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CGAL 4.3 - 3D Mesh Generation
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BisectionGeometricTraits_3 Concept Reference
Secondary Concepts

Definition

The concept BisectionGeometricTraits_3 describes a geometric traits class that provides the basic types and operations to implement a model of MeshDomain_3 based solely on intersection detections. Points in the non-empty intersections are herein computed by bisection.

Such traits class is relevant when intersection detections can be performed efficiently. For instance, when bounding surfaces are implicitly described by a function (such as an isosurface of a 3D function from \( \mathbb{R}^3\) to \( \mathbb{R}\)), the do-intersect predicate with a segment is computed by evaluations of the function values at both end points of the segment.

Has Models:
Any CGAL Kernel.
See Also
ImplicitSurfaceTraits_3
IntersectionGeometricTraits_3
CGAL::Implicit_mesh_domain_3<Function,BGT>
CGAL::Labeled_image_mesh_domain_3<Image,BGT>

Types

typedef unspecified_type FT
 Numerical type. More...
 
typedef unspecified_type Point_3
 The point type. More...
 
typedef unspecified_type Segment_3
 Segment type.
 
typedef unspecified_type Ray_3
 Ray type.
 
typedef unspecified_type Line_3
 Line type.
 
typedef unspecified_type Vector_3
 Vector type.
 
typedef unspecified_type Sphere_3
 Sphere type.
 
typedef unspecified_type Compute_scalar_product_3
 Function object providing the operator. More...
 
typedef unspecified_type Compute_squared_distance_3
 Function object providing the operator. More...
 
typedef unspecified_type Compute_squared_radius_3
 Function object providing the operator. More...
 
typedef unspecified_type Construct_center_3
 Function object providing the operator. More...
 
typedef unspecified_type Construct_midpoint_3
 Function object providing the operator. More...
 
typedef unspecified_type Construct_point_on_3
 Function object providing the following operators: More...
 
typedef unspecified_type Construct_segment_3
 Function object providing the operator. More...
 
typedef unspecified_type Construct_scaled_vector_3
 Function object providing the operator. More...
 
typedef unspecified_type Construct_translated_point_3
 Function object providing the operator. More...
 
typedef unspecified_type Construct_vector_3
 Function object providing the operator. More...
 
typedef unspecified_type Has_on_bounded_side_3
 Function object providing the operator. More...
 

Operations

The following functions give access to the predicate and construction objects:

Compute_scalar_product_3 compute_scalar_product_3_object ()
 
Compute_squared_distance_3 compute_squared_distance_3_object ()
 
Compute_squared_radius_3 compute_squared_radius_3_object ()
 
Construct_center_3 construct_center_3_object ()
 
Construct_midpoint_3 construct_midpoint_3_object ()
 
Construct_point_on_3 construct_point_on_3_object ()
 
Construct_scaled_vector_3 construct_scaled_vector_3_object ()
 
Construct_segment_3 construct_segment_3_object ()
 
Construct_translated_point_3 construct_translated_point_3_object ()
 
Construct_vector_3 construct_vector_3_object ()
 
Has_on_bounded_side_3 has_on_bounded_side_3_object ()
 

Member Typedef Documentation

typedef unspecified_type BisectionGeometricTraits_3::Compute_scalar_product_3

Function object providing the operator.

FT operator()(Vector_3 v, Vector_3 w) which returns the scalar (inner) product of the two vectors v and w.

typedef unspecified_type BisectionGeometricTraits_3::Compute_squared_distance_3

Function object providing the operator.

FT operator()(Point_3, Point_3) which returns the squared distance between two points.

typedef unspecified_type BisectionGeometricTraits_3::Compute_squared_radius_3

Function object providing the operator.

FT operator()(Sphere_3 s) which returns the squared radius of s.

typedef unspecified_type BisectionGeometricTraits_3::Construct_center_3

Function object providing the operator.

Point_3 operator()(Sphere_3 s) which returns the center of the sphere s.

typedef unspecified_type BisectionGeometricTraits_3::Construct_midpoint_3

Function object providing the operator.

Point_3 operator()(Point_3 p, Point_3 q) which computes the midpoint of the segment pq.

typedef unspecified_type BisectionGeometricTraits_3::Construct_point_on_3

Function object providing the following operators:

Point_3 operator()(Line_3 l,int i) which returns an arbitrary point on l. It holds point(i) == point(j), iff i==j. Furthermore, is directed from point(i) to point(j), for all i < j.

Point_3 operator()(Ray_3 r,int i) which returns a point on r. point(0) is the source, point(i), with \( i>0\), is different from the source.

Precondition
\( i \geq0\).

Point_3 operator()(Segment_3 s,int i) which returns either source or target of s: point(0) returns the source of s, point(1) returns the target of s. Parameter i is taken modulo 2, which gives easy access to the other end point.

typedef unspecified_type BisectionGeometricTraits_3::Construct_scaled_vector_3

Function object providing the operator.

Vector_3 operator()(Vector_3 v, FT scale) which returns the vector v scaled by a factor scale.

typedef unspecified_type BisectionGeometricTraits_3::Construct_segment_3

Function object providing the operator.

Segment_3 operator()(Point_3 p, Point_3 q) which returns a segment with source p and target q, directed from the source to the target.

typedef unspecified_type BisectionGeometricTraits_3::Construct_translated_point_3

Function object providing the operator.

Point_3 operator()(Point_3 p, Vector_3 v) which returns the point obtained by translating p by the vector v.

typedef unspecified_type BisectionGeometricTraits_3::Construct_vector_3

Function object providing the operator.

Vector_3 operator()(Point_3 a, Point_3 b) which returns the vector b-a.

typedef unspecified_type BisectionGeometricTraits_3::FT

Numerical type.

Must be a model of FieldWithSqrt and constructible from a double.

typedef unspecified_type BisectionGeometricTraits_3::Has_on_bounded_side_3

Function object providing the operator.

bool operator()(Sphere_3 s, Point_3 p) which returns true iff p lies on the bounded side of `s.

typedef unspecified_type BisectionGeometricTraits_3::Point_3

The point type.

Must have a constructor Point_3(FT, FT, FT).