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//
// Copyright (C) 1998 - 2016 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE at
// the top level of the deal.II distribution.
//
// ---------------------------------------------------------------------
#ifndef dealii__point_h
#define dealii__point_h
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/tensor.h>
#include <cmath>
DEAL_II_NAMESPACE_OPEN
/**
* A class that represents a point in a space with arbitrary dimension
* <tt>dim</tt>.
*
* Objects of this class are used to represent points, i.e., vectors anchored
* at the origin of a Cartesian vector space. They are, among other uses,
* passed to functions that operate on points in spaces of a priori fixed
* dimension: rather than using functions like <tt>double f(double x)</tt> and
* <tt>double f(double x, double y)</tt>, you should use <tt>double
* f(Point<dim> &p)</tt> instead as it allows writing dimension independent
* code.
*
*
* <h3>What's a <code>Point@<dim@></code> and what is a
* <code>Tensor@<1,dim@></code>?</h3>
*
* 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
* <tt>Point</tt> class to denote points in space, i.e., for vectors (rank-1
* tensors) that are <em>anchored at the origin</em>. On the other hand,
* vectors that are anchored elsewhere (and consequently do not represent
* <em>points</em> 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
* <code>std_cxx11::array</code> class. Alternatively, as in the case of
* vector-valued functions, you can use objects of type Vector or
* <code>std::vector</code>.
*
*
* @tparam 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.
* @tparam Number The data type in which the coordinates values are to be
* stored. This will, in almost all cases, simply be the default @p 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.
*
*
* @ingroup geomprimitives
* @author Wolfgang Bangerth, 1997
*/
template <int dim, typename Number = double>
class Point : public Tensor<1,dim,Number>
{
public:
/**
* Standard constructor. Creates an object that corresponds to the origin,
* i.e., all coordinates are set to zero.
*/
Point ();
/**
* Convert a tensor to a point.
*/
explicit Point (const Tensor<1,dim,Number> &);
/**
* Constructor for one dimensional points. This function is only implemented
* for <tt>dim==1</tt> since the usage is considered unsafe for points with
* <tt>dim!=1</tt> as it would leave some components of the point
* coordinates uninitialized.
*/
explicit Point (const Number x);
/**
* Constructor for two dimensional points. This function is only implemented
* for <tt>dim==2</tt> since the usage is considered unsafe for points with
* <tt>dim!=2</tt> as it would leave some components of the point
* coordinates uninitialized (if dim>2) or would not use some arguments (if
* dim<2).
*/
Point (const Number x,
const Number y);
/**
* Constructor for three dimensional points. This function is only
* implemented for <tt>dim==3</tt> since the usage is considered unsafe for
* points with <tt>dim!=3</tt> as it would leave some components of the
* point coordinates uninitialized (if dim>3) or would not use some
* arguments (if dim<3).
*/
Point (const Number x,
const Number y,
const Number z);
/**
* Return a unit vector in coordinate direction <tt>i</tt>, i.e., a vector
* that is zero in all coordinates except for a single 1 in the <tt>i</tt>th
* coordinate.
*/
static Point<dim,Number> unit_vector(const unsigned int i);
/**
* Read access to the <tt>index</tt>th coordinate.
*/
Number operator () (const unsigned int index) const;
/**
* Read and write access to the <tt>index</tt>th coordinate.
*/
Number &operator () (const unsigned int index);
/*
* @name Addition and subtraction of points.
* @{
*/
/**
* Add an offset given as Tensor<1,dim,Number> to a point.
*/
Point<dim,Number> operator + (const Tensor<1,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@>.
*/
Tensor<1,dim,Number> operator - (const Point<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@>.
*/
Point<dim,Number> operator - (const Tensor<1,dim,Number> &) const;
/**
* The opposite vector.
*/
Point<dim,Number> operator - () const;
/**
* @}
*/
/*
* @name Multiplication and scaling of points. Dot products. Norms.
* @{
*/
/**
* Multiply the current point by a factor.
*
* @relates EnableIfScalar
*/
template <typename OtherNumber>
Point<dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator * (const OtherNumber) const;
/**
* Divide the current point by a factor.
*/
template <typename OtherNumber>
Point<dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator / (const OtherNumber) const;
/**
* Return the scalar product of the vectors representing two points.
*/
Number operator * (const Tensor<1,dim,Number> &p) 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.
*/
typename numbers::NumberTraits<Number>::real_type square () const;
/**
* Return the Euclidean distance of <tt>this</tt> point to the point
* <tt>p</tt>, i.e. the <tt>l_2</tt> norm of the difference between the
* vectors representing the two points.
*/
typename numbers::NumberTraits<Number>::real_type distance (const Point<dim,Number> &p) const;
/**
* @}
*/
/**
* Read or write the data of this object to or from a stream for the purpose
* of serialization
*/
template <class Archive>
void serialize(Archive &ar, const unsigned int version);
};
/*------------------------------- Inline functions: Point ---------------------------*/
#ifndef DOXYGEN
template <int dim, typename Number>
inline
Point<dim,Number>::Point ()
{}
template <int dim, typename Number>
inline
Point<dim,Number>::Point (const Tensor<1,dim,Number> &t)
:
Tensor<1,dim,Number>(t)
{}
template <int dim, typename Number>
inline
Point<dim,Number>::Point (const Number x)
{
switch (dim)
{
case 1:
this->values[0] = x;
default:
Assert (dim==1, StandardExceptions::ExcInvalidConstructorCall());
}
}
template <int dim, typename Number>
inline
Point<dim,Number>::Point (const Number x, const Number y)
{
switch (dim)
{
case 2:
this->values[0] = x;
this->values[1] = y;
default:
Assert (dim==2, StandardExceptions::ExcInvalidConstructorCall());
}
}
template <int dim, typename Number>
inline
Point<dim,Number>::Point (const Number x, const Number y, const Number z)
{
switch (dim)
{
case 3:
this->values[0] = x;
this->values[1] = y;
this->values[2] = z;
default:
Assert (dim==3, StandardExceptions::ExcInvalidConstructorCall());
}
}
template <int dim, typename Number>
inline
Point<dim,Number>
Point<dim,Number>::unit_vector(unsigned int i)
{
Point<dim,Number> p;
p[i] = 1.;
return p;
}
template <int dim, typename Number>
inline
Number
Point<dim,Number>::operator () (const unsigned int index) const
{
AssertIndexRange((int) index, dim);
return this->values[index];
}
template <int dim, typename Number>
inline
Number &
Point<dim,Number>::operator () (const unsigned int index)
{
AssertIndexRange((int) index, dim);
return this->values[index];
}
template <int dim, typename Number>
inline
Point<dim,Number>
Point<dim,Number>::operator + (const Tensor<1,dim,Number> &p) const
{
Point<dim,Number> tmp = *this;
tmp += p;
return tmp;
}
template <int dim, typename Number>
inline
Tensor<1,dim,Number>
Point<dim,Number>::operator - (const Point<dim,Number> &p) const
{
return (Tensor<1,dim,Number>(*this) -= p);
}
template <int dim, typename Number>
inline
Point<dim,Number>
Point<dim,Number>::operator - (const Tensor<1,dim,Number> &p) const
{
Point<dim,Number> tmp = *this;
tmp -= p;
return tmp;
}
template <int dim, typename Number>
inline
Point<dim,Number>
Point<dim,Number>::operator - () const
{
Point<dim,Number> result;
for (unsigned int i=0; i<dim; ++i)
result.values[i] = -this->values[i];
return result;
}
template <int dim, typename Number>
template<typename OtherNumber>
inline
Point<dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
Point<dim,Number>::operator * (const OtherNumber factor) const
{
Point<dim,typename ProductType<Number, OtherNumber>::type> tmp;
for (unsigned int i=0; i<dim; ++i)
tmp[i] = this->operator[](i) * factor;
return tmp;
}
template <int dim, typename Number>
template<typename OtherNumber>
inline
Point<dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
Point<dim,Number>::operator / (const OtherNumber factor) const
{
Point<dim,typename ProductType<Number, OtherNumber>::type> tmp;
for (unsigned int i=0; i<dim; ++i)
tmp[i] = this->operator[](i) / factor;
return tmp;
}
template <int dim, typename Number>
inline
Number
Point<dim,Number>::operator * (const Tensor<1,dim,Number> &p) const
{
Number res = Number();
for (unsigned int i=0; i<dim; ++i)
res += this->operator[](i) * p[i];
return res;
}
template <int dim, typename Number>
inline
typename numbers::NumberTraits<Number>::real_type
Point<dim,Number>::square () const
{
return this->norm_square();
}
template <int dim, typename Number>
inline
typename numbers::NumberTraits<Number>::real_type
Point<dim,Number>::distance (const Point<dim,Number> &p) const
{
Number sum = Number();
for (unsigned int i=0; i<dim; ++i)
{
const Number diff=this->values[i]-p(i);
sum += numbers::NumberTraits<Number>::abs_square (diff);
}
return std::sqrt(sum);
}
template <int dim, typename Number>
template <class Archive>
inline
void
Point<dim,Number>::serialize(Archive &ar, const unsigned int)
{
// forward to serialization
// function in the base class
ar &static_cast<Tensor<1,dim,Number> &>(*this);
}
#endif // DOXYGEN
/*------------------------------- Global functions: Point ---------------------------*/
/**
* Global operator scaling a point vector by a scalar.
*
* @relates Point
* @relates EnableIfScalar
*/
template <int dim, typename Number, typename OtherNumber>
inline
Point<dim,typename ProductType<Number, typename EnableIfScalar<OtherNumber>::type>::type>
operator * (const OtherNumber factor,
const Point<dim,Number> &p)
{
return p * factor;
}
/**
* Output operator for points. Print the elements consecutively, with a space
* in between.
* @relates Point
*/
template <int dim, typename Number>
inline
std::ostream &operator << (std::ostream &out,
const Point<dim,Number> &p)
{
for (unsigned int i=0; i<dim-1; ++i)
out << p[i] << ' ';
out << p[dim-1];
return out;
}
/**
* Output operator for points. Print the elements consecutively, with a space
* in between.
* @relates Point
*/
template <int dim, typename Number>
inline
std::istream &operator >> (std::istream &in,
Point<dim,Number> &p)
{
for (unsigned int i=0; i<dim; ++i)
in >> p[i];
return in;
}
#ifndef DOXYGEN
/**
* Output operator for points of dimension 1. This is implemented specialized
* from the general template in order to avoid a compiler warning that the
* loop is empty.
*/
template <typename Number>
inline
std::ostream &operator << (std::ostream &out,
const Point<1,Number> &p)
{
out << p[0];
return out;
}
#endif // DOXYGEN
DEAL_II_NAMESPACE_CLOSE
#endif
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