DGtal: DGtal::ImplicitPolynomial3Shape< TSpace

Aim: model of CEuclideanOrientedShape concepts to create a shape from a polynomial. More...

#include <DGtal/shapes/implicit/ImplicitPolynomial3Shape.h>

Public Types
typedef ImplicitPolynomial3Shape< TSpace >	Self

typedef TSpace	Space

typedef Space::RealPoint	RealPoint

typedef Space::RealVector	RealVector

typedef RealPoint::Coordinate	Ring

typedef Space::Integer	Integer

typedef MPolynomial< 3, Ring >	Polynomial3

typedef Ring	Value

Public Member Functions
	BOOST_STATIC_ASSERT ((Space::dimension==3))

	ImplicitPolynomial3Shape (const Polynomial3 &poly)

ImplicitPolynomial3Shape &	operator= (const ImplicitPolynomial3Shape &other)

	~ImplicitPolynomial3Shape ()

void	init (const Polynomial3 &poly)

double	operator() (const RealPoint &aPoint) const

bool	isInside (const RealPoint &aPoint) const

Orientation	orientation (const RealPoint &aPoint) const

RealVector	gradient (const RealPoint &aPoint) const

double	meanCurvature (const RealPoint &aPoint) const

double	gaussianCurvature (const RealPoint &aPoint) const

void	principalCurvatures (const RealPoint &aPoint, double &k1, double &k2) const

void	principalDirections (const RealPoint &aPoint, RealVector &d1, RealVector &d2) const

RealPoint	nearestPoint (const RealPoint &aPoint, const double accuracy=0.0001, const int maxIter=20, const double gamma=0.5) const

void	selfDisplay (std::ostream &out) const

bool	isValid () const

Protected Member Functions
	ImplicitPolynomial3Shape ()

Private Attributes
Polynomial3	myPolynomial
	The 3-polynomial defining the implicit shape. More...

Polynomial3	myFx

Polynomial3	myFy

Polynomial3	myFz

Polynomial3	myFxx

Polynomial3	myFxy

Polynomial3	myFxz

Polynomial3	myFyx

Polynomial3	myFyy

Polynomial3	myFyz

Polynomial3	myFzx

Polynomial3	myFzy

Polynomial3	myFzz

Polynomial3	myUpPolynome

Polynomial3	myLowPolynome

Detailed Description

template<typename TSpace>
class DGtal::ImplicitPolynomial3Shape< TSpace >

Aim: model of CEuclideanOrientedShape concepts to create a shape from a polynomial.

Description of template class 'ImplicitPolynomial3Shape'

Model of CImplicitFunction

Template Parameters

TSpace the Digital space definition.

Definition at line 67 of file ImplicitPolynomial3Shape.h.

Member Typedef Documentation

◆ Integer

template<typename TSpace >

typedef Space::Integer DGtal::ImplicitPolynomial3Shape< TSpace >::Integer

Definition at line 76 of file ImplicitPolynomial3Shape.h.

◆ Polynomial3

template<typename TSpace >

typedef MPolynomial< 3, Ring > DGtal::ImplicitPolynomial3Shape< TSpace >::Polynomial3

Definition at line 77 of file ImplicitPolynomial3Shape.h.

◆ RealPoint

template<typename TSpace >

typedef Space::RealPoint DGtal::ImplicitPolynomial3Shape< TSpace >::RealPoint

Definition at line 73 of file ImplicitPolynomial3Shape.h.

◆ RealVector

template<typename TSpace >

typedef Space::RealVector DGtal::ImplicitPolynomial3Shape< TSpace >::RealVector

Definition at line 74 of file ImplicitPolynomial3Shape.h.

◆ Ring

template<typename TSpace >

typedef RealPoint::Coordinate DGtal::ImplicitPolynomial3Shape< TSpace >::Ring

Definition at line 75 of file ImplicitPolynomial3Shape.h.

◆ Self

template<typename TSpace >

typedef ImplicitPolynomial3Shape<TSpace> DGtal::ImplicitPolynomial3Shape< TSpace >::Self

Definition at line 71 of file ImplicitPolynomial3Shape.h.

◆ Space

template<typename TSpace >

typedef TSpace DGtal::ImplicitPolynomial3Shape< TSpace >::Space

Definition at line 72 of file ImplicitPolynomial3Shape.h.

◆ Value

template<typename TSpace >

typedef Ring DGtal::ImplicitPolynomial3Shape< TSpace >::Value

Definition at line 78 of file ImplicitPolynomial3Shape.h.

Constructor & Destructor Documentation

◆ ImplicitPolynomial3Shape() [1/2]

template<typename TSpace >

DGtal::ImplicitPolynomial3Shape< TSpace >::ImplicitPolynomial3Shape ( const Polynomial3 & poly )

Constructor from an arbitrary polynomial.

Parameters

poly	any multivariate polynomial (the number of variables is the dimension of the space)

◆ ~ImplicitPolynomial3Shape()

template<typename TSpace >

DGtal::ImplicitPolynomial3Shape< TSpace >::~ImplicitPolynomial3Shape ( )

Destructor.

◆ ImplicitPolynomial3Shape() [2/2]

template<typename TSpace >

DGtal::ImplicitPolynomial3Shape< TSpace >::ImplicitPolynomial3Shape ( )

protected

Constructor. Forbidden by default (protected to avoid g++ warnings).

Member Function Documentation

◆ BOOST_STATIC_ASSERT()

template<typename TSpace >

DGtal::ImplicitPolynomial3Shape< TSpace >::BOOST_STATIC_ASSERT ( (Space::dimension==3) )

◆ gaussianCurvature()

template<typename TSpace >

double DGtal::ImplicitPolynomial3Shape< TSpace >::gaussianCurvature ( const RealPoint & aPoint ) const

inline

Gaussian curvature estimation at aPoint

Precondition: aPoint must be close to the surface.

Uses the formula \( G = -det(M) / \|\nabla f\|^4 \), where \( M := \left[ \begin{array}{cccc} f_{xx}& f_{xy} & f_{xz} & f_x \\ f_{xy}& f_{yy} & f_{yz} & f_y \\ f_{xz}& f_{yz} & f_{zz} & f_z \\ f_x& f_y & f_z & 0 \end{array} \right] \).

Parameters

aPoint any point in the Euclidean space.

Returns: the gaussian curvature value of the polynomial at aPoint.

◆ gradient()

template<typename TSpace >

RealVector DGtal::ImplicitPolynomial3Shape< TSpace >::gradient ( const RealPoint & aPoint ) const

inline

Parameters

aPoint any point in the Euclidean space.

Returns: the gradient vector of the polynomial at aPoint.

◆ init()

template<typename TSpace >

void DGtal::ImplicitPolynomial3Shape< TSpace >::init ( const Polynomial3 & poly )

Initialize from an arbitrary polynomial.

Parameters

poly	any multivariate polynomial (the number of variables is the dimension of the space)

◆ isInside()

template<typename TSpace >

bool DGtal::ImplicitPolynomial3Shape< TSpace >::isInside ( const RealPoint & aPoint ) const

Parameters

aPoint any point in the Euclidean space.

Returns: 'true' if the polynomial value is < 0.

◆ isValid()

template<typename TSpace >

bool DGtal::ImplicitPolynomial3Shape< TSpace >::isValid ( ) const

Checks the validity/consistency of the object.

Returns: 'true' if the object is valid, 'false' otherwise.

◆ meanCurvature()

template<typename TSpace >

double DGtal::ImplicitPolynomial3Shape< TSpace >::meanCurvature ( const RealPoint & aPoint ) const

inline

Mean curvature estimation. This computation is based on the hessian formula of the mean curvature k=(∇F ∗ H (F ) ∗ ∇F T − |∇F |^2 *Trace(H (F ))/2|∇F |^3

Precondition: a Point must be close to the surface.

Parameters

aPoint any point in the Euclidean space.

Returns: the mean curvature value of the polynomial at aPoint.

◆ nearestPoint()

template<typename TSpace >

RealPoint DGtal::ImplicitPolynomial3Shape< TSpace >::nearestPoint	(	const RealPoint &	aPoint,
		const double	accuracy = `0.0001`,
		const int	maxIter = `20`,
		const double	gamma = `0.5`
	)		const

inline

Perform a gradient descent in order to move a point aPoint closer to the implicit surface f(x)=0. More precisely, we use a sequence: x_n = x_(n-1) - gamma * f(x) * gradient(x_(n-1))/ ||gradient(x_(n-1))||^2 The descent is stopped if maxIter is reached or if | f(x_n) | < accuracy.

Parameters

aPoint	any point in the Euclidean space.
accuracy	distance criterion to stop the descent.
maxIter	fixes the maximum number of steps.
gamma	coefficient associated with the gradient.