CGAL 6.1 - 2D Boolean Operations on Nef Polygons

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ExtendedKernelTraits_2 Concept Reference
Reference Manual » Concepts

## Definition

`ExtendedKernelTraits_2` is a kernel concept providing extended geometryIt is called extended geometry for simplicity, though it is not a real geometry in the classical sense. Let `K` be an instance of the data type `ExtendedKernelTraits_2`. The central notion of extended geometry are extended points. An extended point represents either a standard affine point of the Cartesian plane or a non-standard point representing the equivalence class of rays where two rays are equivalent if one is contained in the other.

Let \( R\) be an infinimaximal numberA finite but very large number., \( F\) be the square box with corners \( NW(-R,R)\), \( NE(R,R)\), \( SE(R,-R)\), and \( SW(-R,-R)\). Let \( p\) be a non-standard point and let \( r\) be a ray defining it. If the frame \( F\) contains the source point of \( r\) then let \( p(R)\) be the intersection of \( r\) with the frame \( F\), if \( F\) does not contain the source of \( r\) then \( p(R)\) is undefined. For a standard point let \( p(R)\) be equal to \( p\) if \( p\) is contained in the frame \( F\) and let \( p(R)\) be undefined otherwise. Clearly, for any standard or non-standard point \( p\), \( p(R)\) is defined for any sufficiently large \( R\). Let \( f\) be any function on standard points, say with \( k\) arguments. We call \( f\) extensible if for any \( k\) points \( p_1\), \( \ldots\) , \( p_k\) the function value \( f(p_1(R),\ldots,p_k(R))\) is constant for all sufficiently large \( R\). We define this value as \( f(p_1,\ldots,p_k)\). Predicates like lexicographic order of points, orientation, and incircle tests are extensible.

An extended segment is defined by two extended points such that it is either an affine segment, an affine ray, an affine line, or a segment that is part of the square box. Extended directions extend the affine notion of direction to extended objects.

This extended geometry concept serves two purposes. It offers functionality for changing between standard affine and extended geometry. At the same time it provides extensible geometric primitives on the extended geometric objects.

Has models
  `CGAL::Extended_cartesian<FT>`
  `CGAL::Extended_homogeneous<RT>`
  `CGAL::Filtered_extended_homogeneous<RT>`

## Extended kernel types
enum | Point_type { SWCORNER , LEFTFRAME , NWCORNER , BOTTOMFRAME , STANDARD , TOPFRAME , SECORNER , RIGHTFRAME , NECORNER }
| a type descriptor for extended points. More...
typedef unspecified_type | RT
| the ring type of our extended kernel.
typedef unspecified_type | Point_2
| extended points.
typedef unspecified_type | Segment_2
| extended segments.
typedef unspecified_type | Direction_2
| extended directions.

## Affine kernel types
typedef unspecified_type | Standard_kernel
| the standard affine kernel.
typedef unspecified_type | Standard_RT
| the standard ring type.
typedef unspecified_type | Standard_point_2
| standard points.
typedef unspecified_type | Standard_segment_2
| standard segments.
typedef unspecified_type | Standard_line_2
| standard oriented lines.
typedef unspecified_type | Standard_direction_2
| standard directions.
typedef unspecified_type | Standard_ray_2
| standard rays.
typedef unspecified_type | Standard_aff_transformation_2
| standard affine transformations.

## Interfacing the affine kernel types
Point_2 | construct_point (const Standard_point_2 &p)
| creates an extended point and initializes it to the standard point `p`.
Point_2 | construct_point (const Standard_line_2 &l)
| creates an extended point and initializes it to the equivalence class of all the rays underlying the oriented line `l`.
Point_2 | construct_point (const Standard_point_2 &p1, const Standard_point_2 &p2)
| creates an extended point and initializes it to the equivalence class of all the rays underlying the oriented line `l(p1,p2)`.
Point_2 | construct_point (const Standard_point_2 &p, const Standard_direction_2 &d)
| creates an extended point and initializes it to the equivalence class of all the rays underlying the ray starting in `p` in direction `d`.
Point_2 | construct_opposite_point (const Standard_line_2 &l)
| creates an extended point and initializes it to the equivalence class of all the rays underlying the oriented line opposite to `l`.
Point_type | type (const Point_2 &p)
| determines the type of `p` and returns it.
bool | is_standard (const Point_2 &p)
| returns `true` iff `p` is a standard point.
Standard_point_2 | standard_point (const Point_2 &p)
| returns the standard point represented by `p`.
Standard_line_2 | standard_line (const Point_2 &p)
| returns the oriented line representing the bundle of rays defining `p`.
Standard_ray_2 | standard_ray (const Point_2 &p)
| a ray defining `p`.
Point_2 | NE ()
| returns the point on the northeast frame corner.
Point_2 | SE ()
| returns the point on the southeast frame corner.
Point_2 | NW ()
| returns the point on the northwest frame corner.
Point_2 | SW ()
| returns the point on the southwest frame corner.

## Geometric kernel calls
Point_2 | source (const Segment_2 &s)
| returns the source point of `s`.
Point_2 | target (const Segment_2 &s)
| returns the target point of `s`.
Segment_2 | construct_segment (const Point_2 &p, const Point_2 &q)
| constructs a segment `pq`.
int | orientation (const Segment_2 &s, const Point_2 &p)
| returns the orientation of `p` with respect to the line through `s`.
int | orientation (const Point_2 &p1, const Point_2 &p2, const Point_2 &p3)
| returns the orientation of `p3` with respect to the line through `p1p2`.
bool | left_turn (const Point_2 &p1, const Point_2 &p2, const Point_2 &p3)
| return true iff the `p3` is left of the line through `p1p2`.
bool | is_degenerate (const Segment_2 &s)
| return true iff `s` is degenerate.
int | compare_xy (const Point_2 &p1, const Point_2 &p2)
| returns the lexicographic order of `p1` and `p2`.
int | compare_x (const Point_2 &p1, const Point_2 &p2)
| returns the order on the \( x\)-coordinates of `p1` and `p2`.
int | compare_y (const Point_2 &p1, const Point_2 &p2)
| returns the order on the \( y\)-coordinates of `p1` and `p2`.
Point_2 | intersection (const Segment_2 &s1, const Segment_2 &s2)
| returns the point of intersection of the lines supported by `s1` and `s2`.
Direction_2 | construct_direction (const Point_2 &p1, const Point_2 &p2)
| returns the direction of the vector `p2` - `p1`.
bool | strictly_ordered_ccw (const Direction_2 &d1, const Direction_2 &d2, const Direction_2 &d3)
| returns `true` iff `d2` is in the interior of the counterclockwise angular sector between `d1` and `d3`.
bool | strictly_ordered_along_line (const Point_2 &p1, const Point_2 &p2, const Point_2 &p3)
| returns `true` iff `p2` is in the relative interior of the segment `p1p3`.
bool | contains (const Segment_2 &s, const Point_2 &p)
| returns true iff `s` contains `p`.
bool | first_pair_closer_than_second (const Point_2 &p1, const Point_2 &p2, const Point_2 &p3, const Point_2 &p4)
| returns true iff \( \|p1-p2\| < \|p3-p4\|\).
const char * | output_identifier ()
| returns a unique identifier for kernel object Input/Output.

## Member Enumeration Documentation

## ◆ Point_type

enum ExtendedKernelTraits_2::Point_type

a type descriptor for extended points.

Enumerator
SWCORNER |
LEFTFRAME |
NWCORNER |
BOTTOMFRAME |
STANDARD |
TOPFRAME |
SECORNER |
RIGHTFRAME |
NECORNER |

## Member Function Documentation

## ◆ intersection()

Point_2 ExtendedKernelTraits_2::intersection | ( | const Segment_2 & | s1,
|  | const Segment_2 & | s2
| ) |  |

returns the point of intersection of the lines supported by `s1` and `s2`.

Precondition
  The intersection point exists.

## ◆ output_identifier()

const char * ExtendedKernelTraits_2::output_identifier | ( |  | ) |

returns a unique identifier for kernel object Input/Output.

Usually this should be the name of the model.

## ◆ standard_line()

Standard_line_2 ExtendedKernelTraits_2::standard_line | ( | const Point_2 & | p | ) |

returns the oriented line representing the bundle of rays defining `p`.

Precondition
  `!K.is_standard(p)`.

## ◆ standard_point()

Standard_point_2 ExtendedKernelTraits_2::standard_point | ( | const Point_2 & | p | ) |

returns the standard point represented by `p`.

Precondition
  `K.is_standard(p)`.

## ◆ standard_ray()

Standard_ray_2 ExtendedKernelTraits_2::standard_ray | ( | const Point_2 & | p | ) |

a ray defining `p`.

Precondition
  `!K.is_standard(p)`.