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Reference documentation for deal.II version 9.1.1
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#include <deal.II/base/point.h>
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 |
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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) |
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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) |
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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 | |
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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 > |
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static constexpr unsigned int | dimension |
static constexpr unsigned int | rank |
static constexpr unsigned int | n_independent_components |
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).
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
.
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. |
Standard constructor. Creates an object that corresponds to the origin, i.e., all coordinates are set to zero.
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explicit |
Convert a tensor to a point.
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< 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< 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< 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.
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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 i
th coordinate.
Number Point< dim, Number >::operator() | ( | const unsigned int | index | ) | const |
Read access to the index
th coordinate.
Number& Point< dim, Number >::operator() | ( | const unsigned int | index | ) |
Read and write access to the index
th coordinate.
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.
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>.
Point<dim, Number> Point< dim, Number >::operator- | ( | ) | const |
The opposite vector.
Point<dim, typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type> Point< dim, Number >::operator/ | ( | const OtherNumber | ) | const |
Divide the current point by a factor.
Number Point< dim, Number >::operator* | ( | const Tensor< 1, dim, Number > & | p | ) | const |
Return the scalar product of the vectors representing two points.
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.
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.
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
.
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
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related |
Multiply the current point by a factor.
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related |
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