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

CGAL 4.8 - 3D Triangulations

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RegularTriangulationCellBase_3 Concept Reference
Concepts

## Definition

The regular triangulation of a set of weighted points does not necessarily have one vertex for each of the input points. Some of the input weighted points have no cell in the dual power diagrams and therefore do not correspond to a vertex of the regular triangulation. Those weighted points are said to be hidden points. A point which is hidden at a given time may appear later as a vertex of the regular triangulation upon removal on some other weighted point. Therefore, hidden points have to be stored somewhere. The regular triangulation stores those hidden points in its cells.

A hidden point can appear as vertex of the triangulation only when the three dimensional cell where its point component is located (the cell which hides it) is removed. Therefore we decided to store in each cell of a regular triangulation the list of hidden points that are located in the face. Thus points hidden by a face are easily reinserted in the triangulation when the face is removed.

The base cell of a regular triangulation has to be a model of the concept `RegularTriangulationCellBase_3`, which refines the concept `TriangulationCellBase_3` by adding in the cell a container to store hidden points and an operator to compute its weighted circumcenter.

Refines:
  `TriangulationCellBase_3`

Has Models:

  CGAL::Regular_triangulation_cell_base_3

  CGAL::Regular_triangulation_cell_base_with_weighted_circumcenter_3

See Also
  `RegularTriangulationTraits_3`

## Types
typedef unspecified_type | Weighted_point
| Must be the same as the point type `RegularTriangulationTraits_3::Weighted_point_3` defined by the geometric traits class of the triangulation.
typedef unspecified_type | Point_iterator
| Iterator of value type Point.

## Access functions
Point_iterator | hidden_points_begin ()
| Returns an iterator pointing to the first hidden point.
Point_iterator | hidden_points_end ()
| Returns a past-the-end iterator.
const RegularTriangulationTraits_3::Bare_point & | weighted_circumcenter (const RegularTriangulationTraits_3 &gt=RegularTriangulationTraits_3()) const
| Returns the weighted circumcenter of the cell, with no weight. More...

## Setting
void | hide_point (const Point &p)
| Adds `p` to the set of hidden points of the cell.

## Member Function Documentation

const RegularTriangulationTraits_3::Bare_point& RegularTriangulationCellBase_3::weighted_circumcenter | ( | const RegularTriangulationTraits_3 & | gt = `RegularTriangulationTraits_3()`) |  | const

Returns the weighted circumcenter of the cell, with no weight.

`RegularTriangulationTraits_3` is the geometric traits class of the triangulation. This operator is required only when the dual functions are called.