\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - dD Geometry Kernel

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Functions
Global Kernel Functions
dD Geometry Kernel Reference

##

## Functions
template<class ForwardIterator >
Point_d< R > | CGAL::center_of_sphere (ForwardIterator first, ForwardIterator last)
| returns the center of the sphere spanned by the points in `A = tuple[first,last)`. More...
Point_d< R > | CGAL::lift_to_paraboloid (const Point_d< R > &p)
| returns the projection of \( p = (x_0,\ldots,x_{d-1})\) onto the paraboloid of revolution which is the point \( (p_0, \ldots,p_{d-1},\sum_{0 \le i < d}p_i^2)\) in \( (d+1)\)-space.
template<class ForwardIterator , class OutputIterator >
OutputIterator | CGAL::linear_base (ForwardIterator first, ForwardIterator last, OutputIterator result)
| computes a basis of the linear space spanned by the vectors in `A = tuple [first,last)` and returns it via an iterator range starting in `result`. More...
Point_d< R > | CGAL::midpoint (const Point_d< R > &p, const Point_d< R > &q)
| computes the midpoint of the segment \( pq\). More...
Point_d< R > | CGAL::project_along_d_axis (const Point_d< R > &p)
| returns \( p\) projected along the \( d\)-axis onto the hyperspace spanned by the first \( d-1\) standard base vectors.
FT | CGAL::squared_distance (Point_d< R > p, Point_d< R > q)
| computes the square of the Euclidean distance between the two points \( p\) and \( q\). More...
bool | CGAL::do_intersect (Type1< R > obj1, Type2< R > obj2)
| checks whether `obj1` and `obj2` intersect. More...
cpp11::result_of < R::Intersect_d(Type1< R > , Type2< R >)>::type | CGAL::intersection (Type1< R > f1, Type2< R > f2)
| returns the intersection between `f1` and `f2`. More...
template<class ForwardIterator >
bool | CGAL::affinely_independent (ForwardIterator first, ForwardIterator last)
| returns true iff the points in `A = tuple [first,last)` are affinely independent. More...
template<class ForwardIterator >
int | CGAL::affine_rank (ForwardIterator first, ForwardIterator last)
| computes the affine rank of the points in `A = tuple [first,last)`. More...
Comparison_result | CGAL::compare_lexicographically (const Point_d< R > &p, const Point_d< R > &q)
| Compares the Cartesian coordinates of points `p` and `q` lexicographically in ascending order of its Cartesian components `p[i]` and `q[i]` for \( i = 0,\ldots,d-1\). More...
template<class ForwardIterator >
bool | CGAL::contained_in_affine_hull (ForwardIterator first, ForwardIterator last, const Point_d< R > &p)
| determines whether \( p\) is contained in the affine hull of the points in `A = tuple [first,last)`. More...
template<class ForwardIterator >
bool | CGAL::contained_in_linear_hull (ForwardIterator first, ForwardIterator last, const Vector_d< R > &v)
| determines whether \( v\) is contained in the linear hull of the vectors in `A = tuple [first,last)`. More...
template<class ForwardIterator >
bool | CGAL::contained_in_simplex (ForwardIterator first, ForwardIterator last, const Point_d< R > &p)
| determines whether \( p\) is contained in the simplex of the points in `A = tuple [first,last)`. More...
bool | CGAL::lexicographically_smaller (const Point_d< R > &p, const Point_d< R > &q)
| returns `true` iff `p` is lexicographically smaller than `q` with respect to Cartesian lexicographic order of points. More...
bool | CGAL::lexicographically_smaller_or_equal (const Point_d< R > &p, const Point_d< R > &q)
| returns `true` iff \( p\) is lexicographically smaller than \( q\) with respect to Cartesian lexicographic order of points or equal to \( q\). More...
template<class ForwardIterator >
bool | CGAL::linearly_independent (ForwardIterator first, ForwardIterator last)
| decides whether the vectors in `A = tuple [first,last)` are linearly independent. More...
template<class ForwardIterator >
int | CGAL::linear_rank (ForwardIterator first, ForwardIterator last)
| computes the linear rank of the vectors in `A = tuple [first,last)`. More...
template<class ForwardIterator >
Orientation | CGAL::orientation (ForwardIterator first, ForwardIterator last)
| determines the orientation of the points of the tuple `A = tuple [first,last)` where \( A\) consists of \( d+1\) points in \( d\)-space. More...
template<class ForwardIterator >
Bounded_side | CGAL::side_of_bounded_sphere (ForwardIterator first, ForwardIterator last, const Point_d< R > &p)
| returns the relative position of point `p` to the sphere defined by `A = tuple [first,last)`. More...
template<class ForwardIterator >
Oriented_side | CGAL::side_of_oriented_sphere (ForwardIterator first, ForwardIterator last, const Point_d< R > &p)
| returns the relative position of point `p` to the oriented sphere defined by the points in `A = tuple [first,last)` The order of the points in \( A\) is important, since it determines the orientation of the implicitly constructed sphere. More...

## Function Documentation

template<class ForwardIterator >

int CGAL::affine_rank | ( | ForwardIterator | first,
|  | ForwardIterator | last
| ) |  |

computes the affine rank of the points in `A = tuple [first,last)`.

Precondition
  The objects in \( A\) are of the same dimension.

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

bool CGAL::affinely_independent | ( | ForwardIterator | first,
|  | ForwardIterator | last
| ) |  |

returns true iff the points in `A = tuple [first,last)` are affinely independent.

Precondition
  The objects are of the same dimension.

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

Point_d<R> CGAL::center_of_sphere | ( | ForwardIterator | first,
|  | ForwardIterator | last
| ) |  |

returns the center of the sphere spanned by the points in `A = tuple[first,last)`.

Precondition
  \( A\) contains \( d+1\) affinely independent points of dimension \( d\).

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/constructions_d.h>`

Comparison_result CGAL::compare_lexicographically | ( | const Point_d< R > & | p,
|  | const Point_d< R > & | q
| ) |  |

Compares the Cartesian coordinates of points `p` and `q` lexicographically in ascending order of its Cartesian components `p[i]` and `q[i]` for \( i = 0,\ldots,d-1\).

Precondition
  `p.dimension() == q.dimension()`.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

bool CGAL::contained_in_affine_hull | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Point_d< R > & | p
| ) |  |

determines whether \( p\) is contained in the affine hull of the points in `A = tuple [first,last)`.

Precondition
  The objects in `A` are of the same dimension.

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

bool CGAL::contained_in_linear_hull | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Vector_d< R > & | v
| ) |  |

determines whether \( v\) is contained in the linear hull of the vectors in `A = tuple [first,last)`.

Precondition
  The objects in \( A\) are of the same dimension.

Template Parameters

  ForwardIterator | has `Vector_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

bool CGAL::contained_in_simplex | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Point_d< R > & | p
| ) |  |

determines whether \( p\) is contained in the simplex of the points in `A = tuple [first,last)`.

Precondition
  The objects in \( A\) are of the same dimension and affinely independent.

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

bool CGAL::do_intersect | ( | Type1< R > | obj1,
|  | Type2< R > | obj2
| ) |  |

checks whether `obj1` and `obj2` intersect.

Two objects `obj1` and `obj2` intersect if there is a point `p` that is part of both `obj1` and `obj2`. The intersection region of those two objects is defined as the set of all points `p` that are part of both `obj1` and `obj2`.

Precondition
  The objects are of the same dimension.

The types `Type1` and `Type2` can be any of the following:

- `Point_d<R>`
- `Line_d<R>`
- `Ray_d<R>`
- `Segment_d<R>`
- `Hyperplane_d<R>`

See Also
  `intersection`

`#include <CGAL/intersections_d.h>`

cpp11::result_of<R::Intersect_d(Type1<R>, Type2<R>)>::type CGAL::intersection | ( | Type1< R > | f1,
|  | Type2< R > | f2
| ) |  |

returns the intersection between `f1` and `f2`.

Precondition
  The objects are of the same dimension.

The same functionality is also available through the functor `Kernel::Intersect_d`.

The following table gives the possible values for `Type1` and `Type2`.

The return type can be obtained through `cpp11::result_of<Kernel::Intersect_d(A, B)>::type`. It is equivalent to `boost::optional< boost::variant< T... > >`, the last column in the table providing the template parameter pack.

Type1 | Type2 | `T...`
--- | --- | ---
Line_d | Line_d |
Point_d
Line_d
Segment_d | Line_d |
Point_d
Segment_d
Segment_d | Segment_d |
Point_d
Segment_d
Ray_d | Line_d |
Point_d
Ray_d
Ray_d | Segment_d |
Point_d
Segment_d
Ray_d | Ray_d |
Point_d
Segment_d
Ray_d
Hyperplane_d | Line_d |
Point_d
Line_d
Hyperplane_d | Ray_d |
Point_d
Ray_d
Hyperplane_d | Segment_d |
Point_d
Segment_d

Example

The following example demonstrates the most common use of `intersection` routines.

#include <CGAL/intersections_d.h>
template<typename R>
struct Intersection_visitor {
typedef void result_type;
void operator()(const Point_d<R>& p) const {
// handle point
}
void operator()(const Segment_d<R>& s) const {
// handle segment
}
};
template <class R>
void foo(Segment_d<R> seg, Line_d<R> lin)
{
// with C++11 support
// auto result = intersection(seg, lin);
// without C++11 support
typename cpp11::result_of<R::Intersect_d(Segment_d<R>, Line_d<R>)>::type
result = intersection(seg, lin);
if(result) { boost::apply_visitor(Intersection_visitor<R>(), *result); }
else { // no intersection
}
}

See Also
  `do_intersect`
  `Kernel_d::Intersect_d`
  CGAL_INTERSECTION_VERSION
  `boost::optional`
  `boost::variant`
  `cpp11::result_of`

`#include <CGAL/intersections_d.h>`

bool CGAL::lexicographically_smaller | ( | const Point_d< R > & | p,
|  | const Point_d< R > & | q
| ) |  |

returns `true` iff `p` is lexicographically smaller than `q` with respect to Cartesian lexicographic order of points.

Precondition
  `p.dimension() == q.dimension()`.

`#include <CGAL/predicates_d.h>`

bool CGAL::lexicographically_smaller_or_equal | ( | const Point_d< R > & | p,
|  | const Point_d< R > & | q
| ) |  |

returns `true` iff \( p\) is lexicographically smaller than \( q\) with respect to Cartesian lexicographic order of points or equal to \( q\).

Precondition
  `p.dimension() == q.dimension()`.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator , class OutputIterator >

OutputIterator CGAL::linear_base | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | OutputIterator | result
| ) |  |

computes a basis of the linear space spanned by the vectors in `A = tuple [first,last)` and returns it via an iterator range starting in `result`.

The returned iterator marks the end of the output.

Precondition
  \( A\) contains vectors of the same dimension \( d\).

Template Parameters

  ForwardIterator | has `Vector_d<R>` as value type.

`#include <CGAL/constructions_d.h>`

template<class ForwardIterator >

int CGAL::linear_rank | ( | ForwardIterator | first,
|  | ForwardIterator | last
| ) |  |

computes the linear rank of the vectors in `A = tuple [first,last)`.

Precondition
  The objects are of the same dimension.

Template Parameters

  ForwardIterator | has `Vector_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

bool CGAL::linearly_independent | ( | ForwardIterator | first,
|  | ForwardIterator | last
| ) |  |

decides whether the vectors in `A = tuple [first,last)` are linearly independent.

Precondition
  The objects in `A` are of the same dimension.

Template Parameters

  ForwardIterator | has `Vector_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

Point_d<R> CGAL::midpoint | ( | const Point_d< R > & | p,
|  | const Point_d< R > & | q
| ) |  |

computes the midpoint of the segment \( pq\).

Precondition
  `p.dimension() == q.dimension()`.

`#include <CGAL/constructions_d.h>`

template<class ForwardIterator >

Orientation CGAL::orientation | ( | ForwardIterator | first,
|  | ForwardIterator | last
| ) |  |

determines the orientation of the points of the tuple `A = tuple [first,last)` where \( A\) consists of \( d+1\) points in \( d\)-space.

This is the sign of the determinant

\[ \left| \begin{array}{cccc} 1 & 1 & 1 & 1 \\ A[0] & A[1] & \dots& A[d] \end{array} \right| \]

where `A[i]` denotes the Cartesian coordinate vector of the \( i\)-th point in \( A\).

Precondition
  `size [first,last) == d+1` and `A[i].dimension() == d` \( \forall0 \leq i \leq d\).

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

Bounded_side CGAL::side_of_bounded_sphere | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Point_d< R > & | p
| ) |  |

returns the relative position of point `p` to the sphere defined by `A = tuple [first,last)`.

The order of the points of \( A\) does not matter.

Precondition
  `orientation(first,last)` is not `ZERO`.

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

template<class ForwardIterator >

Oriented_side CGAL::side_of_oriented_sphere | ( | ForwardIterator | first,
|  | ForwardIterator | last,
|  | const Point_d< R > & | p
| ) |  |

returns the relative position of point `p` to the oriented sphere defined by the points in `A = tuple [first,last)` The order of the points in \( A\) is important, since it determines the orientation of the implicitly constructed sphere.

If the points in \( A\) are positively oriented, the positive side is the bounded interior of the sphere.

Precondition
  `A` contains \( d+1\) points in \( d\)-space.

Template Parameters

  ForwardIterator | has `Point_d<R>` as value type.

`#include <CGAL/predicates_d.h>`

FT CGAL::squared_distance | ( | Point_d< R > | p,
|  | Point_d< R > | q
| ) |  |

computes the square of the Euclidean distance between the two points \( p\) and \( q\).

Precondition
  The dimensions of \( p\) and \( q\) are the same.

`#include <CGAL/constructions_d.h>`