FreeTrajectoryState Class Reference

#include <FreeTrajectoryState.h>

Public Member Functions
CartesianTrajectoryError	cartesianError () const
TrackCharge	charge () const
const CurvilinearTrajectoryError &	curvilinearError () const
	FreeTrajectoryState ()
	FreeTrajectoryState (const GlobalTrajectoryParameters &aGlobalParameters)
	FreeTrajectoryState (const GlobalPoint &aX, const GlobalVector &aP, TrackCharge aCharge, const MagneticField *fieldProvider)
	FreeTrajectoryState (const GlobalPoint &aX, const GlobalVector &aP, TrackCharge aCharge, const MagneticField *fieldProvider, GlobalVector fieldValue)
	FreeTrajectoryState (const GlobalTrajectoryParameters &aGlobalParameters, const CurvilinearTrajectoryError &aCurvilinearError)
	FreeTrajectoryState (const GlobalTrajectoryParameters &aGlobalParameters, const CartesianTrajectoryError &aCartesianError)
	FreeTrajectoryState (const GlobalTrajectoryParameters &aGlobalParameters, const CartesianTrajectoryError &, const CurvilinearTrajectoryError &aCurvilinearError)
bool	hasCurvilinearError () const
bool	hasError () const
GlobalVector	momentum () const
const GlobalTrajectoryParameters &	parameters () const
GlobalPoint	position () const
void	rescaleError (double factor)
void	setCartesianError (const CartesianTrajectoryError &err)
void	setCartesianError (const AlgebraicSymMatrix66 &err)
CurvilinearTrajectoryError &	setCurvilinearError ()
void	setCurvilinearError (const CurvilinearTrajectoryError &err)
double	signedInverseMomentum () const
double	transverseCurvature () const

Private Member Functions
void	createCartesianError (CartesianTrajectoryError &aCartesianError) const
void	createCurvilinearError (CartesianTrajectoryError const &aCartesianError) const
void	missingError () const

Private Attributes
CurvilinearTrajectoryError	theCurvilinearError
GlobalTrajectoryParameters	theGlobalParameters

Detailed Description

A 6-dimensional state vector of a helix given at some point in space along the helix, and the associated error matrix. The error can be obtained in two different parametrizations: CurvilinearTrajectoryError and CartesianTrajectoryError (see descriptions of corresponding classes). The FreeTrajectoryState can be instantiated with either of these error parametrisations; it converts from one to the other internally.

Definition at line 29 of file FreeTrajectoryState.h.

Constructor & Destructor Documentation

FreeTrajectoryState::FreeTrajectoryState ( )

inline

Definition at line 34 of file FreeTrajectoryState.h.

                       :  
    theCurvilinearError(InvalidError()) {}

FreeTrajectoryState::FreeTrajectoryState ( const GlobalTrajectoryParameters &  aGlobalParameters )

inline

Definition at line 37 of file FreeTrajectoryState.h.

                                                                           :
  theGlobalParameters(aGlobalParameters),  
  theCurvilinearError(InvalidError())
  {}

FreeTrajectoryState::FreeTrajectoryState ( const GlobalPoint &  aX, const GlobalVector &  aP, TrackCharge  aCharge, const MagneticField *  fieldProvider  )

inline

Definition at line 42 of file FreeTrajectoryState.h.

                                                          :
    theGlobalParameters(aX, aP, aCharge, fieldProvider),  
    theCurvilinearError(InvalidError())
  {}

FreeTrajectoryState::FreeTrajectoryState ( const GlobalPoint &  aX, const GlobalVector &  aP, TrackCharge  aCharge, const MagneticField *  fieldProvider, GlobalVector  fieldValue  )

inline

Definition at line 51 of file FreeTrajectoryState.h.

                                       :
    theGlobalParameters(aX, aP, aCharge, fieldProvider,fieldValue),
    theCurvilinearError(InvalidError())
  {}

FreeTrajectoryState::FreeTrajectoryState ( const GlobalTrajectoryParameters &  aGlobalParameters, const CurvilinearTrajectoryError &  aCurvilinearError  )

inline

Definition at line 62 of file FreeTrajectoryState.h.

                                                                           :
    theGlobalParameters(aGlobalParameters),
    theCurvilinearError(aCurvilinearError)
  {}

FreeTrajectoryState::FreeTrajectoryState ( const GlobalTrajectoryParameters &  aGlobalParameters, const CartesianTrajectoryError &  aCartesianError  )

inline

Definition at line 70 of file FreeTrajectoryState.h.

References createCurvilinearError().

                                                                       :
    theGlobalParameters(aGlobalParameters)
  { createCurvilinearError(aCartesianError);  }

FreeTrajectoryState::FreeTrajectoryState ( const GlobalTrajectoryParameters &  aGlobalParameters, const CartesianTrajectoryError &  , const CurvilinearTrajectoryError &  aCurvilinearError  )

inline

Definition at line 75 of file FreeTrajectoryState.h.

                                                                           :
    theGlobalParameters(aGlobalParameters),
    theCurvilinearError(aCurvilinearError)  
  {}

Member Function Documentation

CartesianTrajectoryError FreeTrajectoryState::cartesianError ( ) const

inline

Definition at line 114 of file FreeTrajectoryState.h.

References createCartesianError(), hasError(), missingError(), and unlikely.

Referenced by BasicTrajectoryState::cartesianError(), HTrack::computePull(), CSCEfficiency::getFromFTS(), SimpleCosmicBONSeeder::seeds(), TrajectoryStateClosestToBeamLine::transverseImpactParameter(), and SimpleNavigableLayer::wellInside().

                                                  {
    if unlikely(!hasError()) missingError();
    CartesianTrajectoryError aCartesianError;
    createCartesianError(aCartesianError);
    return aCartesianError;
  }

TrackCharge FreeTrajectoryState::charge ( void  ) const

inline

Definition at line 90 of file FreeTrajectoryState.h.

References GlobalTrajectoryParameters::charge(), and theGlobalParameters.

Referenced by TrackProducerAlgorithm< reco::Track >::buildTrack(), TrackProducerAlgorithm< reco::GsfTrack >::buildTrack(), MuonTrackLoader::buildTrackAtPCA(), MuonSeedTrack::buildTrackAtPCA(), MuonTrackLoader::buildTrackUpdatedAtPCA(), TangentApproachInRPhi::calculate(), reco::TransientTrackFromFTS::charge(), BasicTrajectoryState::charge(), HTrack::computePull(), HTrack::computeResolution(), HTrack::computeTDRResolution(), BasicTrajectoryState::createLocalParameters(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), HTrack::Fill(), GsfTrackProducerBase::fillMode(), PerigeeConversions::ftsToPerigeeParameters(), MatcherUsingTracksAlgorithm::getChi2(), CSCEfficiency::getFromFTS(), TrackAssociatorByChi2::parametersAtClosestApproach(), KinematicState::particleCharge(), OutInConversionSeedFinder::startSeed(), InOutConversionSeedFinder::startSeed(), SteppingHelixStateInfo::SteppingHelixStateInfo(), and reco::TransientTrackFromFTS::track().

                             {
    return theGlobalParameters.charge();
  }

void FreeTrajectoryState::createCartesianError ( CartesianTrajectoryError &  aCartesianError ) const

private

Definition at line 29 of file FreeTrajectoryState.cc.

References JacobianCurvilinearToCartesian::jacobian(), CurvilinearTrajectoryError::matrix(), theCurvilinearError, and theGlobalParameters.

Referenced by cartesianError().

                                                                                              {
  
  JacobianCurvilinearToCartesian curv2Cart(theGlobalParameters);
  const AlgebraicMatrix65& jac = curv2Cart.jacobian();

  aCartesianError = 
    ROOT::Math::Similarity(jac, theCurvilinearError.matrix());
}

void FreeTrajectoryState::createCurvilinearError ( CartesianTrajectoryError const &  aCartesianError ) const

private

Definition at line 39 of file FreeTrajectoryState.cc.

References JacobianCartesianToCurvilinear::jacobian(), CartesianTrajectoryError::matrix(), theCurvilinearError, and theGlobalParameters.

Referenced by FreeTrajectoryState(), and setCartesianError().

                                                                                                     {
  
  JacobianCartesianToCurvilinear cart2Curv(theGlobalParameters);
  const AlgebraicMatrix56& jac = cart2Curv.jacobian();
  
  theCurvilinearError = 
    ROOT::Math::Similarity(jac, aCartesianError.matrix());

} 

const CurvilinearTrajectoryError& FreeTrajectoryState::curvilinearError ( ) const

inline

Definition at line 121 of file FreeTrajectoryState.h.

References hasError(), missingError(), theCurvilinearError, and unlikely.

Referenced by MuonErrorMatrix::adjust(), TSGFromPropagation::adjust(), FastTSGFromPropagation::adjust(), analyticalErrorPropagation(), TrackProducerAlgorithm< reco::Track >::buildTrack(), TrackProducerAlgorithm< reco::GsfTrack >::buildTrack(), MuonTrackLoader::buildTrackAtPCA(), MuonSeedTrack::buildTrackAtPCA(), MuonTrackLoader::buildTrackUpdatedAtPCA(), HTrack::computePull(), BasicTrajectoryState::createLocalErrorFromCurvilinearError(), BasicTrajectoryState::curvilinearError(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), GsfTrackProducerBase::fillMode(), PerigeeConversions::ftsToPerigeeError(), MatcherUsingTracksAlgorithm::getChi2(), TrajectoryStateAccessor::inversePtError(), TSCBLBuilderNoMaterial::operator()(), operator<<(), TrackKinematicStatePropagator::propagateToTheTransversePCACharged(), SteppingHelixStateInfo::SteppingHelixStateInfo(), reco::TransientTrackFromFTS::track(), and MuonNavigableLayer::trackingRange().

                                                             {
    if  unlikely(!hasError()) missingError();
    return theCurvilinearError;
  }

bool FreeTrajectoryState::hasCurvilinearError ( ) const

inline

Definition at line 102 of file FreeTrajectoryState.h.

References theCurvilinearError, and CurvilinearTrajectoryError::valid().

Referenced by BasicTrajectoryState::checkCurvilinError(), BasicTrajectoryState::createLocalError(), and hasError().

{return theCurvilinearError.valid();}

bool FreeTrajectoryState::hasError ( void  ) const

inline

Definition at line 104 of file FreeTrajectoryState.h.

References hasCurvilinearError().

Referenced by analyticalErrorPropagation(), cartesianError(), curvilinearError(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), MatcherUsingTracksAlgorithm::getChi2(), CSCEfficiency::getFromFTS(), BasicTrajectoryState::hasError(), operator<<(), MuonUpdatorAtVertex::propagate(), rescaleError(), SteppingHelixStateInfo::SteppingHelixStateInfo(), and TrajectoryStateClosestToPoint::TrajectoryStateClosestToPoint().

                        {
    return hasCurvilinearError();
  }

void FreeTrajectoryState::missingError ( ) const

private

Definition at line 14 of file FreeTrajectoryState.cc.

References CurvilinearTrajectoryError::matrix(), theCurvilinearError, and CurvilinearTrajectoryError::valid().

Referenced by cartesianError(), and curvilinearError().

                                             {
  std::stringstream form;
  form<< "FreeTrajectoryState: attempt to access errors when none available" <<
    "\nCurvilinear error valid/values :"<< theCurvilinearError.valid() << "\n" 
      <<  theCurvilinearError.matrix();
    edm::LogWarning("FreeTrajectoryState") << "(was exception) " << form.str();
    //  throw TrajectoryStateException(form.str());
    // bPoint();
}

GlobalVector FreeTrajectoryState::momentum ( ) const

inline

Definition at line 87 of file FreeTrajectoryState.h.

References GlobalTrajectoryParameters::momentum(), and theGlobalParameters.

Referenced by MuonErrorMatrix::adjust(), TSGFromPropagation::adjust(), FastTSGFromPropagation::adjust(), ElectronSeedAnalyzer::analyze(), ValidationMisalignedTracker::analyze(), TestOutliers::analyze(), HLTPMMassFilter::approxMomAtVtx(), CosmicMuonTrajectoryBuilder::build(), TrackProducerAlgorithm< reco::Track >::buildTrack(), TrackProducerAlgorithm< reco::GsfTrack >::buildTrack(), MuonTrackLoader::buildTrackAtPCA(), MuonSeedTrack::buildTrackAtPCA(), MuonTrackLoader::buildTrackUpdatedAtPCA(), TangentApproachInRPhi::calculate(), DirectMuonNavigation::checkCompatible(), DirectMuonNavigation::compatibleEndcapLayers(), PixelHitMatcher::compatibleHits(), DirectMuonNavigation::compatibleLayers(), PixelHitMatcher::compatibleSeeds(), HTrack::computePull(), HTrack::computeResolution(), HTrack::computeTDRResolution(), ReferenceTrajectory::construct(), BasicTrajectoryState::createLocalParameters(), TransverseImpactPointExtrapolator::doExtrapolation(), MuonPatternRecoDumper::dumpFTS(), CSCEfficiency::efficienciesPerChamber(), PropagateToMuon::extrapolate(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), HTrack::Fill(), TrackDetectorAssociator::fillCaloTruth(), GsfTrackProducerBase::fillMode(), MuonTrackAnalyzer::fillPlots(), PerigeeConversions::ftsToPerigeeParameters(), MatcherUsingTracksAlgorithm::getChi2(), CSCEfficiency::getFromFTS(), HDetIdAssociator::getTrajectory(), BasicTrajectoryState::globalDirection(), KinematicState::globalMomentum(), BasicTrajectoryState::globalMomentum(), TrackDetMatchInfo::insideCone(), TrajectoryStateAccessor::inversePtError(), MuonNavigableLayer::isInsideOut(), PerigeeConversions::jacobianCurvilinear2Perigee(), MuonErrorMatrixAdjuster::makeTrack(), MatcherUsingTracksAlgorithm::matchByDirectComparison(), MatcherUsingTracksAlgorithm::matchWithPropagation(), CosmicParametersDefinerForTP::momentum(), StateOnTrackerBound::operator()(), PropagationDirectionChooser::operator()(), TrimmedTrackFilter::operator()(), CrossingPtBasedLinearizationPointFinder::CompareTwoTracks::operator()(), CosmicTrackingParticleSelector::operator()(), DirectMuonNavigation::outward(), DirectTrackerNavigation::outward(), TrackAssociatorByChi2::parametersAtClosestApproach(), TrajectorySeedProducer::produce(), TrackCandidateProducer::produce(), reco::modules::DuplicateTrackMerger::produce(), StandAloneMuonTrajectoryBuilder::propagateTheSeedTSOS(), TrackKinematicStatePropagator::propagateToTheTransversePCANeutral(), TrackClassifier::reconstructionInformation(), StandAloneMuonFilter::refit(), MuonTrackingRegionBuilder::region(), SeedFilter::seeds(), TrackAlgoCompareUtil::SetTrackingParticleD0Dz(), SteppingHelixStateInfo::SteppingHelixStateInfo(), CSCEfficiency::stripWire_Efficiencies(), ThresholdPtTrajectoryFilter::test(), reco::TransientTrackFromFTS::track(), and StandAloneMuonTrajectoryBuilder::trajectories().

                                {
    return theGlobalParameters.momentum();
  }

const GlobalTrajectoryParameters& FreeTrajectoryState::parameters ( void  ) const

inline

Definition at line 109 of file FreeTrajectoryState.h.

References theGlobalParameters.

Referenced by MuonErrorMatrix::adjust(), TSGFromPropagation::adjust(), FastTSGFromPropagation::adjust(), LayerCollector::allLayers(), analyticalErrorPropagation(), TangentApproachInRPhi::calculate(), MuonPatternRecoDumper::dumpFTS(), TrajectoryExtrapolatorToLine::extrapolate(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), PerigeeConversions::ftsToPerigeeParameters(), AddTvTrack::getSecondaryVertices(), BasicTrajectoryState::globalParameters(), PerigeeConversions::jacobianCurvilinear2Perigee(), BasicTrajectoryState::magneticField(), operator<<(), PerigeeKinematicState::PerigeeKinematicState(), MuonUpdatorAtVertex::propagate(), rescaleError(), StartingLayerFinder::startingLayers(), ThresholdPtTrajectoryFilter::test(), KinematicState::trajectoryParameters(), TrajectoryStateClosestToPoint::TrajectoryStateClosestToPoint(), PerigeeRefittedTrackState::trajectoryStateOnSurface(), and BasicTrajectoryState::update().

                                                       {
    return theGlobalParameters;
  }

GlobalPoint FreeTrajectoryState::position ( ) const

inline

Definition at line 84 of file FreeTrajectoryState.h.

References GlobalTrajectoryParameters::position(), and theGlobalParameters.

Referenced by TrackDetectorAssociator::addTAMuonSegmentMatch(), ElectronSeedAnalyzer::analyze(), ValidationMisalignedTracker::analyze(), HcalCorrPFCalculation::analyze(), MuonAlignmentAnalyzer::analyze(), TestOutliers::analyze(), TrackDetectorAssociator::associate(), CosmicMuonTrajectoryBuilder::build(), SeedForPhotonConversionFromQuadruplets::buildSeedBool(), TrackProducerAlgorithm< reco::Track >::buildTrack(), TrackProducerAlgorithm< reco::GsfTrack >::buildTrack(), MuonTrackLoader::buildTrackAtPCA(), MuonSeedTrack::buildTrackAtPCA(), MuonTrackLoader::buildTrackUpdatedAtPCA(), TangentApproachInRPhi::calculate(), reco::TransientTrackFromFTS::calculateTSOSAtVertex(), reco::GsfTransientTrack::calculateTSOSAtVertex(), DirectMuonNavigation::checkCompatible(), DirectTrackerNavigation::checkCompatible(), PixelHitMatcher::compatibleHits(), DirectMuonNavigation::compatibleLayers(), DirectTrackerNavigation::compatibleLayers(), PixelHitMatcher::compatibleSeeds(), BasicTrajectoryState::createLocalParameters(), SimpleNavigableLayer::crossingState(), TransverseImpactPointExtrapolator::doExtrapolation(), MuonPatternRecoDumper::dumpFTS(), PropagateToMuon::extrapolate(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), CSCEfficiency::filter(), PerigeeConversions::ftsToPerigeeParameters(), MatcherUsingTracksAlgorithm::getChi2(), CSCEfficiency::getFromFTS(), SteppingHelixStateInfo::getStateOnSurface(), TrackDetectorAssociator::getTAMuonChamberMatches(), HDetIdAssociator::getTrajectory(), KinematicState::globalPosition(), BasicTrajectoryState::globalPosition(), reco::TrackTransientTrack::impactPointState(), reco::TransientTrackFromFTS::impactPointTSCP(), reco::TrackTransientTrack::impactPointTSCP(), reco::GsfTransientTrack::impactPointTSCP(), MuonNavigableLayer::isInsideOut(), MatcherUsingTracksAlgorithm::matchByDirectComparison(), MatcherUsingTracksAlgorithm::matchWithPropagation(), StateOnTrackerBound::operator()(), PropagationDirectionChooser::operator()(), TransientTrackKinematicStateBuilder::operator()(), CosmicTrackingParticleSelector::operator()(), DirectMuonNavigation::outward(), DirectTrackerNavigation::outward(), TrackAssociatorByChi2::parametersAtClosestApproach(), reco::modules::DuplicateTrackMerger::produce(), OutsideInMuonSeeder::produce(), MuonErrorMatrixAdjuster::produce(), TrackKinematicStatePropagator::propagateToTheTransversePCANeutral(), MuonForwardNavigableLayer::pushResult(), MuonBarrelNavigableLayer::pushResult(), CSCEfficiency::recHitSegment_Efficiencies(), TrackClassifier::reconstructionInformation(), TrackAlgoCompareUtil::SetTrackingParticleD0Dz(), SteppingHelixStateInfo::SteppingHelixStateInfo(), reco::TransientTrackFromFTS::track(), MuonNavigableLayer::trackingRange(), TrajectoryStateClosestToBeamLine::transverseImpactParameter(), CosmicParametersDefinerForTP::vertex(), and SimpleNavigableLayer::wellInside().

                               {
    return theGlobalParameters.position();
  }

void FreeTrajectoryState::rescaleError

(

double

factor

)

Definition at line 50 of file FreeTrajectoryState.cc.

References hasError(), GlobalTrajectoryParameters::magneticField(), MagneticField::nominalValue(), parameters(), theCurvilinearError, unlikely, and CurvilinearTrajectoryError::zeroFieldScaling().

Referenced by TSGFromPropagation::adjust(), FastTSGFromPropagation::adjust(), BasicTrajectoryState::rescaleError(), and SimpleCosmicBONSeeder::seeds().

                                                    {
  if unlikely(!hasError()) return;
  bool zeroField = (parameters().magneticField().nominalValue()==0);  
  if unlikely(zeroField)  theCurvilinearError.zeroFieldScaling(factor*factor);
  else theCurvilinearError *= (factor*factor);
}

void FreeTrajectoryState::setCartesianError ( const CartesianTrajectoryError &  err )

inline

Definition at line 129 of file FreeTrajectoryState.h.

References createCurvilinearError().

Referenced by TwoBodyDecayTrajectoryState::setError().

                                                              {
    createCurvilinearError(err);
  }

void FreeTrajectoryState::setCartesianError ( const AlgebraicSymMatrix66 &  err )

inline

Definition at line 132 of file FreeTrajectoryState.h.

References createCurvilinearError().

                                                          {
    createCurvilinearError(CartesianTrajectoryError(err)); 
  }

CurvilinearTrajectoryError& FreeTrajectoryState::setCurvilinearError ( )

inline

Definition at line 136 of file FreeTrajectoryState.h.

References theCurvilinearError.

Referenced by BasicTrajectoryState::checkCurvilinError(), and BasicTrajectoryState::setCurvilinearError().

                                                     {
        return theCurvilinearError;
  }

void FreeTrajectoryState::setCurvilinearError ( const CurvilinearTrajectoryError &  err )

inline

Definition at line 140 of file FreeTrajectoryState.h.

References theCurvilinearError.

                                                                  {
        theCurvilinearError = err;
  }

double FreeTrajectoryState::signedInverseMomentum ( ) const

inline

Definition at line 93 of file FreeTrajectoryState.h.

References GlobalTrajectoryParameters::signedInverseMomentum(), and theGlobalParameters.

Referenced by HTrack::computeTDRResolution(), BasicTrajectoryState::createLocalParameters(), PerigeeConversions::jacobianCurvilinear2Perigee(), reco::modules::DuplicateTrackMerger::produce(), and BasicTrajectoryState::signedInverseMomentum().

                                       {
    return theGlobalParameters.signedInverseMomentum();
  }

double FreeTrajectoryState::transverseCurvature ( ) const

inline

Definition at line 96 of file FreeTrajectoryState.h.

References theGlobalParameters, and GlobalTrajectoryParameters::transverseCurvature().

Referenced by SiPixelTrackResidualSource::analyze(), TransverseImpactPointExtrapolator::doExtrapolation(), AnalyticalImpactPointExtrapolator::extrapolateSingleState(), AnalyticalTrajectoryExtrapolatorToLine::extrapolateSingleState(), ConversionFastHelix::helixStateAtVertex(), PerigeeConversions::jacobianCurvilinear2Perigee(), ConversionFastHelix::straightLineStateAtVertex(), and BasicTrajectoryState::transverseCurvature().

                                     {
    return theGlobalParameters.transverseCurvature();
  }

Member Data Documentation

CurvilinearTrajectoryError FreeTrajectoryState::theCurvilinearError

mutableprivate

Definition at line 162 of file FreeTrajectoryState.h.

Referenced by createCartesianError(), createCurvilinearError(), curvilinearError(), hasCurvilinearError(), missingError(), rescaleError(), and setCurvilinearError().

GlobalTrajectoryParameters FreeTrajectoryState::theGlobalParameters

private

Definition at line 161 of file FreeTrajectoryState.h.

Referenced by charge(), createCartesianError(), createCurvilinearError(), momentum(), parameters(), position(), signedInverseMomentum(), and transverseCurvature().