DGtal  1.5.beta
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< RealPointRealPoints
 
typedef std::vector< RealVectorRealVectors
 
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
TRealPointany model of 3D RealPoint.
TRealVectorany 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
aany point
bany point
rightthe normal vector at face xba where x is some vertex(ices)
leftthe 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.

162  {
163  const RealVector diedre = right.crossProduct( left );
164  const Scalar length = std::max( 0.0, std::min( 1.0, diedre.norm() ) );
165  const Scalar angle = ( diedre.dot( b - a) > 0.0 )
166  ? asin( length ) : - asin( length );
167  RealVector e_p = right + left;
168  RealVector e_m = right - left;
169  const Scalar norm_e_p = e_p.norm();
170  const Scalar norm_e_m = e_m.norm();
171  e_p = norm_e_p > 1e-10 ? e_p / norm_e_p : RealVector::zero;
172  e_m = norm_e_m > 1e-10 ? e_m / norm_e_m : RealVector::zero;
173  const RealTensor T_p =
174  { e_p[ 0 ] * e_p[ 0 ], e_p[ 0 ] * e_p[ 1 ], e_p[ 0 ] * e_p[ 2 ],
175  e_p[ 1 ] * e_p[ 0 ], e_p[ 1 ] * e_p[ 1 ], e_p[ 1 ] * e_p[ 2 ],
176  e_p[ 2 ] * e_p[ 0 ], e_p[ 2 ] * e_p[ 1 ], e_p[ 2 ] * e_p[ 2 ] };
177  const RealTensor T_m =
178  { e_m[ 0 ] * e_m[ 0 ], e_m[ 0 ] * e_m[ 1 ], e_m[ 0 ] * e_m[ 2 ],
179  e_m[ 1 ] * e_m[ 0 ], e_m[ 1 ] * e_m[ 1 ], e_m[ 1 ] * e_m[ 2 ],
180  e_m[ 2 ] * e_m[ 0 ], e_m[ 2 ] * e_m[ 1 ], e_m[ 2 ] * e_m[ 2 ] };
181  return 0.5 * ( b - a ).norm()
182  * ( ( angle - sin( angle ) ) * T_p + ( angle + sin( angle ) ) * T_m );
183  }
SimpleMatrix< Scalar, 3, 3 > RealTensor
int max(int a, int b)

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
aany point
bany point
rightthe normal vector at face xba where x is some vertex(ices)
leftthe 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.

195  {
196  const RealVector diedre = right.crossProduct( left );
197  const Scalar length = std::max( 0.0, std::min( 1.0, diedre.norm() ) );
198  const Scalar angle = ( diedre.dot( b - a) > 0.0 )
199  ? asin( length ) : - asin( length );
200  const Scalar norm_ab = (b - a).norm();
201  const RealVector e = norm_ab > 1e-10 ? (b - a) / norm_ab : RealVector::zero;
202  const RealTensor T =
203  { e[ 0 ] * e[ 0 ], e[ 0 ] * e[ 1 ], e[ 0 ] * e[ 2 ],
204  e[ 1 ] * e[ 0 ], e[ 1 ] * e[ 1 ], e[ 1 ] * e[ 2 ],
205  e[ 2 ] * e[ 0 ], e[ 2 ] * e[ 1 ], e[ 2 ] * e[ 2 ] };
206  return ( 0.5 * norm_ab * angle ) * T; // JOL * 0.5
207  }

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
aany point
bany point
cany point
Returns
the area of triangle abc

Definition at line 245 of file NormalCycleFormula.h.

246  {
247  return 0.5 * ( ( b - a ).crossProduct( c - a ) ).norm();
248  }
auto crossProduct(PointVector< 3, LeftEuclideanRing, LeftContainer > const &lhs, PointVector< 3, RightEuclideanRing, RightContainer > const &rhs) -> decltype(DGtal::constructFromArithmeticConversion(lhs, rhs))
Cross product of two 3D Points/Vectors.

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
ptsthe (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.

86  {
87  if ( pts.size() < 3 ) return 0.0;
88  if ( pts.size() == 3 )
89  return area( pts[ 0 ], pts[ 1 ], pts[ 2 ] );
90  const RealPoint b = barycenter( pts );
91  Scalar a = 0.0;
92  for ( Index i = 0; i < pts.size(); i++ )
93  a += area( b, pts[ i ], pts[ (i+1)%pts.size() ] );
94  return a;
95  }
SMesh::Index Index
static Scalar area(const RealPoints &pts)
static RealPoint barycenter(const RealPoints &pts)
PointVector< 3, double > RealPoint

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
vecsany vector of vectors.
Returns
the average unit vector.

Definition at line 254 of file NormalCycleFormula.h.

255  {
256  RealVector avg;
257  for ( auto v : vecs ) avg += v;
258  auto avg_norm = avg.norm();
259  return avg_norm != 0.0 ? avg / avg_norm : avg;
260  }

◆ 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
ptsany vector of points
Returns
the barycenter of these points.

Definition at line 219 of file NormalCycleFormula.h.

220  {
221  RealPoint b;
222  for ( auto p : pts ) b += p;
223  b /= pts.size();
224  return b;
225  }

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
aany point
vtcsa range of points
Returns
the Gaussian curvature according to Normal Cycle formula.

Definition at line 124 of file NormalCycleFormula.h.

126  {
127  Scalar angle_sum = 0.0;
128  for ( Size i = 0; i < vtcs.size(); i++ )
129  angle_sum += acos( (vtcs[i] - a).getNormalized()
130  .dot( ( vtcs[(i+1)%vtcs.size()] - a ).getNormalized() ) );
131  return 2.0 * M_PI - angle_sum;
132  }
HalfEdgeDataStructure::Size Size

◆ 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
aany point
pairsa 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.

144  {
145  Scalar angle_sum = 0.0;
146  for ( Size i = 0; i < pairs.size(); i += 2 )
147  angle_sum += acos( ( pairs[i] - a ).getNormalized()
148  .dot( ( pairs[i+1] - a ).getNormalized() ) );
149  return 2.0 * M_PI - angle_sum;
150  }

◆ 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
aany point
bany point
cany point
Returns
the unit normal vector to abc, ( ab x ac ) / || ab x ac ||.

Definition at line 234 of file NormalCycleFormula.h.

235  {
236  return ( ( b - a ).crossProduct( c - a ) ).getNormalized();
237  }

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
aany point
bany point
rightthe normal vector at face xba where x is some vertex(ices)
leftthe 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.

109  {
110  const RealVector diedre = right.crossProduct( left );
111  const Scalar n = std::min( 1.0, std::max( diedre.norm(), 0.0 ) );
112  const Scalar angle = ( diedre.dot( b - a) < 0.0 )
113  ? asin( n ) : - asin( n );
114  return ( b - a ).norm() * angle;
115  }
double angle(const DGtal::Z2i::RealPoint &point)

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.


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