BisectionGeometricTraits_3 Concept Reference

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

All models of the CGAL concept Kernel

See also

ImplicitSurfaceTraits_3

IntersectionGeometricTraits_3

CGAL::Labeled_mesh_domain_3<BGT>

Types
typedef unspecified_type	FT
	Numerical type.
typedef unspecified_type	Point_3
	The point type.
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
	Model of Kernel::ComputeScalarProduct_3.
typedef unspecified_type	Compute_squared_distance_3
	Model of Kernel::ComputeSquaredDistance_3.
typedef unspecified_type	Compute_squared_radius_3
	Model of Kernel::ComputeSquaredRadius_3.
typedef unspecified_type	Construct_center_3
	Model of Kernel::ConstructCenter_3.
typedef unspecified_type	Construct_midpoint_3
	Model of Kernel::ConstructMidpoint_3.
typedef unspecified_type	Construct_point_on_3
	Model of Kernel::ConstructPoint_3.
typedef unspecified_type	Construct_segment_3
	Model of Kernel::ConstructSegment_3.
typedef unspecified_type	Construct_scaled_vector_3
	Model of Kernel::ConstructScaledVector_3.
typedef unspecified_type	Construct_translated_point_3
	Model of Kernel::ConstructTranslatedPoint_3.
typedef unspecified_type	Construct_vector_3
	Model of Kernel::ConstructVertex_3.
typedef unspecified_type	Has_on_bounded_side_3
	Model of Kernel::HasOnBoundedSide_3.

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

Compute_scalar_product_3

typedef unspecified_type BisectionGeometricTraits_3::Compute_scalar_product_3

Model of Kernel::ComputeScalarProduct_3.

That function object must provide the operator:

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

Compute_squared_distance_3

typedef unspecified_type BisectionGeometricTraits_3::Compute_squared_distance_3

Model of Kernel::ComputeSquaredDistance_3.

That function object must provide the operator:

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

Compute_squared_radius_3

typedef unspecified_type BisectionGeometricTraits_3::Compute_squared_radius_3

Model of Kernel::ComputeSquaredRadius_3.

That function object must provide the operator:

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

Construct_center_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_center_3

Model of Kernel::ConstructCenter_3.

That function object must provide the operator:

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

Construct_midpoint_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_midpoint_3

Model of Kernel::ConstructMidpoint_3.

That function object must provide the operator:

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

Construct_point_on_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_point_on_3

Model of Kernel::ConstructPoint_3.

That function object must provide 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.

Construct_scaled_vector_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_scaled_vector_3

Model of Kernel::ConstructScaledVector_3.

That function object must provide the operator:

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

Construct_segment_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_segment_3

Model of Kernel::ConstructSegment_3.

That function object must provide 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.

Construct_translated_point_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_translated_point_3

Model of Kernel::ConstructTranslatedPoint_3.

That function object must provide the operator:

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

Construct_vector_3

typedef unspecified_type BisectionGeometricTraits_3::Construct_vector_3

Model of Kernel::ConstructVertex_3.

That function object must provide the operator:

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

FT

typedef unspecified_type BisectionGeometricTraits_3::FT

Numerical type.

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

Has_on_bounded_side_3

typedef unspecified_type BisectionGeometricTraits_3::Has_on_bounded_side_3

Model of Kernel::HasOnBoundedSide_3.

That function object must provide the operator:

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

Point_3

typedef unspecified_type BisectionGeometricTraits_3::Point_3

The point type.

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