CGAL 5.6.3 - 3D Periodic Mesh Generation

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CGAL::Periodic_3_function_wrapper< Function, BGT > Class Template Reference
3D Periodic Mesh Generation Reference » Domain Classes

`#include <CGAL/Periodic_3_function_wrapper.h>`

## Definition

template<class Function, class BGT>
class CGAL::Periodic_3_function_wrapper< Function, BGT >

The class `Periodic_3_function_wrapper` is a helper class designed to wrap an (a priori non-periodic) implicit function describing a domain through the relationship [`p` is inside if `f(p)<0`] and defined over the canonical cuboid to a function defined over the whole Euclidean space and periodic, with the same period as the canonical cuboid.

More precisely, if `f` is the real function defined either over \( \mathbb R^3\) or over the canonical cuboid, we construct the periodic real function \( f^{\ast} \) defined over \( \mathbb R^3\) as follows:

- For any point \((x,y,z)\) in the canonical cuboid, \( f^{\ast}(x,y,z) = f(x,y,z)\)
- For any point \((x,y,z)\) outside the canonical cuboid, there is a unique canonical representative \((x_0,y_0,z_0)\) of \((x,y,z)\) in the canonical cuboid, i.e., \((x,y,z)=(x_0 + k_x c, y_0 + k_y c, z_0 + k_z c)\) with \((k_x,k_y,k_z)\) in \( \mathbb Z^3\), and \( f^{\ast}(x,y,z) = f(x_0,y_0,z_0) \).

For example, if considering the unit cube as canonical cuboid, an oracle answering a query such as "what is the value of the implicit function at this point?" at the point `(2.5, 2.5, 2.5)` will be in fact evaluated at the canonical representative, that is `(0.5, 0.5, 0.5)`. Consequently, it is then not required to provide an input domain that is defined over the whole space or periodic, but only defined over the canonical cuboid.

Figure 61.1 Illustration in 2D (cut view) of a domain defined by an implicit function that is transformed into a periodic implicit function. Only the values of the implicit function that are in the canonical cuboid are used: the values of the implicit function at \( P \) and \( Q \) are obtained by evaluating instead at \( P' \) and \( Q' \), as shown on the right.

In practice, the implicit function provided by the user is likely defined over a larger domain than the canonical cuboid (in general, it is \( \mathbb R^3\)). Note that – when constructing artificially periodic functions – all the values of the implicit function for points outside this canonical cuboid are unused since queries are always answered by looking at the canonical representative. Figure 61.2 gives an example of such domain where some information is discarded.

Figure 61.2 Illustration in 2D (cut view) of a domain defined by an implicit function artificially made periodic. Any value of the function outside of the canonical cuboid is ignored.

Note also that when constructing artificially periodic functions, it is the responsibility of the user to provide an input function that is compatible with the canonical cuboid (that is, whose isovalues are periodically continuous and without intersections). Figure 61.3 is an example of a bad choice of input function and canonical cuboid: there is no continuity of the isovalues at the border of the canonical cuboid. In such configuration, the mesher might or might not finish and the result is likely to be non-manifold and to contain self-intersections.

Figure 61.3 Illustration in 2D (cut view) of a domain defined by an implicit function artificially made periodic. The zero isovalue of the implicit function does not form a continuous curve.

Template Parameters

  Function | provides the definition of the function. This parameter stands for a model of the concept `ImplicitFunction` described in the surface mesh generation package. The number types `Function::FT` and `BGT::FT` are required to match.
  BGT | is a geometric traits class that provides the basic operations to implement intersection tests and intersection computations through a bisection method. This parameter must be instantiated with a model of the concept `BisectionGeometricTraits_3`.

Examples
  Periodic_3_mesh_3/mesh_implicit_multi_domain.cpp, and Periodic_3_mesh_3/mesh_implicit_shape_with_features.cpp.

## Types
typedef BGT::FT | FT
typedef BGT::Point_3 | Point_3
typedef BGT::Iso_cuboid_3 | Iso_cuboid_3

## Creation
| Periodic_3_function_wrapper (Function f, const Iso_cuboid_3 &domain)
| Construction from an implicit function and the canonical cuboid.

## Operations
FT | operator() (const Point_3 &p) const
| Evaluates the function \( f \) passed in input at the canonical representative of \( p \).