DGtal::NormalCycleFormula< TRealPoint, TRealVector > Struct Template Reference

Aim: A helper class that provides static methods to compute normal cycle formulas of curvatures. More...

#include <DGtal/geometry/meshes/NormalCycleFormula.h>

Public Types
typedef TRealPoint	RealPoint
typedef TRealVector	RealVector
typedef RealVector::Component	Scalar
typedef std::vector< RealPoint >	RealPoints
typedef std::vector< RealVector >	RealVectors
typedef SimpleMatrix< Scalar, 3, 3 >	RealTensor
typedef std::size_t	Size
typedef std::size_t	Index

Static Public Member Functions
Formulas for curvature
static Scalar	area (const RealPoints &pts)
static Scalar	twiceMeanCurvature (const RealPoint &a, const RealPoint &b, const RealVector &right, const RealVector &left)
static Scalar	gaussianCurvature (const RealPoint &a, const RealPoints &vtcs)
static Scalar	gaussianCurvatureWithPairs (const RealPoint &a, const RealPoints &pairs)
static RealTensor	anisotropicCurvatureH1 (const RealPoint &a, const RealPoint &b, const RealVector &right, const RealVector &left)
static RealTensor	anisotropicCurvatureH2 (const RealPoint &a, const RealPoint &b, const RealVector &right, const RealVector &left)
Other geometric services
static RealPoint	barycenter (const RealPoints &pts)
static RealVector	normal (const RealPoint &a, const RealPoint &b, const RealPoint &c)
static Scalar	area (const RealPoint &a, const RealPoint &b, const RealPoint &c)
static RealVector	averageUnitVector (const RealVectors &vecs)

Static Public Attributes
static const Dimension	dimension = RealPoint::dimension

Detailed Description

template<typename TRealPoint, typename TRealVector>
struct DGtal::NormalCycleFormula< TRealPoint, TRealVector >

Aim: A helper class that provides static methods to compute normal cycle formulas of curvatures.

Description of class 'NormalCycleFormula'

Template Parameters

TRealPoint	any model of 3D RealPoint.
TRealVector	any model of 3D RealVector.

Definition at line 64 of file NormalCycleFormula.h.

Member Typedef Documentation

Index

template<typename TRealPoint , typename TRealVector >

typedef std::size_t DGtal::NormalCycleFormula< TRealPoint, TRealVector >::Index

Definition at line 73 of file NormalCycleFormula.h.

RealPoint

template<typename TRealPoint , typename TRealVector >

typedef TRealPoint DGtal::NormalCycleFormula< TRealPoint, TRealVector >::RealPoint

Definition at line 66 of file NormalCycleFormula.h.

RealPoints

template<typename TRealPoint , typename TRealVector >

typedef std::vector< RealPoint > DGtal::NormalCycleFormula< TRealPoint, TRealVector >::RealPoints

Definition at line 69 of file NormalCycleFormula.h.

RealTensor

template<typename TRealPoint , typename TRealVector >

typedef SimpleMatrix< Scalar, 3, 3 > DGtal::NormalCycleFormula< TRealPoint, TRealVector >::RealTensor

Definition at line 71 of file NormalCycleFormula.h.

RealVector

template<typename TRealPoint , typename TRealVector >

typedef TRealVector DGtal::NormalCycleFormula< TRealPoint, TRealVector >::RealVector

Definition at line 67 of file NormalCycleFormula.h.

RealVectors

template<typename TRealPoint , typename TRealVector >

typedef std::vector< RealVector > DGtal::NormalCycleFormula< TRealPoint, TRealVector >::RealVectors

Definition at line 70 of file NormalCycleFormula.h.

Scalar

template<typename TRealPoint , typename TRealVector >

typedef RealVector::Component DGtal::NormalCycleFormula< TRealPoint, TRealVector >::Scalar

Definition at line 68 of file NormalCycleFormula.h.

Size

template<typename TRealPoint , typename TRealVector >

typedef std::size_t DGtal::NormalCycleFormula< TRealPoint, TRealVector >::Size

Definition at line 72 of file NormalCycleFormula.h.

Member Function Documentation

anisotropicCurvatureH1()

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::NormalCycleFormula< TRealPoint, TRealVector >::anisotropicCurvatureH1 ( const RealPoint &  a, const RealPoint &  b, const RealVector &  right, const RealVector &  left  )

inlinestatic

Computes the anisotropic measure \( \bar{H} \) at edge ab.

Parameters

a	any point
b	any point
right	the normal vector at face xba where x is some vertex(ices)
left	the normal vector at face yab where y is some vertex(ices)

Returns

the anisotropic curvature measure \( \bar{H} \) at edge ab.

Definition at line 159 of file NormalCycleFormula.h.

    {
      const RealVector diedre = right.crossProduct( left );
      const Scalar     length = std::max( 0.0, std::min( 1.0, diedre.norm() ) );
      const Scalar      angle = ( diedre.dot( b - a) > 0.0 )
        ? asin( length ) : - asin( length );
      RealVector          e_p = right + left;
      RealVector          e_m = right - left;
      const Scalar norm_e_p   = e_p.norm();
      const Scalar norm_e_m   = e_m.norm();
      e_p = norm_e_p > 1e-10 ? e_p / norm_e_p : RealVector::zero;
      e_m = norm_e_m > 1e-10 ? e_m / norm_e_m : RealVector::zero;
      const RealTensor T_p  =
        { e_p[ 0 ] * e_p[ 0 ], e_p[ 0 ] * e_p[ 1 ], e_p[ 0 ] * e_p[ 2 ],
          e_p[ 1 ] * e_p[ 0 ], e_p[ 1 ] * e_p[ 1 ], e_p[ 1 ] * e_p[ 2 ],
          e_p[ 2 ] * e_p[ 0 ], e_p[ 2 ] * e_p[ 1 ], e_p[ 2 ] * e_p[ 2 ] };
      const RealTensor T_m  =
        { e_m[ 0 ] * e_m[ 0 ], e_m[ 0 ] * e_m[ 1 ], e_m[ 0 ] * e_m[ 2 ],
          e_m[ 1 ] * e_m[ 0 ], e_m[ 1 ] * e_m[ 1 ], e_m[ 1 ] * e_m[ 2 ],
          e_m[ 2 ] * e_m[ 0 ], e_m[ 2 ] * e_m[ 1 ], e_m[ 2 ] * e_m[ 2 ] };
      return 0.5 * ( b - a ).norm()
        * ( ( angle - sin( angle ) ) * T_p + ( angle + sin( angle ) ) * T_m );
    }

References max().

anisotropicCurvatureH2()

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::NormalCycleFormula< TRealPoint, TRealVector >::anisotropicCurvatureH2 ( const RealPoint &  a, const RealPoint &  b, const RealVector &  right, const RealVector &  left  )

inlinestatic

Computes the anisotropic measure \( \bar{\tilde{H}} \) at edge ab.

Parameters

a	any point
b	any point
right	the normal vector at face xba where x is some vertex(ices)
left	the normal vector at face yab where y is some vertex(ices)

Returns

the anisotropic curvature measure \( \bar{\tilde{H}} \) at edge ab.

Definition at line 192 of file NormalCycleFormula.h.

    {
      const RealVector diedre = right.crossProduct( left );
      const Scalar     length = std::max( 0.0, std::min( 1.0, diedre.norm() ) );
      const Scalar      angle = ( diedre.dot( b - a) > 0.0 )
        ? asin( length ) : - asin( length );
      const Scalar norm_ab = (b - a).norm();
      const RealVector   e = norm_ab > 1e-10 ? (b - a) / norm_ab : RealVector::zero;
      const RealTensor   T =
        { e[ 0 ] * e[ 0 ], e[ 0 ] * e[ 1 ], e[ 0 ] * e[ 2 ],
          e[ 1 ] * e[ 0 ], e[ 1 ] * e[ 1 ], e[ 1 ] * e[ 2 ],
          e[ 2 ] * e[ 0 ], e[ 2 ] * e[ 1 ], e[ 2 ] * e[ 2 ] };
      return ( 0.5 * norm_ab * angle ) * T; // JOL * 0.5
    }

References max().

area() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::NormalCycleFormula< TRealPoint, TRealVector >::area ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c  )

inlinestatic

Computes triangle area

Parameters

a	any point
b	any point
c	any point

Returns

the area of triangle abc

Definition at line 245 of file NormalCycleFormula.h.

    {
      return 0.5 * ( ( b - a ).crossProduct( c - a ) ).norm();
    }    

References DGtal::crossProduct().

area() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::NormalCycleFormula< TRealPoint, TRealVector >::area ( const RealPoints &  pts )

inlinestatic

Computes area of polygonal face pts.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.

Returns

the area of the given polygonal face.

Definition at line 85 of file NormalCycleFormula.h.

    {
      if ( pts.size() <  3 ) return 0.0;
      if ( pts.size() == 3 )
        return area( pts[ 0 ], pts[ 1 ], pts[ 2 ] );
      const RealPoint b = barycenter( pts );
      Scalar          a = 0.0;
      for ( Index i = 0; i < pts.size(); i++ )
        a += area( b, pts[ i ], pts[ (i+1)%pts.size() ] );
      return a;
    }

References DGtal::NormalCycleFormula< TRealPoint, TRealVector >::barycenter().

averageUnitVector()

template<typename TRealPoint , typename TRealVector >

static RealVector DGtal::NormalCycleFormula< TRealPoint, TRealVector >::averageUnitVector ( const RealVectors &  vecs )

inlinestatic

Given a vector of unit vectors, returns their average unit vector.

Parameters

vecs	any vector of vectors.

Returns

the average unit vector.

Definition at line 254 of file NormalCycleFormula.h.

    {
      RealVector avg;
      for ( auto v : vecs ) avg += v;
      auto avg_norm = avg.norm();
      return avg_norm != 0.0 ? avg / avg_norm : avg;
    }

barycenter()

template<typename TRealPoint , typename TRealVector >

static RealPoint DGtal::NormalCycleFormula< TRealPoint, TRealVector >::barycenter ( const RealPoints &  pts )

inlinestatic

Given a vector of points, returns its barycenter.

Parameters

pts	any vector of points

Returns

the barycenter of these points.

Definition at line 219 of file NormalCycleFormula.h.

    {
      RealPoint b;
      for ( auto p : pts ) b += p;
      b /= pts.size();
      return b;
    }

Referenced by DGtal::NormalCycleFormula< TRealPoint, TRealVector >::area().

gaussianCurvature()

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::NormalCycleFormula< TRealPoint, TRealVector >::gaussianCurvature ( const RealPoint &  a, const RealPoints &  vtcs  )

inlinestatic

Computes the Gaussian curvature at point a with incident vertices vtcs.

Parameters

a	any point
vtcs	a range of points

Returns

the Gaussian curvature according to Normal Cycle formula.

Definition at line 124 of file NormalCycleFormula.h.

    {
      Scalar angle_sum = 0.0;
      for ( Size i = 0; i < vtcs.size(); i++ )
        angle_sum += acos( (vtcs[i] - a).getNormalized()
                           .dot( ( vtcs[(i+1)%vtcs.size()] - a ).getNormalized() ) );
      return 2.0 * M_PI - angle_sum;
    }

gaussianCurvatureWithPairs()

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::NormalCycleFormula< TRealPoint, TRealVector >::gaussianCurvatureWithPairs ( const RealPoint &  a, const RealPoints &  pairs  )

inlinestatic

Computes the Gaussian curvature at point a with incident pairs of points pairs.

Parameters

a	any point
pairs	a range of points [x_0, y_0, x_1, y_1, etc] such that (a,x_i,y_i) is an incident face to a.

Returns

the Gaussian curvature according to Normal Cycle formula.

Definition at line 142 of file NormalCycleFormula.h.

    {
      Scalar angle_sum = 0.0;
      for ( Size i = 0; i < pairs.size(); i += 2 )
        angle_sum += acos( ( pairs[i] - a ).getNormalized()
                           .dot( ( pairs[i+1] - a ).getNormalized() ) );
      return 2.0 * M_PI - angle_sum;
    }

normal()

template<typename TRealPoint , typename TRealVector >

static RealVector DGtal::NormalCycleFormula< TRealPoint, TRealVector >::normal ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c  )

inlinestatic

Computes a unit normal vector to triangle abc

Parameters

a	any point
b	any point
c	any point

Returns

the unit normal vector to abc, ( ab x ac ) / || ab x ac ||.

Definition at line 234 of file NormalCycleFormula.h.

    {
      return ( ( b - a ).crossProduct( c - a ) ).getNormalized();
    }    

References DGtal::crossProduct().

twiceMeanCurvature()

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::NormalCycleFormula< TRealPoint, TRealVector >::twiceMeanCurvature ( const RealPoint &  a, const RealPoint &  b, const RealVector &  right, const RealVector &  left  )

inlinestatic

Computes twice the mean curvature on edge ab given normal vectors right, left.

Parameters

a	any point
b	any point
right	the normal vector at face xba where x is some vertex(ices)
left	the normal vector at face yab where y is some vertex(ices)

Returns

twice the mean curvature according to Normal Cycle formula.

Definition at line 106 of file NormalCycleFormula.h.

    {
      const RealVector diedre = right.crossProduct( left );
      const Scalar n = std::min( 1.0, std::max( diedre.norm(), 0.0 ) );
      const Scalar angle = ( diedre.dot( b - a) < 0.0 )
        ? asin( n ) : - asin( n );
      return ( b - a ).norm() * angle;
    }

References angle(), and max().

Field Documentation

dimension

template<typename TRealPoint , typename TRealVector >

const Dimension DGtal::NormalCycleFormula< TRealPoint, TRealVector >::dimension = RealPoint::dimension

static

Definition at line 74 of file NormalCycleFormula.h.

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The documentation for this struct was generated from the following file:

• NormalCycleFormula.h