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

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

#include <DGtal/geometry/meshes/CorrectedNormalCurrentFormula.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	Index

Static Public Member Functions
Formulas for mu0 measure
static Scalar	mu0ConstantU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &u)
static Scalar	mu0InterpolatedU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &ua, const RealVector &ub, const RealVector &uc, bool unit_u=false)
static Scalar	mu0ConstantU (const RealPoints &pts, const RealVector &u)
static Scalar	mu0InterpolatedU (const RealPoints &pts, const RealVectors &u, bool unit_u=false)
Formulas for mu1 measure
static Scalar	mu1ConstantUAtEdge (const RealPoint &a, const RealPoint &b, const RealVector &ur, const RealVector &ul)
static Scalar	mu1ConstantU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &u)
static Scalar	mu1InterpolatedU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &ua, const RealVector &ub, const RealVector &uc, bool unit_u=false)
static Scalar	mu1ConstantU (const RealPoints &pts, const RealVector &u)
static Scalar	mu1InterpolatedU (const RealPoints &pts, const RealVectors &u, bool unit_u=false)
Formulas for mu2 measure
static Scalar	mu2ConstantU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &u)
static Scalar	mu2ConstantUAtVertex (const RealPoint &a, const RealVectors &vu)
static Scalar	mu2InterpolatedU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &ua, const RealVector &ub, const RealVector &uc, bool unit_u=false)
static Scalar	mu2ConstantU (const RealPoints &pts, const RealVector &u)
static Scalar	mu2InterpolatedU (const RealPoints &pts, const RealVectors &u, bool unit_u=false)
Formulas for muXY measure
static RealTensor	muXYConstantU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &u)
static RealTensor	muXYConstantUAtEdge (const RealPoint &a, const RealPoint &b, const RealVector &ur, const RealVector &ul)
static RealTensor	muXYInterpolatedU (const RealPoint &a, const RealPoint &b, const RealPoint &c, const RealVector &ua, const RealVector &ub, const RealVector &uc, bool unit_u=false)
static RealTensor	muXYConstantU (const RealPoints &pts, const RealVector &u)
static RealTensor	muXYInterpolatedU (const RealPoints &pts, const RealVectors &u, bool unit_u=false)
Other geometric services
static RealPoint	barycenter (const RealPoints &pts)
static RealVector	averageUnitVector (const RealVectors &vecs)

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

Detailed Description

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

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

Description of class 'CorrectedNormalCurrentFormula'

Template Parameters

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

Formula for interpolated measures on a triangle with vertices A, B, C, and normal vectors uA, uB, uC:

MU0=-1/6*((uAz + uBz + uCz)*Bx - (uAz + uBz + uCz)*Cx)*Ay + 1/6*((uAz + uBz + uCz)*Ax - (uAz + uBz + uCz)*Cx)*By - 1/6*((uAz + uBz + uCz)*Ax - (uAz + uBz + uCz)*Bx)*Cy + 1/6*((uAy + uBy + uCy)*Bx - (uAy + uBy + uCy)*Cx - (uAx + uBx + uCx)*By + (uAx + uBx + uCx)*Cy)*Az - 1/6*((uAy + uBy + uCy)*Ax - (uAy + uBy + uCy)*Cx - (uAx + uBx + uCx)*Ay + (uAx + uBx + uCx)*Cy)*Bz + 1/6*((uAy + uBy + uCy)*Ax - (uAy + uBy + uCy)*Bx - (uAx + uBx + uCx)*Ay + (uAx + uBx + uCx)*By)*Cz

Let UM=uA+uB+uC.

MU0=-1/6*(uMz*Bx - uMz*Cx)*Ay + 1/6*(uMz*Ax - uMz*Cx)*By - 1/6*(uMz*Ax - uMz*Bx)*Cy + 1/6*(uMy*Bx - uMy*Cx - uMx*By + uMx*Cy)*Az - 1/6*(uMy*Ax - uMy*Cx - uMx*Ay + uMx*Cy)*Bz + 1/6*(uMy*Ax - uMy*Bx - uMx*Ay + uMx*By)*Cz

We see by simple computations that MU0 can be written as (uM = UM/3)

MU0=1/2*det( uM, B-A, C-A )

MU1=1/6*((uBy - uCy)*uAz - (uAy + 2*uCy)*uBz + (uAy + 2*uBy)*uCz)*Ax + 1/6*((uBy + 2*uCy)*uAz - (uAy - uCy)*uBz - (2*uAy + uBy)*uCz)*Bx - 1/6*((2*uBy + uCy)*uAz - (2*uAy + uCy)*uBz - (uAy - uBy)*uCz)*Cx - 1/6*((uBx - uCx)*uAz - (uAx + 2*uCx)*uBz + (uAx + 2*uBx)*uCz)*Ay - 1/6*((uBx + 2*uCx)*uAz - (uAx - uCx)*uBz - (2*uAx + uBx)*uCz)*By + 1/6*((2*uBx + uCx)*uAz - (2*uAx + uCx)*uBz - (uAx - uBx)*uCz)*Cy + 1/6*((uBx - uCx)*uAy - (uAx + 2*uCx)*uBy + (uAx + 2*uBx)*uCy)*Az + 1/6*((uBx + 2*uCx)*uAy - (uAx - uCx)*uBy - (2*uAx + uBx)*uCy)*Bz - 1/6*((2*uBx + uCx)*uAy - (2*uAx + uCx)*uBy - (uAx - uBx)*uCy)*Cz

This formula can also be written in a clearer form with determinants:

6*MU1 = det(u_A+u_B+u_C,u_C-u_B,A) + det(u_A+u_B+u_C,u_A-u_C,B) + det(u_A+u_B+u_C,u_B-u_A,C)

It follows that MU1=1/2( det(uM, u_C-u_B, A) + det(uM, u_A-u_C, B) + det(uM, u_B-u_A, C)

Gaussian curvature measure is

MU2=-1/2*uCx*uBy*uAz + 1/2*uBx*uCy*uAz + 1/2*uCx*uAy*uBz - 1/2*uAx*uCy*uBz - 1/2*uBx*uAy*uCz + 1/2*uAx*uBy*uCz

which is simply MU2=1/2*det( uA, uB, uC )

Anisotropic curvature measure is written as (for X, Y arbitrary vectors, <.|.> the standard Euclidean scalar product).

MUXY = 1/2 det( uM, < Y | uC-uA > X, (B-A)) - det( uM, < Y | uB-uA > X, (C-A))

Definition at line 99 of file CorrectedNormalCurrentFormula.h.

Member Typedef Documentation

Index

template<typename TRealPoint , typename TRealVector >

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

Definition at line 107 of file CorrectedNormalCurrentFormula.h.

RealPoint

template<typename TRealPoint , typename TRealVector >

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

Definition at line 101 of file CorrectedNormalCurrentFormula.h.

RealPoints

template<typename TRealPoint , typename TRealVector >

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

Definition at line 104 of file CorrectedNormalCurrentFormula.h.

RealTensor

template<typename TRealPoint , typename TRealVector >

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

Definition at line 106 of file CorrectedNormalCurrentFormula.h.

RealVector

template<typename TRealPoint , typename TRealVector >

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

Definition at line 102 of file CorrectedNormalCurrentFormula.h.

RealVectors

template<typename TRealPoint , typename TRealVector >

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

Definition at line 105 of file CorrectedNormalCurrentFormula.h.

Scalar

template<typename TRealPoint , typename TRealVector >

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

Definition at line 103 of file CorrectedNormalCurrentFormula.h.

Member Function Documentation

averageUnitVector()

template<typename TRealPoint , typename TRealVector >

static RealVector DGtal::CorrectedNormalCurrentFormula< 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 627 of file CorrectedNormalCurrentFormula.h.

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

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0InterpolatedU(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1InterpolatedU(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2InterpolatedU(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYInterpolatedU().

barycenter()

template<typename TRealPoint , typename TRealVector >

static RealPoint DGtal::CorrectedNormalCurrentFormula< 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 615 of file CorrectedNormalCurrentFormula.h.

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

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0ConstantU(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0InterpolatedU(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1InterpolatedU(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2InterpolatedU(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYInterpolatedU().

mu0ConstantU() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0ConstantU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  u  )

inlinestatic

Computes mu0 measure (area) of triangle abc given a constant corrected normal vector u.

Parameters

a	any point
b	any point
c	any point
u	the constant corrected normal vector to triangle abc

Returns

the mu0-measure of triangle abc, i.e. its area.

Definition at line 123 of file CorrectedNormalCurrentFormula.h.

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

References DGtal::crossProduct().

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0ConstantU().

mu0ConstantU() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0ConstantU ( const RealPoints &  pts, const RealVector &  u  )

inlinestatic

Computes mu0 measure (area) of polygonal face pts given a constant corrected normal vector u.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the constant corrected normal vector to this polygonal face.

Returns

the mu0-measure of the given polygonal face, i.e. its area.

Definition at line 165 of file CorrectedNormalCurrentFormula.h.

    {
      if ( pts.size() <  3 ) return 0.0;
      if ( pts.size() == 3 )
        return mu0ConstantU( pts[ 0 ], pts[ 1 ], pts[ 2 ], u );
      const RealPoint b = barycenter( pts );
      Scalar          a = 0.0;
      for ( Index i = 0; i < pts.size(); i++ )
        a += mu0ConstantU( b, pts[ i ], pts[ (i+1)%pts.size() ], u );
      return a;
    }

References DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::barycenter(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0ConstantU().

mu0InterpolatedU() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0InterpolatedU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  ua, const RealVector &  ub, const RealVector &  uc, bool  unit_u = false  )

inlinestatic

Computes mu0 measure (area) of triangle abc given an interpolated corrected normal vector ua, ub, uc.

Parameters

a	any point
b	any point
c	any point
ua	the corrected normal vector at point a
ub	the corrected normal vector at point b
uc	the corrected normal vector at point c
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the mu0-measure of triangle abc, i.e. its area.

Definition at line 143 of file CorrectedNormalCurrentFormula.h.

    {
      // MU0=1/2*det( uM, B-A, C-A )
      //    =  1/2 < ( (u_A + u_B + u_C)/3.0 ) | (AB x AC ) >
      RealVector uM = ( ua+ub+uc ) / 3.0;
      if ( unit_u )
        {
          auto uM_norm = uM.norm();
          uM = uM_norm == 0.0 ? uM : uM / uM_norm;
        }
      return 0.5 * ( b - a ).crossProduct( c - a ).dot( uM );
    }

References DGtal::crossProduct().

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0InterpolatedU().

mu0InterpolatedU() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0InterpolatedU ( const RealPoints &  pts, const RealVectors &  u, bool  unit_u = false  )

inlinestatic

Computes area of polygonal face pts given an interpolated corrected normal vector ua, ub, uc.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the (ccw ordered) normal vectors at the corresponding vertices in pts.
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the mu0-measure of the given polygonal face, i.e. its area.

Definition at line 186 of file CorrectedNormalCurrentFormula.h.

    {
      ASSERT( pts.size() == u.size() );
      if ( pts.size() <  3 ) return 0.0;
      if ( pts.size() == 3 )
        return mu0InterpolatedU( pts[ 0 ], pts[ 1 ], pts[ 2 ],
                                 u[ 0 ], u[ 1 ], u[ 2 ], unit_u );
      const RealPoint   b = barycenter( pts );
      const RealVector ub = averageUnitVector( u );
      Scalar          a = 0.0;
      for ( Index i = 0; i < pts.size(); i++ )
        a += mu0InterpolatedU( b,  pts[ i ], pts[ (i+1)%pts.size() ],
                               ub,   u[ i ],   u[ (i+1)%pts.size() ], unit_u );
      return a;
    }

References DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::averageUnitVector(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::barycenter(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu0InterpolatedU().

mu1ConstantU() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1ConstantU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  u  )

inlinestatic

Computes mu1 measure (twice the mean curvature) of triangle abc given a constant corrected normal vector u.

Parameters

a	any point
b	any point
c	any point
u	the constant corrected normal vector to triangle abc

Returns

the mu1-measure of triangle abc, i.e. twice its mean curvature, always 0.0.

Definition at line 248 of file CorrectedNormalCurrentFormula.h.

    {
      (void)a; (void)b; (void)c; (void)u;
      return 0.0;
    }

mu1ConstantU() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1ConstantU ( const RealPoints &  pts, const RealVector &  u  )

inlinestatic

Computes mu1 measure (twice the mean curvature) of polygonal face pts given a constant corrected normal vector u.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the constant corrected normal vector to this polygonal face.

Returns

the mu1-measure of the given polygonal face, i.e. twice its mean curvature, always 0.0.

Definition at line 291 of file CorrectedNormalCurrentFormula.h.

    {
      return 0.0;
    }

mu1ConstantUAtEdge()

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1ConstantUAtEdge ( const RealPoint &  a, const RealPoint &  b, const RealVector &  ur, const RealVector &  ul  )

inlinestatic

Computes mu1 measure (twice the mean curvature) at edge ab given a constant corrected normal vector ur to the right of ab, and a constant corrected normal vector ul to the left of ab.

The formula is \( int_{ab} \Psi \langle e | e_1 \rangle dH^1 \), where e is the unit vector parallel to ab, and \( e_1 \) is the unit vector orthogonal to ur and ul, and \( \Psi \) is the angle between these two vectors.

Parameters

a	any point
b	any point
ur	the constant corrected normal vector to the right of oriented edge ab
ul	the constant corrected normal vector to the left of oriented edge ab

Returns

the mu1-measure of edge ab, i.e. twice its mean curvature.

Definition at line 226 of file CorrectedNormalCurrentFormula.h.

    {
      RealVector   e  = b - a;
      RealVector   e1 = ur.crossProduct( ul );
      const Scalar l1 = std::min( e1.norm(), 1.0 );
      if ( l1 < 1e-10 ) return 0.0;
      e1 /= l1;
      const Scalar Psi = asin( l1 );
      return -Psi * e.dot( e1 );
    }

mu1InterpolatedU() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1InterpolatedU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  ua, const RealVector &  ub, const RealVector &  uc, bool  unit_u = false  )

inlinestatic

Computes mu1 measure (twice the mean curvature) of triangle abc given an interpolated corrected normal vector ua, ub, uc.

Parameters

a	any point
b	any point
c	any point
ua	the corrected normal vector at point a
ub	the corrected normal vector at point b
uc	the corrected normal vector at point c
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the mu1-measure of triangle abc, i.e. twice its mean curvature.

Definition at line 271 of file CorrectedNormalCurrentFormula.h.

    {
      // MU1=1/2( | uM u_C-u_B A | + | uM u_A-u_C B | + | uM u_B-u_A C |
      RealVector uM = ( ua+ub+uc ) / 3.0;
      if ( unit_u ) uM /= uM.norm();
      return 0.5 * ( uM.crossProduct( uc - ub ).dot( a )
                     + uM.crossProduct( ua - uc ).dot( b )
                     + uM.crossProduct( ub - ua ).dot( c ) );
    }

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1InterpolatedU().

mu1InterpolatedU() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1InterpolatedU ( const RealPoints &  pts, const RealVectors &  u, bool  unit_u = false  )

inlinestatic

Computes mean curvature of polygonal face pts given an interpolated corrected normal vector ua, ub, uc.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the (ccw ordered) normal vectors at the corresponding vertices in pts.
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the mu1-measure of the given polygonal face, i.e. its mean curvature.

Definition at line 305 of file CorrectedNormalCurrentFormula.h.

    {
      ASSERT( pts.size() == u.size() );
      if ( pts.size() <  3 ) return 0.0;
      if ( pts.size() == 3 )
        return mu1InterpolatedU( pts[ 0 ], pts[ 1 ], pts[ 2 ],
                                 u[ 0 ], u[ 1 ], u[ 2 ], unit_u );
      const RealPoint   b = barycenter( pts );
      const RealVector ub = averageUnitVector( u );
      Scalar          a = 0.0;
      for ( Index i = 0; i < pts.size(); i++ )
        a += mu1InterpolatedU( b,  pts[ i ], pts[ (i+1)%pts.size() ],
                               ub,   u[ i ],   u[ (i+1)%pts.size() ], unit_u );
      return a;
    }

References DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::averageUnitVector(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::barycenter(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu1InterpolatedU().

mu2ConstantU() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2ConstantU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  u  )

inlinestatic

Computes mu2 measure (Gaussian curvature) of triangle abc given a constant corrected normal vector u.

Parameters

a	any point
b	any point
c	any point
u	the constant corrected normal vector to triangle abc

Returns

the mu2-measure of triangle abc, i.e. its Gaussian curvature, always 0.0.

Definition at line 337 of file CorrectedNormalCurrentFormula.h.

    {
      (void)a; (void)b; (void)c; (void)u;
      return 0.0;
    }

mu2ConstantU() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2ConstantU ( const RealPoints &  pts, const RealVector &  u  )

inlinestatic

Computes mu2 measure (Gaussian curvature) of polygonal face pts given a constant corrected normal vector u.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the constant corrected normal vector to this polygonal face.

Returns

the mu2-measure of the given polygonal face, i.e. its Gaussian curvature, always 0.0.

Definition at line 425 of file CorrectedNormalCurrentFormula.h.

    {
      (void) pts; (void) u;
      return 0.0;
    }

mu2ConstantUAtVertex()

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2ConstantUAtVertex ( const RealPoint &  a, const RealVectors &  vu  )

inlinestatic

Computes mu2 measure (Gaussian curvature) at given vertex a, surrounded by faces with constant corrected normal vectors vu.

Parameters

a	any point
vu	the vector of constant corrected normal vectors of the faces incident to a.

Returns

the mu2-measure at this vertex, i.e. its Gaussian curvature.

Precondition

the unit normal vectors should lie in the same hemisphere of the Gauss sphere.

Definition at line 358 of file CorrectedNormalCurrentFormula.h.

    {
      typedef SpaceND< dimension >     Space;
      typedef SphericalTriangle<Space> ST;
      RealVector avg_u;
      for ( const auto& u : vu ) avg_u += u;
      const Scalar l = avg_u.norm();
      if ( l < 1e-10 )
        {
          trace.warning() << "[CorrectedNormalCurrentFormula::mu2ConstantUAtVertex]"
                          << " Invalid surrounding normal vectors at vertex "
                          << a << "."
                          << " Unit normal vectors should lie strictly in the same"
                          << " hemisphere." << std::endl;
          return 0.0;
        }
      avg_u /= l;
      Scalar mu2 = 0.0;
      const auto n = vu.size();
      for ( auto i = 0; i < n; ++i )
        {
          ST st( avg_u, vu[ i ], vu[ (i+1) % n ] );
          mu2 += st.algebraicArea();
        }
      return mu2;
    }

References DGtal::trace, and DGtal::Trace::warning().

mu2InterpolatedU() [1/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2InterpolatedU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  ua, const RealVector &  ub, const RealVector &  uc, bool  unit_u = false  )

inlinestatic

Computes mu2 measure (Gaussian curvature) of triangle abc given an interpolated corrected normal vector ua, ub, uc.

Parameters

a	any point
b	any point
c	any point
ua	the corrected normal vector at point a
ub	the corrected normal vector at point b
uc	the corrected normal vector at point c
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the mu2-measure of triangle abc, i.e. its Gaussian curvature.

Definition at line 401 of file CorrectedNormalCurrentFormula.h.

    {
      // Using non unitary interpolated normals give
      // MU2=1/2*det( uA, uB, uC )
      // When normals are unitary, it is the area of a spherical triangle.
      if ( unit_u )
        {
          typedef SpaceND< dimension > Space;
          SphericalTriangle<Space> ST( ua, ub, uc ); // check order.
          return ST.algebraicArea();
        }
      else
        return 0.5 * ( ua.crossProduct( ub ).dot( uc ) );
    }

References DGtal::SphericalTriangle< TSpace >::algebraicArea().

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2InterpolatedU().

mu2InterpolatedU() [2/2]

template<typename TRealPoint , typename TRealVector >

static Scalar DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2InterpolatedU ( const RealPoints &  pts, const RealVectors &  u, bool  unit_u = false  )

inlinestatic

Computes Gaussian curvature of polygonal face pts given an interpolated corrected normal vector ua, ub, uc.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the (ccw ordered) normal vectors at the corresponding vertices in pts.
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the mu2-measure of the given polygonal face, i.e. its Gaussian curvature.

Definition at line 440 of file CorrectedNormalCurrentFormula.h.

    {
      ASSERT( pts.size() == u.size() );
      if ( pts.size() <  3 ) return 0.0;
      if ( pts.size() == 3 )
        return mu2InterpolatedU( pts[ 0 ], pts[ 1 ], pts[ 2 ],
                                 u[ 0 ], u[ 1 ], u[ 2 ], unit_u );
      const RealPoint   b = barycenter( pts );
      const RealVector ub = averageUnitVector( u );
      Scalar          a = 0.0;
      for ( Index i = 0; i < pts.size(); i++ )
        a += mu2InterpolatedU( b,  pts[ i ], pts[ (i+1)%pts.size() ],
                               ub,   u[ i ],   u[ (i+1)%pts.size() ], unit_u );
      return a;
    }

References DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::averageUnitVector(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::barycenter(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::mu2InterpolatedU().

muXYConstantU() [1/2]

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYConstantU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  u  )

inlinestatic

Computes muXY measure (anisotropic curvature) of triangle abc given a constant corrected normal vector u.

Parameters

a	any point
b	any point
c	any point
u	the constant corrected normal vector to triangle abc

Returns

the muXY-measure of triangle abc, i.e. its anisotropic curvature, always 0.0.

Definition at line 472 of file CorrectedNormalCurrentFormula.h.

    {
      (void)a; (void)b; (void)c; (void)u;
      return RealTensor { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
    }

muXYConstantU() [2/2]

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYConstantU ( const RealPoints &  pts, const RealVector &  u  )

inlinestatic

Computes muXY measure (anisotropic curvature) of polygonal face pts given a constant corrected normal vector u.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the constant corrected normal vector to this polygonal face.

Returns

the muXY-measure of the given polygonal face, i.e. its anisotropic curvature, always 0.0.

Definition at line 571 of file CorrectedNormalCurrentFormula.h.

    {
      (void)pts; (void)u;
      return RealTensor { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
    }

muXYConstantUAtEdge()

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYConstantUAtEdge ( const RealPoint &  a, const RealPoint &  b, const RealVector &  ur, const RealVector &  ul  )

inlinestatic

Computes muXY measure (anisotropic measure) at edge ab given a constant corrected normal vector ur to the right of ab, and a constant corrected normal vector ul to the left of ab.

The formula is \( int_{ab} \Psi \langle e | e_1 \rangle dH^1 \), where e is the unit vector parallel to ab, and \( e_1 \) is the unit vector orthogonal to ur and ul, and \( \Psi \) is the angle between these two vectors.

Parameters

a	any point
b	any point
ur	the constant corrected normal vector to the right of oriented edge ab
ul	the constant corrected normal vector to the left of oriented edge ab

Returns

the mu1-measure of edge ab, i.e. twice its mean curvature.

Definition at line 496 of file CorrectedNormalCurrentFormula.h.

    {
      RealTensor M { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
      RealVector   e  = b - a;
      RealVector   e1 = ur.crossProduct( ul );
      const Scalar l1 = std::min( e1.norm(), 1.0 );
      if ( l1 < 1e-10 ) return M;
      e1 /= l1;
      const Scalar             psi = asin( l1 );
      const Scalar         sin_psi = sin( psi );
      const Scalar        sin_2psi = sin( 2.0 * psi );
      const RealVector      e_x_ur = e. crossProduct( ur );
      const RealVector     e1_x_ur = e1.crossProduct( ur );
      const RealVector e_x_e1_x_ur = e. crossProduct( e1_x_ur );
      for ( Dimension i = 0; i < 3; i++ )
        for ( Dimension j = 0; j < 3; j++ )
          {
            Scalar v = (-psi - 0.5*sin_2psi) * e_x_ur[ i ] * e1_x_ur[ j ]
              + (sin_psi*sin_psi) * ( e_x_ur[ i ] * ur[ j ]
                                      - e_x_e1_x_ur[ i ] * e1_x_ur[ j ] )
              + (psi - 0.5*sin_2psi) * e_x_e1_x_ur[ i ] * ur[ j ];
            M.setComponent( i, j, -0.5 * v );
          }
      return M;
    }

References DGtal::crossProduct().

muXYInterpolatedU() [1/2]

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYInterpolatedU ( const RealPoint &  a, const RealPoint &  b, const RealPoint &  c, const RealVector &  ua, const RealVector &  ub, const RealVector &  uc, bool  unit_u = false  )

inlinestatic

Computes muXY measure (anisotropic curvature) of triangle abc given an interpolated corrected normal vector ua, ub, uc.

Parameters

a	any point
b	any point
c	any point
ua	the corrected normal vector at point a
ub	the corrected normal vector at point b
uc	the corrected normal vector at point c
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the muXY-measure of triangle abc, i.e. its anisotropic curvature.

Definition at line 537 of file CorrectedNormalCurrentFormula.h.

    {
      RealTensor T;
      //  MUXY = 1/2 < uM | < Y | uc-ua > X x (b-a) - < Y | ub-ua > X x (c-a) >
      //  MUXY = 1/2 ( < Y | ub-ua > | X uM (c-a) | - < Y | uc-ua > | X uM (b-a) | )
      RealVector uM = ( ua+ub+uc ) / 3.0;
      if ( unit_u ) uM /= uM.norm();
      const RealVector uac = uc - ua;
      const RealVector uab = ub - ua;
      const RealVector  ab = b - a;
      const RealVector  ac = c - a;
      for ( Dimension i = 0; i < 3; ++i ) {
        RealVector X = RealVector::base( i, 1.0 );
        for ( Dimension j = 0; j < 3; ++j ) {
          // Since RealVector Y = RealVector::base( j, 1.0 );
          // < Y | uac > = uac[ j ]
          const Scalar tij =
            0.5 * uM.dot( uac[ j ] * X.crossProduct( ab )
                          - uab[ j ] * X.crossProduct( ac ) );
          T.setComponent( i, j, tij );
        }
      }
      return T;
    }

References DGtal::PointVector< dim, TEuclideanRing, TContainer >::base(), and DGtal::SimpleMatrix< TComponent, TM, TN >::setComponent().

Referenced by DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYInterpolatedU().

muXYInterpolatedU() [2/2]

template<typename TRealPoint , typename TRealVector >

static RealTensor DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYInterpolatedU ( const RealPoints &  pts, const RealVectors &  u, bool  unit_u = false  )

inlinestatic

Computes anisotropic curvature of polygonal face pts given an interpolated corrected normal vector ua, ub, uc.

Parameters

pts	the (ccw ordered) points forming the vertices of a polygonal face.
u	the (ccw ordered) normal vectors at the corresponding vertices in pts.
unit_u	when 'true' considers that interpolated corrected normals should be made unitary, otherwise interpolated corrected normals may have smaller norms.

Returns

the muXY-measure of the given polygonal face, i.e. its anisotropic curvature.

Definition at line 586 of file CorrectedNormalCurrentFormula.h.

    {
      ASSERT( pts.size() == u.size() );
      if ( pts.size() <  3 ) return RealTensor { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
      if ( pts.size() == 3 )
        return muXYInterpolatedU( pts[ 0 ], pts[ 1 ], pts[ 2 ],
                                 u[ 0 ], u[ 1 ], u[ 2 ], unit_u );
      const RealPoint   b = barycenter( pts );
      const RealVector ub = averageUnitVector( u );
      RealTensor        T = { 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0 };
      for ( Index i = 0; i < pts.size(); i++ )
        T += muXYInterpolatedU( b,  pts[ i ], pts[ (i+1)%pts.size() ],
                                ub,   u[ i ],   u[ (i+1)%pts.size() ], unit_u );
      return T;
    }

References DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::averageUnitVector(), DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::barycenter(), and DGtal::CorrectedNormalCurrentFormula< TRealPoint, TRealVector >::muXYInterpolatedU().

Field Documentation

dimension

template<typename TRealPoint , typename TRealVector >

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

static

Definition at line 108 of file CorrectedNormalCurrentFormula.h.

---

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

• CorrectedNormalCurrentFormula.h