dr) / (1 + dr * omega()));
  const double arcLength = arcLengthFactor(directDistance) * directDistance;
  return arcLength;
}

atArcLength()

Vector2D atArcLength

(

double

arcLength

)

const

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

Definition at line 352 of file GeneralizedCircle.cc.

{
  double chi = arcLength * curvature();
  double chiHalf = chi / 2.0;
 
  double atX = arcLength * sinc(chiHalf) * sin(chiHalf) + impact();
  double atY = -arcLength * sinc(chi);
  return Vector2D::compose(-n12().unit(), atX, atY);
}

atCylindricalR()

std::pair< Vector2D, Vector2D > atCylindricalR

(

double

cylindricalR

)

const

Calculates the two points with the given cylindrical radius on the generalised circle.

Two versions in this case

Definition at line 209 of file GeneralizedCircle.cc.

{
  // extraploted to r
  // solve
  //  n0 + n1*x + n2*y + n3*r*r == 0
  //  and r = cylindricalR
  // search for x and y
 
  // solve the equation in a coordinate system parallel and orthogonal to the reduced circle center
  const Vector2D nUnit = n12().unit();
 
  // parallel component
  const double sameCylindricalRParallel = -(n0() + n3() * square(cylindricalR)) / n12().norm();
 
  // orthogonal component
  const double sameCylindricalROrthogonal = sqrt(square(cylindricalR) - square(sameCylindricalRParallel));
 
  Vector2D sameCylindricalR1 =
    Vector2D::compose(nUnit, sameCylindricalRParallel, -sameCylindricalROrthogonal);
 
  Vector2D sameCylindricalR2 =
    Vector2D::compose(nUnit, sameCylindricalRParallel, sameCylindricalROrthogonal);
 
  std::pair<Vector2D, Vector2D> result(sameCylindricalR1, sameCylindricalR2);
  return result;
}

atCylindricalRForwardOf()

Vector2D atCylindricalRForwardOf	(	const Vector2D &	startPoint,
		double	cylindricalR
	)		const

Approach on the circle with the given cylindrical radius that lies in the forward direction of a start point.

Calculates the point on the circle with cylindrical radius cylindricalR, which is closest following the circle in the direction of positive forward orientation This is particularly useful to extraplotate into a certain layer. In case no intersection with this cylindrical radius exists the function returns Vector2D(NAN,NAN)

Parameters

startPoint	Start point from which to follow in the circle in the forward direction
cylindricalR	Cylindrical radius of interest

Returns

Close point in forward direction with same cylindrical radius on the circle.

Definition at line 237 of file GeneralizedCircle.cc.

{
  std::pair<Vector2D, Vector2D> candidatePoints = atCylindricalR(cylindricalR);
  return chooseNextForwardOf(startPoint, candidatePoints.first, candidatePoints.second);
}

center()

Vector2D center ( ) const

inline

Gives the center of the circle. If it was a line both components will be infinity.

Definition at line 595 of file GeneralizedCircle.h.

      {
        return n12().divided(-2 * n3());
      }

chooseNextForwardOf()

Vector2D chooseNextForwardOf	(	const Vector2D &	start,
		const Vector2D &	end1,
		const Vector2D &	end2
	)		const

Returns the end point which is first reached if one follows the forward direction of the circle starting from the start point.

Evaluates which of the given end points end1 and end2 is closer to start This especially treats the discontinuity on the far side of the circle correctly. If the generalized circle is truly a line none of the points might lie in the forward direction and Vector2D(NAN,NAN) is returned.

Parameters

start	Point to start the traversal
end1	One possible end point
end2	Other possible end point

Returns

end1 or end2 depending, which lies closer to start in the forward direction or Vector2D(NAN,NAN) if neither end1 nor end2 are reachable in the forward direction (line case only)

Definition at line 175 of file GeneralizedCircle.cc.

{
  double lengthOnCurve1 = arcLengthBetween(start, end1);
  double lengthOnCurve2 = arcLengthBetween(start, end2);
 
  if (lengthOnCurve1 >= 0.0) {
 
    if (lengthOnCurve2 >= 0.0) {
      return lengthOnCurve1 < lengthOnCurve2 ? end1 : end2;
    } else {
      return end1;
    }
  } else {
 
    if (lengthOnCurve2 >= 0.0) {
      return end2;
    } else {
 
      // both lengths on curve have a negative sign
      // the candidate with the lesser length on curve wins because of the discontinuaty
      // unless the generalized circle is a line
      // in this case their is no forward intersection with the same cylindrical radius
      if (isLine()) {
        return Vector2D(NAN, NAN);
      } else {
        return lengthOnCurve1 < lengthOnCurve2 ? end1 : end2;
      }
    }
  }
  return Vector2D(NAN, NAN); // just avoid a compiler warning
}

closest()

Vector2D closest

(

const Vector2D &

point

)

const

Closest approach on the circle to the point.

Calculates the closest point on the circle relative to the given point.

Parameters

point

Point in the plane to calculate the closest approach to

Returns

Point of closest approach on the circle.

Definition at line 116 of file GeneralizedCircle.cc.

{
  if (fastDistance(point) == 0) return point;
 
  // solve the minimization | point - pointOnCirlce | ^2 subjected to the on circle constraint
  //                       1.0                   * m_n0 +
  //                       point.x()             * m_n1 +
  //                       point.y()             * m_n2 +
  //                       point.cylindricalRSquared() * m_n3 == 0
  // solved with a Lagrangian multiplicator for the constraint
 
  Vector2D gradientAtPoint = gradient(point);
  Vector2D coordinateSystem = gradientAtPoint.unit();
 
  // component of closest approach orthogonal to coordinateSystem
  double closestOrthogonal = n12().cross(point) / gradientAtPoint.norm();
 
  // component of closest approach parallel to coordinateSystem - two solutions expected
  double nOrthogonal = n12().unnormalizedOrthogonalComp(coordinateSystem);
  double nParallel = n12().unnormalizedParallelComp(coordinateSystem);
 
  double closestParallel = 0.0;
  if (isLine()) {
    closestParallel = -(nOrthogonal * closestOrthogonal + n0()) / nParallel;
 
  } else {
    const double a = n3();
    const double b = nParallel;
    const double c = n0() + (nOrthogonal + n3() * closestOrthogonal) * closestOrthogonal;
 
    const std::pair<double, double> closestParallel12 = solveQuadraticABC(a, b, c);
 
    // take the solution with smaller distance to point
    const double pointParallel = point.unnormalizedParallelComp(coordinateSystem);
 
    const double criterion1 =
      closestParallel12.first * (closestParallel12.first - 2 * pointParallel);
    const double criterion2 =
      closestParallel12.second * (closestParallel12.second - 2 * pointParallel);
 
    closestParallel = criterion1 < criterion2 ? closestParallel12.first : closestParallel12.second;
  }
  return Vector2D::compose(coordinateSystem, closestParallel, closestOrthogonal);
}

conformalTransform()

void conformalTransform ( )

inline

Transforms the generalized circle to conformal space inplace Applies the conformal map in the self-inverse form X = x / (x^2 + y^2) and Y = y / (x^2 +y^2) inplace It works most easily by the exchange of the circle parameters n0 <-> n3.

Definition at line 229 of file GeneralizedCircle.h.

      {
        std::swap(m_n0, m_n3);
        reverse(); // Correct orientation
      }

conformalTransformed()

GeneralizedCircle conformalTransformed ( ) const

inline

Returns a copy of the circle in conformal space.

Applies the conformal map in the self-inverse from X = x / (x^2 + y^2) and Y = y / (x^2 +y^2) and returns the result as a new GeneralizedCircle. It works most easily by the exchange of the circle parameters n0 <-> n3

Definition at line 335 of file GeneralizedCircle.h.

      {
        return GeneralizedCircle(-n3(), -n12(), -n0());
      }

curvature()

double curvature ( ) const

inline

Gives the signed curvature of the generalized circle.

Definition at line 580 of file GeneralizedCircle.h.

      {
        return 2 * n3();
      }

d0()

double d0 ( ) const

inline

Getter for the absolute distance to the z axes at the support point.

Definition at line 499 of file GeneralizedCircle.h.

      {
        return -impact();
      }

distance() [1/2]

double distance

(

const Vector2D &

point

)

const

Gives the proper distance of the point to the circle line retaining the sign of the fast distance.

Definition at line 296 of file GeneralizedCircle.cc.

{
  const double fastD = fastDistance(point);
  return distance(fastD);
}

distance() [2/2]

double distance

(

double

fastDistance

)

const

Helper function to translate the fast linearized distance measure of the generalized circle to the proper distance from the circle retaining the sign of the distance.

Definition at line 302 of file GeneralizedCircle.cc.

{
  if (fastDistance == 0.0 or isLine()) {
    // special case for unfitted state
    // and line
    return fastDistance;
  } else {
 
    const double a = n3();
    const double b = 1;
    const double c = -fastDistance;
 
    std::pair<double, double> distance12 = solveQuadraticABC(a, b, c);
 
    // take the small solution which has always the same sign of the fastDistance
    return distance12.second;
  }
}

fastDistance() [1/2]

double fastDistance ( const double  distance ) const

inline

Helper function to translate the proper distance to the linearized distance measure of the generalized circle retaining the sign of the distance.

Definition at line 487 of file GeneralizedCircle.h.

      {
        return (distance * n3() + 1.0) * distance;
      }

fastDistance() [2/2]

double fastDistance ( const Vector2D &  point ) const

inline

Approximate distance.

Gives a fast signed approximation of the distance to the circle. The absolute value of the fast distance is accurate up to first order of distance/circle radius, hence best used with big circles and small distances. The sign of the fast distance indicates if the point is to the right or to the left of the circle.

Definition at line 459 of file GeneralizedCircle.h.

      {
        return n0() + point.dot(n12()) + point.normSquared() * n3();
      }

fastImpact()

double fastImpact ( ) const

inline

Approximate distance to the origin.

Definition at line 465 of file GeneralizedCircle.h.

      {
        return n0();
      }

fromCenterAndRadius()

GeneralizedCircle fromCenterAndRadius ( const Vector2D &  center, double  absRadius, ERotation  orientation = ERotation::c_CounterClockwise  )

static

Constructor from center, radius and a optional orientation.

The center and radius alone do not carry any orientation. However the generalized circle does. This constructor makes an orientated representation from them. If not given the orientation defaults to mathematical positive counterclockwise.

Definition at line 67 of file GeneralizedCircle.cc.

{
  GeneralizedCircle generalizedCircle;
  generalizedCircle.setCenterAndRadius(center, absRadius, orientation);
  return generalizedCircle;
}

fromPerigeeParameters() [1/2]

GeneralizedCircle fromPerigeeParameters ( double  curvature, const Vector2D &  tangential, double  impact  )

static

Constructor of a generalized circle from perigee parameters.

Tangential at perigee given as two dimensional vector.

Definition at line 76 of file GeneralizedCircle.cc.

{
  GeneralizedCircle generalizedCircle;
  generalizedCircle.setPerigeeParameters(curvature, tangential, impact);
  return generalizedCircle;
}

fromPerigeeParameters() [2/2]

GeneralizedCircle fromPerigeeParameters ( double  curvature, double  tangentialPhi, double  impact  )

static

Constructor of a generalized circle from perigee parameters.

Tangential at perigee given as azimuth angle.

Definition at line 85 of file GeneralizedCircle.cc.

{
  GeneralizedCircle generalizedCircle;
  generalizedCircle.setPerigeeParameters(curvature, tangentialPhi, impact);
  return generalizedCircle;
}

gradient()

Vector2D gradient ( const Vector2D &  point ) const

inline

Gradient of the distance field Gives the gradient of the approximated distance field for the given point.

Parameters

point

Point in the plane to calculate the gradient

Returns

Gradient of the distance field

Definition at line 347 of file GeneralizedCircle.h.

      {
        return point * (2.0 * n3()) + n12();
      }

impact()

double impact ( ) const

inline

Gives the signed distance of the origin to the circle.

Definition at line 493 of file GeneralizedCircle.h.

      {
        return distance(fastImpact());
      }

intersections()

std::pair< Vector2D, Vector2D > intersections

(

const GeneralizedCircle &

generalizedCircle

)

const

Calculates the two points common to both circles.

If the two points coincide both returned points are the same. If there is no common point both returned points will be made of NAN.

Definition at line 322 of file GeneralizedCircle.cc.

{
  const double m0 = generalizedCircle.n0();
  const Vector2D& m12 = generalizedCircle.n12();
  const double m3 = generalizedCircle.n3();
 
  const double n0 = this->n0();
  const Vector2D& n12 = this->n12();
  const double n3 = this->n3();
 
  Vector2D unitC = n12 * m3 - m12 * n3;
  double absC = unitC.normalize();
 
  double xParallel = (m0 * n3 - m3 * n0) / absC;
 
  // Use symmetric solution and use all input parameters
  Vector2D mn12 = n12 + m12;
  double mn12Parallel = unitC.unnormalizedParallelComp(mn12);
  double mn12Orthogonal = unitC.unnormalizedOrthogonalComp(mn12);
 
  double a = m3 + n3;
  double b = mn12Orthogonal;
  double c = (a * xParallel + mn12Parallel) * xParallel + m0 + n0;
 
  std::pair<double, double> xOrthogonal = solveQuadraticABC(a, b, c);
 
  return std::make_pair(Vector2D::compose(unitC, xParallel, xOrthogonal.first),
                        Vector2D::compose(unitC, xParallel, xOrthogonal.second));
}

invalidate()

void invalidate ( )

inline

Sets all circle parameters to zero.

Definition at line 212 of file GeneralizedCircle.h.

      {
        setN(0.0, 0.0, 0.0, 0.0);
      }

isCircle()

bool isCircle ( ) const

inline

Indicates if the generalized circle is actually a circle.

Definition at line 562 of file GeneralizedCircle.h.

      {
        return n3() != 0.0;
      }

isForwardOrBackwardOf()

EForwardBackward isForwardOrBackwardOf ( const Vector2D &  from, const Vector2D &  to  ) const

inline

Calculates if the to vector is closer to the from vector following the along orientation of the circle or against.

Returns

• EForwardBackward::c_Forward in case the to vector is closer following the along the orientation
• EForwardBackward::c_Backward in case the to vector is closer against the orientation.
• EForwardBackward::c_Unknown if neither can be determined.

Definition at line 408 of file GeneralizedCircle.h.

      {
        Vector2D difference = to - from;
        Vector2D tangentialAtFrom = tangential(from);
        return tangentialAtFrom.isForwardOrBackwardOf(difference);
      }

isInvalid()

bool isInvalid ( ) const

inline

Indicates if all circle parameters are zero.

Definition at line 300 of file GeneralizedCircle.h.

      {
        return n0() == 0 and n12().isNull() and n3() == 0;
      }

isLeft()

bool isLeft ( const Vector2D &  rhs ) const

inline

Return if the point given is left of the line.

Definition at line 544 of file GeneralizedCircle.h.

      {
        return isRightOrLeft(rhs) == ERightLeft::c_Left;
      }

isLine()

bool isLine ( ) const

inline

Indicates if the generalized circle is actually a line.

Definition at line 556 of file GeneralizedCircle.h.

      {
        return n3() == 0.0;
      }

isRight()

bool isRight ( const Vector2D &  rhs ) const

inline

Return if the point given is right of the line.

Definition at line 550 of file GeneralizedCircle.h.

      {
        return isRightOrLeft(rhs) == ERightLeft::c_Right;
      }

isRightOrLeft()

ERightLeft isRightOrLeft ( const Vector2D &  point ) const

inline

Indicates if the point is on the right or left side of the circle.

This is also referred to as alpha.

Definition at line 538 of file GeneralizedCircle.h.

      {
        return static_cast<ERightLeft>(sign(fastDistance(point)));
      }

isValid()

bool isValid ( ) const

inline

Indicates if the combination of the circle parameters makes up a valid circle.

Definition at line 306 of file GeneralizedCircle.h.

      {
        return not isInvalid();
      }

maximalCylindricalR()

double maximalCylindricalR ( ) const

inline

Gives the maximal cylindrical radius the circle reaches.

Definition at line 523 of file GeneralizedCircle.h.

      {
        return std::fabs(impact() + 2 * radius());
      }

minimalCylindricalR()

double minimalCylindricalR ( ) const

inline

Gives the minimal cylindrical radius the circle reaches (unsigned)

Definition at line 517 of file GeneralizedCircle.h.

      {
        return std::fabs(impact());
      }

n0()

double n0 ( ) const

inline

Getter for the first circle parameter.

Definition at line 269 of file GeneralizedCircle.h.

      {
        return m_n0;
      }

n1()

double n1 ( ) const

inline

Getter for the second circle parameter.

Definition at line 275 of file GeneralizedCircle.h.

      {
        return m_n12.x();
      }

n12()

const Vector2D & n12 ( ) const

inline

Getter for the second and third circle parameter which natuarally from a vector.

Definition at line 287 of file GeneralizedCircle.h.

      {
        return m_n12;
      }

n2()

double n2 ( ) const

inline

Getter for the third circle parameter.

Definition at line 281 of file GeneralizedCircle.h.

      {
        return m_n12.y();
      }

n3()

double n3 ( ) const

inline

Getter for the fourth circle parameter.

Definition at line 293 of file GeneralizedCircle.h.

      {
        return m_n3;
      }

normal()

Vector2D normal ( const Vector2D &  point ) const

inline

Normal vector to the circle near the given position.

Gives the unit normal vector to the circle line. It points to the outside of the circle for counterclockwise orientation, and inwards for the clockwise orientation. It is essentially the gradient normalized to unit length

Parameters

point

Point in the plane to calculate the tangential

Returns

Unit normal vector to the circle line

Definition at line 361 of file GeneralizedCircle.h.

      {
        return gradient(point).unit();
      }

normalizationSquared()

double normalizationSquared ( ) const

inline

Calculates the generalized circle specific squared norm.

Correctly normalized this should give one.

Definition at line 315 of file GeneralizedCircle.h.

      {
        return n12().normSquared() - 4 * n0() * n3();
      }

normalize()

void normalize ( )

inlineprotected

Normalizes the circle parameters.

The normalization is only made if the circle parameters have a valid positive norm. If the normalization of the circle is negative or zero all circle parameters are not changed.

Definition at line 252 of file GeneralizedCircle.h.

      {
        double normalization_squared = normalizationSquared();
        if (normalization_squared > 0) scaleN(1.0 / std::sqrt(normalization_squared));
      }

omega()

double omega ( ) const

inline

Gives the omega parameter as used by the framework helix.

It is the signed curvature of the generalized circle with the opposite sign.

Definition at line 589 of file GeneralizedCircle.h.

      {
        return -curvature();
      }

orientation()

ERotation orientation ( ) const

inline

Gives the orientation of the circle.

The circle can be either orientated counterclockwise or clockwise.

Returns

ERotation::c_CounterClockwise for counterclockwise travel, ERotation::c_Clockwise for clockwise.

Definition at line 612 of file GeneralizedCircle.h.

      {
        return static_cast<ERotation>(sign(n3()));
      }

passiveMoveBy()

void passiveMoveBy ( const Vector2D &  by )

inline

Moves the coordinate system by the given vector.

Updates the circle parameters inplace.

Definition at line 239 of file GeneralizedCircle.h.

      {
        setN(fastDistance(by), gradient(by), n3());
      }

perigee()

Vector2D perigee

(

)

const

Calculates the closest approach to the two dimensional origin.

Definition at line 161 of file GeneralizedCircle.cc.

{
  Vector2D result(n12());
  result.setCylindricalR(-impact());
  return result;
}

perimeter()

double perimeter ( ) const

inline

Gives the perimeter of the circle.

Definition at line 601 of file GeneralizedCircle.h.

      {
        return 2 * M_PI * radius();
      }

radius()

double radius ( ) const

inline

Gives the signed radius of the circle. If it was a line this will be infinity.

Definition at line 568 of file GeneralizedCircle.h.

      {
        return 1 / curvature();
      }

reverse()

void reverse ( )

inline

Flips the orientation of the circle in place.

Definition at line 218 of file GeneralizedCircle.h.

      {
        scaleN(-1);
      }

reversed()

GeneralizedCircle reversed ( ) const

inline

Returns a copy of the circle with opposite orientation.

Definition at line 322 of file GeneralizedCircle.h.

      {
        return GeneralizedCircle(-n0(), -n12(), -n3());
      }

scaleN()

void scaleN ( const double  factor )

inlineprivate

Scales the circle parameters by a common factor.

Definition at line 260 of file GeneralizedCircle.h.

      {
        m_n0 *= factor;
        m_n12 *= factor;
        m_n3 *= factor;
      }

setCenterAndRadius()

void setCenterAndRadius	(	const Vector2D &	center,
		double	absRadius,
		ERotation	orientation = ERotation::c_CounterClockwise
	)

Setter for the circle center and radius.

Definition at line 94 of file GeneralizedCircle.cc.

{
  double curvature = static_cast<double>(orientation) / fabs(absRadius);
  setN0((center.normSquared() - absRadius * absRadius) * curvature / 2.0);
  setN1(-center.x() * curvature);
  setN2(-center.y() * curvature);
  setN3(curvature / 2.0);
  normalize(); // the call to normalize should be superfluous
}

setN() [1/4]

void setN ( const double  n0, const double  n1, const double  n2, const double  n3 = 0.0  )

inline

Setter for all four circle parameters.

Makes a normalization after setting. The normal representation of a line leave out the last parameter.

Definition at line 178 of file GeneralizedCircle.h.

      {
        setN0(n0);
        setN12(n1, n2);
        setN3(n3);
        normalize();
      }

setN() [2/4]

void setN ( const double  n0, const Vector2D &  n12, const double  n3 = 0.0  )

inline

Setter for all four circle parameters.

Makes a normalization after setting. The normal representation of a line leave out the last parameter.

Definition at line 191 of file GeneralizedCircle.h.

      {
        setN0(n0);
        setN12(n12);
        setN3(n3);
        normalize();
      }

setN() [3/4]

void setN ( const GeneralizedCircle &  n0123 )

inline

Setter for all four circle parameters from another circle.

Definition at line 206 of file GeneralizedCircle.h.

      {
        setN(n0123.n0(), n0123.n12(), n0123.n3());
      }

setN() [4/4]

void setN ( const Line2D &  n012 )

inline

Setter for all four circle parameters from another circle.

Definition at line 200 of file GeneralizedCircle.h.

      {
        setN(n012.n0(), n012.n12());
      }

setN0()

void setN0 ( const double  n0 )

inlineprotected

Setter for first circle parameter.

Makes no normalization after setting. Use is discouraged.

Definition at line 102 of file GeneralizedCircle.h.

      {
        m_n0 = n0;
      }

setN1()

void setN1 ( const double  n1 )

inlineprotected

Setter for second circle parameter.

Makes no normalization after setting. Use is discouraged.

Definition at line 112 of file GeneralizedCircle.h.

      {
        m_n12.setX(n1);
      }

setN12() [1/2]

void setN12 ( const double  n1, const double  n2  )

inlineprotected

Setter for second and third circle parameter.

Makes no normalization after setting. Use is discouraged.

Definition at line 132 of file GeneralizedCircle.h.

      {
        m_n12.setXY(n1, n2);
      }

setN12() [2/2]

void setN12 ( const Vector2D &  n12 )

inlineprotected

Setter for second and third circle parameter.

Makes no normalization after setting. Use is discouraged.

Definition at line 142 of file GeneralizedCircle.h.

      {
        m_n12.setXY(n12);
      }

setN2()

void setN2 ( const double  n2 )

inlineprotected

Setter for third circle parameter.

Makes no normalization after setting. Use is discouraged.

Definition at line 122 of file GeneralizedCircle.h.

      {
        m_n12.setY(n2);
      }

setN3()

void setN3 ( const double  n3 )

inlineprotected

Setter for fourth circle parameter.

Makes no normalization after setting. Use is discouraged.

Definition at line 152 of file GeneralizedCircle.h.

      {
        m_n3 = n3;
      }

setPerigeeParameters() [1/2]

void setPerigeeParameters ( const double  curvature, const double  tangentialPhi, const double  impact  )

inline

Setter for the perigee parameters.

Definition at line 168 of file GeneralizedCircle.h.

      {
        setPerigeeParameters(curvature, Vector2D::Phi(tangentialPhi), impact);
      }

setPerigeeParameters() [2/2]

void setPerigeeParameters	(	double	curvature,
		const Vector2D &	tangential,
		double	impact
	)

Setter for the perigee parameters.

Definition at line 106 of file GeneralizedCircle.cc.

{
  double n0 = impact * (impact * curvature / 2.0 + 1.0);
  Vector2D n12 = -tangential.orthogonal() * (1 + curvature * impact);
  double n3 = curvature / 2.0;
  setN(n0, n12, n3);
}

tangential() [1/2]

Vector2D tangential ( ) const

inline

Gives the tangential vector at the closest approach to the origin / at the perigee.

Definition at line 505 of file GeneralizedCircle.h.

      {
        return tangential(Vector2D(0.0, 0.0)).unit();
      }

tangential() [2/2]

Vector2D tangential ( const Vector2D &  point ) const

inline

Tangential vector to the circle near the given position.

Gives the unit tangential vector to the circle line. It always points in the direction of positive advance at the point of closest approach from the given point.

Parameters

point

Point in the plane to calculate the tangential

Returns

Unit tangential vector to the circle line

Definition at line 374 of file GeneralizedCircle.h.

      {
        return normal(point).orthogonal();
      }

tangentialPhi()

double tangentialPhi ( ) const

inline

Gives to azimuth angle phi of the direction of flight at the perigee.

Definition at line 511 of file GeneralizedCircle.h.

      {
        return tangential().phi();
      }

Member Data Documentation

m_n0

double m_n0

private

Memory for the first parameter.

Definition at line 691 of file GeneralizedCircle.h.

m_n12

Vector2D m_n12

private

Memory for the second and third parameter.

Definition at line 688 of file GeneralizedCircle.h.

m_n3

double m_n3

private

Memory for the fourth parameter.

Definition at line 685 of file GeneralizedCircle.h.

---

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

• tracking/trackFindingCDC/geometry/include/GeneralizedCircle.h
• tracking/trackFindingCDC/geometry/src/GeneralizedCircle.cc