Point< dim, Number > Class Template Reference

#include <deal.II/base/point.h>

Inheritance diagram for Point< dim, Number >:

Public Member Functions
	Point ()
	Point (const Tensor< 1, dim, Number > &)
	Point (const Number x)
	Point (const Number x, const Number y)
	Point (const Number x, const Number y, const Number z)
template<std::size_t dummy_dim, typename std::enable_if<(dim==dummy_dim) &&(dummy_dim !=0), int >::type = 0>
	Point (const boost::geometry::model::point< Number, dummy_dim, boost::geometry::cs::cartesian > &boost_pt)
Number	operator() (const unsigned int index) const
Number &	operator() (const unsigned int index)
template<class Archive >
void	serialize (Archive &ar, const unsigned int version)
Addition and subtraction of points.
Point< dim, Number >	operator+ (const Tensor< 1, dim, Number > &) const
Tensor< 1, dim, Number >	operator- (const Point< dim, Number > &) const
Point< dim, Number >	operator- (const Tensor< 1, dim, Number > &) const
Point< dim, Number >	operator- () const
Public Member Functions inherited from Tensor< 1, dim, Number >
constexpr	Tensor ()=default
	Tensor (const array_type &initializer)
	Tensor (const Tensor< rank_, dim, OtherNumber > &initializer)
	Tensor (const Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> &initializer)
constexpr	operator Tensor< 1, dim, Tensor< rank_ - 1, dim, OtherNumber >> () const
value_type &	operator[] (const unsigned int i)
constexpr const value_type &	operator[] (const unsigned int i) const
const Number &	operator[] (const TableIndices< rank_ > &indices) const
Number &	operator[] (const TableIndices< rank_ > &indices)
Number *	begin_raw ()
constexpr const Number *	begin_raw () const
Number *	end_raw ()
constexpr const Number *	end_raw () const
Tensor &	operator= (const Tensor< rank_, dim, OtherNumber > &rhs)
Tensor &	operator= (const Number &d)
bool	operator== (const Tensor< rank_, dim, OtherNumber > &) const
constexpr bool	operator!= (const Tensor< rank_, dim, OtherNumber > &) const
Tensor &	operator+= (const Tensor< rank_, dim, OtherNumber > &)
Tensor &	operator-= (const Tensor< rank_, dim, OtherNumber > &)
Tensor &	operator*= (const OtherNumber &factor)
Tensor &	operator/= (const OtherNumber &factor)
Tensor	operator- () const
void	clear ()
numbers::NumberTraits< Number >::real_type	norm () const
numbers::NumberTraits< Number >::real_type	norm_square () const
void	unroll (Vector< OtherNumber > &result) const
void	serialize (Archive &ar, const unsigned int version)

Static Public Member Functions
static Point< dim, Number >	unit_vector (const unsigned int i)
Static Public Member Functions inherited from Tensor< 1, dim, Number >
static unsigned int	component_to_unrolled_index (const TableIndices< rank_ > &indices)
static TableIndices< rank_ >	unrolled_to_component_indices (const unsigned int i)
static constexpr std::size_t	memory_consumption ()

Related Functions
(Note that these are not member functions.)
template<int dim, typename Number >
std::ostream &	operator<< (std::ostream &out, const Point< dim, Number > &p)
template<int dim, typename Number >
std::istream &	operator>> (std::istream &in, Point< dim, Number > &p)
Related Functions inherited from Tensor< 1, dim, Number >
Tensor< rank, dim, Number >	sum (const Tensor< rank, dim, Number > &local, const MPI_Comm &mpi_communicator)
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type >	operator- (const Tensor< 0, dim, Number > &p, const Tensor< 0, dim, OtherNumber > &q)
Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type >	operator- (const Tensor< rank, dim, Number > &p, const Tensor< rank, dim, OtherNumber > &q)
constexpr ProductType< Other, Number >::type	operator* (const Other &object, const Tensor< 0, dim, Number > &t)
constexpr ProductType< Number, Other >::type	operator* (const Tensor< 0, dim, Number > &t, const Other &object)
constexpr ProductType< Number, OtherNumber >::type	operator* (const Tensor< 0, dim, Number > &src1, const Tensor< 0, dim, OtherNumber > &src2)
Tensor< rank, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type >	operator* (const Tensor< rank, dim, Number > &t, const OtherNumber &factor)
constexpr Tensor< rank, dim, typename ProductType< typename EnableIfScalar< Number >::type, OtherNumber >::type >	operator* (const Number &factor, const Tensor< rank, dim, OtherNumber > &t)
constexpr Tensor< 0, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type >	operator/ (const Tensor< 0, dim, Number > &t, const OtherNumber &factor)
Tensor< rank, dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type >	operator/ (const Tensor< rank, dim, Number > &t, const OtherNumber &factor)
constexpr Tensor< 0, dim, typename ProductType< Number, OtherNumber >::type >	operator+ (const Tensor< 0, dim, Number > &p, const Tensor< 0, dim, OtherNumber > &q)
Tensor< rank, dim, typename ProductType< Number, OtherNumber >::type >	operator+ (const Tensor< rank, dim, Number > &p, const Tensor< rank, dim, OtherNumber > &q)
std::ostream &	operator<< (std::ostream &out, const Tensor< rank_, dim, Number > &p)
std::ostream &	operator<< (std::ostream &out, const Tensor< 0, dim, Number > &p)
Tensor< rank_1+rank_2 - 2, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type	operator* (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
Tensor< rank_1+rank_2 - 2, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type	contract (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
Tensor< rank_1+rank_2 - 4, dim, typename ProductType< Number, OtherNumber >::type >::tensor_type	double_contract (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
ProductType< Number, OtherNumber >::type	scalar_product (const Tensor< rank, dim, Number > &left, const Tensor< rank, dim, OtherNumber > &right)
ProductType< T1, typename ProductType< T2, T3 >::type >::type	contract3 (const TensorT1< rank_1, dim, T1 > &left, const TensorT2< rank_1+rank_2, dim, T2 > &middle, const TensorT3< rank_2, dim, T3 > &right)
Tensor< rank_1+rank_2, dim, typename ProductType< Number, OtherNumber >::type >	outer_product (const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, OtherNumber > &src2)
void	contract (Tensor< 2, dim, Number > &dest, const Tensor< 2, dim, Number > &src1, const unsigned int index1, const Tensor< 2, dim, Number > &src2, const unsigned int index3)
void	contract (Tensor< 2, dim, Number > &dest, const Tensor< 3, dim, Number > &src1, const unsigned int index1, const Tensor< 1, dim, Number > &src2)
void	contract (Tensor< 3, dim, Number > &dest, const Tensor< 3, dim, Number > &src1, const unsigned int index1, const Tensor< 2, dim, Number > &src2, const unsigned int index2)
void	contract (Tensor< rank_1+rank_2 - 2, dim, Number > &dest, const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, Number > &src2)
ProductType< Number, OtherNumber >::type	contract (const Tensor< 1, dim, Number > &src1, const Tensor< 1, dim, OtherNumber > &src2)
Number	double_contract (const Tensor< 2, dim, Number > &src1, const Tensor< 2, dim, Number > &src2)
void	double_contract (Tensor< 2, dim, Number > &dest, const Tensor< 4, dim, Number > &src1, const Tensor< 2, dim, Number > &src2)
void	outer_product (Tensor< rank_1+rank_2, dim, Number > &dst, const Tensor< rank_1, dim, Number > &src1, const Tensor< rank_2, dim, Number > &src2)
void	outer_product (Tensor< 1, dim, Number > &dst, const Number src1, const Tensor< 1, dim, Number > &src2)
void	outer_product (Tensor< 1, dim, Number > &dst, const Tensor< 1, dim, Number > src1, const Number src2)
Number	determinant (const Tensor< rank, 1, Number > &t)
Number	determinant (const Tensor< 1, 1, Number > &t)
void	cross_product (Tensor< 1, dim, Number > &dst, const Tensor< 1, dim, Number > &src)
void	cross_product (Tensor< 1, dim, Number > &dst, const Tensor< 1, dim, Number > &src1, const Tensor< 1, dim, Number > &src2)
Tensor< 1, dim, Number >	cross_product_2d (const Tensor< 1, dim, Number > &src)
Tensor< 1, dim, Number >	cross_product_3d (const Tensor< 1, dim, Number > &src1, const Tensor< 1, dim, Number > &src2)
Number	determinant (const Tensor< 2, dim, Number > &t)
constexpr Number	determinant (const Tensor< 2, 1, Number > &t)
Number	trace (const Tensor< 2, dim, Number > &d)
Tensor< 2, dim, Number >	invert (const Tensor< 2, dim, Number > &)
Tensor< 2, dim, Number >	transpose (const Tensor< 2, dim, Number > &t)
constexpr Tensor< 2, dim, Number >	adjugate (const Tensor< 2, dim, Number > &t)
constexpr Tensor< 2, dim, Number >	cofactor (const Tensor< 2, dim, Number > &t)
Number	l1_norm (const Tensor< 2, dim, Number > &t)
Number	linfty_norm (const Tensor< 2, dim, Number > &t)

Multiplication and scaling of points. Dot products. Norms.
template<typename OtherNumber >
Point< dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type >	operator/ (const OtherNumber) const
Number	operator* (const Tensor< 1, dim, Number > &p) const
numbers::NumberTraits< Number >::real_type	square () const
numbers::NumberTraits< Number >::real_type	distance (const Point< dim, Number > &p) const
numbers::NumberTraits< Number >::real_type	distance_square (const Point< dim, Number > &p) const
template<typename OtherNumber >
Point< dim, typename ProductType< Number, typename EnableIfScalar< OtherNumber >::type >::type >	operator* (const OtherNumber) const

Additional Inherited Members
Public Types inherited from Tensor< 1, dim, Number >
using	value_type = typename Tensor< rank_ - 1, dim, Number >::tensor_type
using	array_type = typename Tensor< rank_ - 1, dim, Number >::array_type[(dim !=0) ? dim :1]
using	tensor_type = Tensor< rank_, dim, Number >
Static Public Attributes inherited from Tensor< 1, dim, Number >
static constexpr unsigned int	dimension
static constexpr unsigned int	rank
static constexpr unsigned int	n_independent_components

Detailed Description

template<int dim, typename Number = double>
class Point< dim, Number >

A class that represents a point in a Cartesian space of dimension dim .

Objects of this class are used to represent points (i.e., vectors anchored at the origin) of a vector space equipped with a Cartesian coordinate system. They are, among other uses, passed to functions that operate on points in spaces of a priori fixed dimension: rather than using functions like double f(const double x) and double f(const double x, const double y), you can use double f(const Point<dim> &p) instead as it allows writing dimension independent code.

deal.II specifically uses Point objects as indicating points that are represented by Cartesian coordinates, i.e., where a point in dim space dimensions is characterized by signed distances along the axes of a coordinate system spanned by dim mutually orthogonal unit vectors (called the "coordinate axes"). This choice of representing a vector makes addition and scaling of vectors particularly simple: one only has to add or multiply each coordinate value. On the other hand, adding or scaling vectors is not nearly as simple when a vector is represented in other kinds of coordinate systems (e.g., spherical coordinate systems).

What's a Point<dim> and what is a Tensor<1,dim>?

The Point class is derived from Tensor<1,dim> and consequently shares the latter's member functions and other attributes. In fact, it has relatively few additional functions itself (the most notable exception being the distance() function to compute the Euclidean distance between two points in space), and these two classes can therefore often be used interchangeably.

Nonetheless, there are semantic differences that make us use these classes in different and well-defined contexts. Within deal.II, we use the Point class to denote points in space, i.e., for vectors (rank-1 tensors) that are anchored at the origin. On the other hand, vectors that are anchored elsewhere (and consequently do not represent points in the common usage of the word) are represented by objects of type Tensor<1,dim>. In particular, this is the case for direction vectors, normal vectors, gradients, and the differences between two points (i.e., what you get when you subtract one point from another): all of these are represented by Tensor<1,dim> objects rather than Point<dim>.

Furthermore, the Point class is only used where the coordinates of an object can be thought to possess the dimension of a length. An object that represents the weight, height, and cost of an object is neither a point nor a tensor (because it lacks the transformation properties under rotation of the coordinate system) and should consequently not be represented by either of these classes. Use an array of size 3 in this case, or the std::array class. Alternatively, as in the case of vector-valued functions, you can use objects of type Vector or std::vector.

Template Parameters

dim

An integer that denotes the dimension of the space in which a point lies. This of course equals the number of coordinates that identify a point.

Number

The data type in which the coordinates values are to be stored. This will, in almost all cases, simply be the default double, but there are cases where one may want to store coordinates in a different (and always scalar) type. An example would be an interval type that can store the value of a coordinate as well as its uncertainty. Another example would be a type that allows for Automatic Differentiation (see, for example, the Sacado type used in step-33) and thereby can generate analytic (spatial) derivatives of a function when passed a Point object whose coordinates are stored in such a type.

Author

Wolfgang Bangerth, 1997

Definition at line 110 of file point.h.

Constructor & Destructor Documentation

Point() [1/6]

template<int dim, typename Number = double>

Point< dim, Number >::Point

(

)

Standard constructor. Creates an object that corresponds to the origin, i.e., all coordinates are set to zero.

Point() [2/6]

template<int dim, typename Number = double>

Point< dim, Number >::Point ( const Tensor< 1, dim, Number > &  )

explicit

Convert a tensor to a point.

Point() [3/6]

template<int dim, typename Number = double>

Point< dim, Number >::Point ( const Number  x )

explicit

Constructor for one dimensional points. This function is only implemented for dim==1 since the usage is considered unsafe for points with dim!=1 as it would leave some components of the point coordinates uninitialized.

Point() [4/6]

template<int dim, typename Number = double>

Point< dim, Number >::Point	(	const Number	x,
		const Number	y
	)

Constructor for two dimensional points. This function is only implemented for dim==2 since the usage is considered unsafe for points with dim!=2 as it would leave some components of the point coordinates uninitialized (if dim>2) or would not use some arguments (if dim<2).

Point() [5/6]

template<int dim, typename Number = double>

Point< dim, Number >::Point	(	const Number	x,
		const Number	y,
		const Number	z
	)

Constructor for three dimensional points. This function is only implemented for dim==3 since the usage is considered unsafe for points with dim!=3 as it would leave some components of the point coordinates uninitialized (if dim>3) or would not use some arguments (if dim<3).

Point() [6/6]

template<int dim, typename Number = double>

template<std::size_t dummy_dim, typename std::enable_if<(dim==dummy_dim) &&(dummy_dim !=0), int >::type = 0>

Point< dim, Number >::Point

(

const boost::geometry::model::point< Number, dummy_dim, boost::geometry::cs::cartesian > &

boost_pt

)

Convert a boost::geometry::point to a Point.

Member Function Documentation

unit_vector()

template<int dim, typename Number = double>

static Point<dim, Number> Point< dim, Number >::unit_vector ( const unsigned int  i )

static

Return a unit vector in coordinate direction i, i.e., a vector that is zero in all coordinates except for a single 1 in the ith coordinate.

operator()() [1/2]

template<int dim, typename Number = double>

Number Point< dim, Number >::operator()

(

const unsigned int

index

)

const

Read access to the indexth coordinate.

operator()() [2/2]

template<int dim, typename Number = double>

Number& Point< dim, Number >::operator()

(

const unsigned int

index

)

Read and write access to the indexth coordinate.

operator+()

template<int dim, typename Number = double>

Point<dim, Number> Point< dim, Number >::operator+

(

const Tensor< 1, dim, Number > &

)

const

Add an offset given as Tensor<1,dim,Number> to a point.

operator-() [1/3]

template<int dim, typename Number = double>

Tensor<1, dim, Number> Point< dim, Number >::operator-

(

const Point< dim, Number > &

)

const

Subtract two points, i.e., obtain the vector that connects the two. As discussed in the documentation of this class, subtracting two points results in a vector anchored at one of the two points (rather than at the origin) and, consequently, the result is returned as a Tensor<1,dim> rather than as a Point<dim>.

operator-() [2/3]

template<int dim, typename Number = double>

Point<dim, Number> Point< dim, Number >::operator-

(

const Tensor< 1, dim, Number > &

)

const

Subtract a difference vector (represented by a Tensor<1,dim>) from the current point. This results in another point and, as discussed in the documentation of this class, the result is then naturally returned as a Point<dim> object rather than as a Tensor<1,dim>.

operator-() [3/3]

template<int dim, typename Number = double>

Point<dim, Number> Point< dim, Number >::operator-

(

)

const

The opposite vector.

operator/()

template<int dim, typename Number = double>

template<typename OtherNumber >

Point<dim, typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type> Point< dim, Number >::operator/

(

const OtherNumber

)

const

Divide the current point by a factor.

operator*()

template<int dim, typename Number = double>

Number Point< dim, Number >::operator*

(

const Tensor< 1, dim, Number > &

p

)

const

Return the scalar product of the vectors representing two points.

square()

template<int dim, typename Number = double>

numbers::NumberTraits<Number>::real_type Point< dim, Number >::square

(

)

const

Return the scalar product of this point vector with itself, i.e. the square, or the square of the norm. In case of a complex number type it is equivalent to the contraction of this point vector with a complex conjugate of itself.

Note

This function is equivalent to Tensor<rank,dim,Number>::norm_square() which returns the square of the Frobenius norm.

distance()

template<int dim, typename Number = double>

numbers::NumberTraits<Number>::real_type Point< dim, Number >::distance

(

const Point< dim, Number > &

p

)

const

Return the Euclidean distance of this point to the point p, i.e. the l_2 norm of the difference between the vectors representing the two points.

distance_square()

template<int dim, typename Number = double>

numbers::NumberTraits<Number>::real_type Point< dim, Number >::distance_square

(

const Point< dim, Number > &

p

)

const

Return the squared Euclidean distance of this point to the point p.

serialize()

template<int dim, typename Number = double>

template<class Archive >

void Point< dim, Number >::serialize	(	Archive &	ar,
		const unsigned int	version
	)

Read or write the data of this object to or from a stream for the purpose of serialization

Friends And Related Function Documentation

operator*()

template<int dim, typename Number = double>

template<typename OtherNumber >

Point<dim, typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type> operator* ( const OtherNumber  ) const

related

Multiply the current point by a factor.

operator<<()

template<int dim, typename Number >

std::ostream & operator<< ( std::ostream &  out, const Point< dim, Number > &  p  )

related

Output operator for points. Print the elements consecutively, with a space in between.

Definition at line 588 of file point.h.

operator>>()

template<int dim, typename Number >

std::istream & operator>> ( std::istream &  in, Point< dim, Number > &  p  )

related

Input operator for points. Inputs the elements consecutively.

Definition at line 605 of file point.h.

---

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

• deal.II/base/point.h