\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.10.2 - 2D and 3D Linear Geometry Kernel
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Kernel Concept Reference

Definition

The concept of a kernel is defined by a set of requirements on the provision of certain types and access member functions to create objects of these types. The types are function object classes to be used within the algorithms and data structures of CGAL. This allows you to use any model of a kernel as a traits class in the CGAL algorithms and data structures, unless they require types beyond those provided by a kernel.

A kernel provides types, construction objects, and generalized predicates. The former replace constructors of the kernel classes and constructive procedures in the kernel. There are also function objects replacing operators, especially for equality testing.

Naming convention of constructions
All constructions which result type is a geometric object are prefixed by Construct_. If the result type is a number type, the name is prefixed by Compute_. When the result type is not determined, no prefix is used.

Has Models:

CGAL::Cartesian<FieldNumberType>

CGAL::Homogeneous<RingNumberType>

CGAL::Simple_cartesian<FieldNumberType>

CGAL::Simple_homogeneous<RingNumberType>

CGAL::Filtered_kernel<CK>

CGAL::Exact_predicates_exact_constructions_kernel

CGAL::Exact_predicates_exact_constructions_kernel_with_sqrt

CGAL::Exact_predicates_inexact_constructions_kernel

See Also
Kernel_d
CGAL::Ambient_dimension
CGAL::Feature_dimension
CGAL::Kernel_traits
Examples:
Kernel_23/exact.cpp, Kernel_23/MyKernel.h, Kernel_23/points_and_segment.cpp, and Kernel_23/surprising.cpp.

Types

The following types describe the return types of predicates.

They typically map to bool and CGAL kernel enum types, except when an interval arithmetic number type is used such as within the filtering kernels, in which case it is Uncertain<bool> or similar.

typedef unspecified_type FT
 a model of FieldNumberType
 
typedef unspecified_type RT
 a model of RingNumberType
 
typedef unspecified_type Boolean
 bool or Uncertain<bool>
 
typedef unspecified_type Sign
 CGAL::Sign or Uncertain<CGAL::Sign>
 
typedef unspecified_type Comparison_result
 CGAL::Comparison_result or Uncertain<CGAL::Comparison_result>
 
typedef unspecified_type Orientation
 CGAL::Orientation or Uncertain<CGAL::Orientation>
 
typedef unspecified_type Oriented_side
 CGAL::Oriented_side or Uncertain<CGAL::Oriented_side>
 
typedef unspecified_type Bounded_side
 CGAL::Bounded_side or Uncertain<CGAL::Bounded_side>
 
typedef unspecified_type Angle
 CGAL::Angle or Uncertain<CGAL::Angle>
 

Constants

static const bool Has_filtered_predicates
 A Boolean value indicating whether the predicates are filtered (as in CGAL::Filtered_kernel). More...
 

Two-dimensional Coordinate Access

typedef unspecified_type Cartesian_const_iterator_2
 a model of Kernel::CartesianConstIterator_2
 

Two-dimensional Geometric Objects

typedef unspecified_type Point_2
 a model of Kernel::Point_2
 
typedef unspecified_type Weighted_point_2
 a model of Kernel::WeightedPoint_2
 
typedef unspecified_type Vector_2
 a model of Kernel::Vector_2
 
typedef unspecified_type Direction_2
 a model of Kernel::Direction_2
 
typedef unspecified_type Line_2
 a model of Kernel::Line_2
 
typedef unspecified_type Ray_2
 a model of Kernel::Ray_2
 
typedef unspecified_type Segment_2
 a model of Kernel::Segment_2
 
typedef unspecified_type Triangle_2
 a model of Kernel::Triangle_2
 
typedef unspecified_type Iso_rectangle_2
 a model of Kernel::IsoRectangle_2
 
typedef unspecified_type Circle_2
 a model of Kernel::Circle_2
 
typedef unspecified_type Object_2
 a model of Kernel::Object_2
 

Two-dimensional Constructions

typedef unspecified_type Construct_point_2
 a model of Kernel::ConstructPoint_2
 
typedef unspecified_type Construct_weighted_point_2
 a model of Kernel::ConstructWeightedPoint_2
 
typedef unspecified_type Construct_vector_2
 a model of Kernel::ConstructVector_2
 
typedef unspecified_type Construct_direction_2
 a model of Kernel::ConstructDirection_2
 
typedef unspecified_type Construct_segment_2
 a model of Kernel::ConstructSegment_2
 
typedef unspecified_type Construct_line_2
 a model of Kernel::ConstructLine_2
 
typedef unspecified_type Construct_ray_2
 a model of Kernel::ConstructRay_2
 
typedef unspecified_type Construct_circle_2
 a model of Kernel::ConstructCircle_2
 
typedef unspecified_type Construct_triangle_2
 a model of Kernel::ConstructTriangle_2
 
typedef unspecified_type Construct_iso_rectangle_2
 a model of Kernel::ConstructIsoRectangle_2
 
typedef unspecified_type Construct_object_2
 a model of Kernel::ConstructObject_2
 
typedef unspecified_type Construct_scaled_vector_2
 a model of Kernel::ConstructScaledVector_2
 
typedef unspecified_type Construct_difference_of_vectors_2
 a model of Kernel::ConstructDifferenceOfVectors_2
 
typedef unspecified_type Construct_sum_of_vectors_2
 a model of Kernel::ConstructSumOfVectors_2
 
typedef unspecified_type Construct_divided_vector_2
 a model of Kernel::ConstructDividedVector_2
 
typedef unspecified_type Construct_translated_point_2
 a model of Kernel::ConstructTranslatedPoint_2
 
typedef unspecified_type Construct_point_on_2
 a model of Kernel::ConstructPointOn_2
 
typedef unspecified_type Construct_projected_point_2
 a model of Kernel::ConstructProjectedPoint_2
 
typedef unspecified_type Construct_projected_xy_point_2
 a model of Kernel::ConstructProjectedXYPoint_2
 
typedef unspecified_type Construct_second_point_2
 a model of Kernel::ConstructSecondPoint_2 More...
 
typedef unspecified_type Construct_cartesian_const_iterator_2
 a model of Kernel::ConstructCartesianConstIterator_2
 
typedef unspecified_type Construct_vertex_2
 a model of Kernel::ConstructVertex_2
 
typedef unspecified_type Construct_bbox_2
 a model of Kernel::ConstructBbox_2
 
typedef unspecified_type Construct_perpendicular_vector_2
 a model of Kernel::ConstructPerpendicularVector_2
 
typedef unspecified_type Construct_perpendicular_direction_2
 a model of Kernel::ConstructPerpendicularDirection_2
 
typedef unspecified_type Construct_perpendicular_line_2
 a model of Kernel::ConstructPerpendicularLine_2
 
typedef unspecified_type Construct_midpoint_2
 a model of Kernel::ConstructMidpoint_2
 
typedef unspecified_type Construct_equidistant_line_3
 a model of Kernel::ConstructEquidistantLine_3
 
typedef unspecified_type Construct_center_2
 a model of Kernel::ConstructCenter_2
 
typedef unspecified_type Construct_centroid_2
 a model of Kernel::ConstructCentroid_2
 
typedef unspecified_type Construct_barycenter_2
 a model of Kernel::ConstructBarycenter_2
 
typedef unspecified_type Construct_circumcenter_2
 a model of Kernel::ConstructCircumcenter_2
 
typedef unspecified_type Construct_weighted_circumcenter_2
 a model of Kernel::ConstructWeightedCircumcenter_2
 
typedef unspecified_type Construct_bisector_2
 a model of Kernel::ConstructBisector_2
 
typedef unspecified_type Construct_radical_axis_2
 a model of Kernel::ConstructRadicalAxis_2
 
typedef unspecified_type Construct_radical_line_2
 a model of Kernel::ConstructRadicalLine_2
 
typedef unspecified_type Construct_opposite_direction_2
 a model of Kernel::ConstructOppositeDirection_2
 
typedef unspecified_type Construct_opposite_segment_2
 a model of Kernel::ConstructOppositeSegment_2
 
typedef unspecified_type Construct_opposite_ray_2
 a model of Kernel::ConstructOppositeRay_2
 
typedef unspecified_type Construct_opposite_line_2
 a model of Kernel::ConstructOppositeLine_2
 
typedef unspecified_type Construct_opposite_triangle_2
 a model of Kernel::ConstructOppositeTriangle_2
 
typedef unspecified_type Construct_opposite_circle_2
 a model of Kernel::ConstructOppositeCircle_2
 
typedef unspecified_type Construct_opposite_vector_2
 a model of Kernel::ConstructOppositeVector_2
 
typedef unspecified_type Construct_source_2
 a model of Kernel::ConstructSource_2
 
typedef unspecified_type Construct_target_2
 a model of Kernel::ConstructTarget_2
 
typedef unspecified_type Construct_max_vertex_2
 a model of Kernel::ConstructMaxVertex_2
 
typedef unspecified_type Construct_min_vertex_2
 a model of Kernel::ConstructMinVertex_2
 
typedef unspecified_type Intersect_2
 a model of Kernel::Intersect_2
 
typedef unspecified_type Assign_2
 a model of Kernel::Assign_2
 
typedef unspecified_type Compute_weight_2
 a model of Kernel::ComputeWeight_2
 
typedef unspecified_type Compute_x_2
 a model of Kernel::ComputeX_2
 
typedef unspecified_type Compute_y_2
 a model of Kernel::ComputeY_2
 
typedef unspecified_type Compute_a_2
 a model of Kernel::ComputeA_2
 
typedef unspecified_type Compute_b_2
 a model of Kernel::ComputeB_2
 
typedef unspecified_type ComputeC_2
 a model of Kernel::ComputeC_2
 
typedef unspecified_type Compute_dx_2
 a model of Kernel::ComputeDx_2
 
typedef unspecified_type Compute_dy_2
 a model of Kernel::ComputeDy_2
 
typedef unspecified_type Compute_hx_2
 a model of Kernel::ComputeHx_2
 
typedef unspecified_type Compute_hy_2
 a model of Kernel::ComputeHy_2
 
typedef unspecified_type Compute_hw_2
 a model of Kernel::ComputeHw_2
 
typedef unspecified_type Compute_L_infinity_distance_2
 a model of Kernel::ComputeLInfinityDistance_2
 
typedef unspecified_type Compute_xmax_2
 a model of Kernel::ComputeXmax_2
 
typedef unspecified_type Compute_xmin_2
 a model of Kernel::ComputeXmin_2
 
typedef unspecified_type Compute_ymax_2
 a model of Kernel::ComputeYmax_2
 
typedef unspecified_type Compute_ymin_2
 a model of Kernel::ComputeYmin_2
 
typedef unspecified_type Compute_y_at_x_2
 a model of Kernel::ComputeYAtX_2
 
typedef unspecified_type Compute_squared_distance_2
 a model of Kernel::ComputeSquaredDistance_2
 
typedef unspecified_type Compute_squared_length_2
 a model of Kernel::ComputeSquaredLength_2
 
typedef unspecified_type Compute_squared_radius_2
 a model of Kernel::ComputeSquaredRadius_2
 
typedef unspecified_type Compute_squared_radius_smallest_orthogonal_circle_2
 a model of Kernel::ComputeSquaredRadiusSmallestOrthogonalCircle_2
 
typedef unspecified_type Compute_power_product_2
 a model of Kernel::ComputePowerProduct_2
 
typedef unspecified_type Compute_area_2
 a model of Kernel::ComputeArea_2
 
typedef unspecified_type Compute_determinant_2
 a model of Kernel::ComputeDeterminant_2
 
typedef unspecified_type Compute_scalar_product_2
 a model of Kernel::ComputeScalarProduct_2
 

Two-dimensional Generalized Predicates

typedef unspecified_type Angle_2
 a model of Kernel::Angle_2
 
typedef unspecified_type Equal_2
 a model of Kernel::Equal_2
 
typedef unspecified_type Equal_x_2
 a model of Kernel::EqualX_2
 
typedef unspecified_type Equal_y_2
 a model of Kernel::EqualY_2
 
typedef unspecified_type Less_x_2
 a model of Kernel::LessX_2
 
typedef unspecified_type Less_y_2
 a model of Kernel::LessY_2
 
typedef unspecified_type Less_xy_2
 a model of Kernel::LessXY_2
 
typedef unspecified_type Less_yx_2
 a model of Kernel::LessYX_2
 
typedef unspecified_type Compare_x_2
 a model of Kernel::CompareX_2
 
typedef unspecified_type Compare_x_at_y_2
 a model of Kernel::CompareXAtY_2
 
typedef unspecified_type Compare_y_2
 a model of Kernel::CompareY_2
 
typedef unspecified_type Compare_xy_2
 a model of Kernel::CompareXY_2
 
typedef unspecified_type Compare_yx_2
 a model of Kernel::CompareYX_2
 
typedef unspecified_type Compare_y_at_x_2
 a model of Kernel::CompareYAtX_2
 
typedef unspecified_type Compare_distance_2
 a model of Kernel::CompareDistance_2
 
typedef unspecified_type Compare_power_distance_2
 a model of Kernel::ComparePowerDistance_2
 
typedef unspecified_type Compare_square_distance_2
 a model of Kernel::CompareSquaredDistance_2
 
typedef unspecified_type Compare_angle_with_x_axis_2
 a model of Kernel::CompareAngleWithXAxis_2
 
typedef unspecified_type Compare_slope_2
 a model of Kernel::CompareSlope_2
 
typedef unspecified_type Less_distance_to_point_2
 a model of Kernel::LessDistanceToPoint_2
 
typedef unspecified_type Less_signed_distance_to_line_2
 a model of Kernel::LessSignedDistanceToLine_2
 
typedef unspecified_type Less_rotate_ccw_2
 a model of Kernel::LessRotateCCW_2
 
typedef unspecified_type Left_turn_2
 a model of Kernel::LeftTurn_2
 
typedef unspecified_type Collinear_2
 a model of Kernel::Collinear_2
 
typedef unspecified_type Orientation_2
 a model of Kernel::Orientation_2
 
typedef unspecified_type Side_of_oriented_circle_2
 a model of Kernel::SideOfOrientedCircle_2
 
typedef unspecified_type Side_of_bounded_circle_2
 a model of Kernel::SideOfBoundedCircle_2
 
typedef unspecified_type Power_side_of_oriented_power_circle_2
 a model of Kernel::PowerSideOfOrientedPowerCircle_2
 
typedef unspecified_type Power_side_of_bounded_power_circle_2
 a model of Kernel::PowerSideOfBoundedPowerCircle_2
 
typedef unspecified_type Is_horizontal_2
 a model of Kernel::IsHorizontal_2
 
typedef unspecified_type Is_vertical_2
 a model of Kernel::IsVertical_2
 
typedef unspecified_type Is_degenerate_2
 a model of Kernel::IsDegenerate_2
 
typedef unspecified_type Has_on_2
 a model of Kernel::HasOn_2
 
typedef unspecified_type Collinear_has_on_2
 a model of Kernel::CollinearHasOn_2
 
typedef unspecified_type Has_on_bounded_side_2
 a model of Kernel::HasOnBoundedSide_2
 
typedef unspecified_type Has_on_unbounded_side_2
 a model of Kernel::HasOnUnboundedSide_2
 
typedef unspecified_type Has_on_boundary_2
 a model of Kernel::HasOnBoundary_2
 
typedef unspecified_type Has_on_positive_side_2
 a model of Kernel::HasOnPositiveSide_2
 
typedef unspecified_type Has_on_negative_side_2
 a model of Kernel::HasOnNegativeSide_2
 
typedef unspecified_type Oriented_side_2
 a model of Kernel::OrientedSide_2
 
typedef unspecified_type Bounded_side_2
 a model of Kernel::BoundedSide_2
 
typedef unspecified_type Are_parallel_2
 a model of Kernel::AreParallel_2
 
typedef unspecified_type Are_ordered_along_line_2
 a model of Kernel::AreOrderedAlongLine_2
 
typedef unspecified_type Are_strictly_ordered_along_line_2
 a model of Kernel::AreStrictlyOrderedAlongLine_2
 
typedef unspecified_type Collinear_are_ordered_along_line_2
 a model of Kernel::CollinearAreOrderedAlongLine_2
 
typedef unspecified_type Collinear_are_strictly_ordered_along_line_2
 a model of Kernel::CollinearAreStrictlyOrderedAlongLine_2
 
typedef unspecified_type Counterclockwise_in_between_2
 a model of Kernel::CounterclockwiseInBetween_2
 
typedef unspecified_type Do_intersect_2
 a model of Kernel::DoIntersect_2
 

Three-dimensional Coordinate Access

typedef unspecified_type Cartesian_const_iterator_3
 a model of Kernel::CartesianConstIterator_3
 

Three-dimensional Geometric Objects

typedef unspecified_type Point_3
 a model of Kernel::Point_3
 
typedef unspecified_type Weighted_point_3
 a model of Kernel::WeightedPoint_3
 
typedef unspecified_type Vector_3
 a model of Kernel::Vector_3
 
typedef unspecified_type Direction_3
 a model of Kernel::Direction_3
 
typedef unspecified_type Iso_cuboid_3
 a model of Kernel::IsoCuboid_3
 
typedef unspecified_type Line_3
 a model of Kernel::Line_3
 
typedef unspecified_type Ray_3
 a model of Kernel::Ray_3
 
typedef unspecified_type Circle_3
 a model of Kernel::Circle_3
 
typedef unspecified_type Sphere_3
 a model of Kernel::Sphere_3
 
typedef unspecified_type Segment_3
 a model of Kernel::Segment_3
 
typedef unspecified_type Plane_3
 a model of Kernel::Plane_3
 
typedef unspecified_type Triangle_3
 a model of Kernel::Triangle_3
 
typedef unspecified_type Tetrahedron_3
 a model of Kernel::Tetrahedron_3
 
typedef unspecified_type Object_3
 a model of Kernel::Object_3
 

Three-dimensional Constructions

typedef unspecified_type Construct_point_3
 a model of Kernel::ConstructPoint_3
 
typedef unspecified_type Construct_weighted_point_3
 a model of Kernel::ConstructWeightedPoint_3
 
typedef unspecified_type Construct_vector_3
 a model of Kernel::ConstructVector_3
 
typedef unspecified_type Construct_direction_3
 a model of Kernel::ConstructDirection_3
 
typedef unspecified_type Construct_plane_3
 a model of Kernel::ConstructPlane_3
 
typedef unspecified_type Construct_iso_cuboid_3
 a model of Kernel::ConstructIsoCuboid_3
 
typedef unspecified_type Construct_line_3
 a model of Kernel::ConstructLine_3
 
typedef unspecified_type Construct_ray_3
 a model of Kernel::ConstructRay_3
 
typedef unspecified_type Construct_sphere_3
 a model of Kernel::ConstructSphere_3
 
typedef unspecified_type Construct_segment_3
 a model of Kernel::ConstructSegment_3
 
typedef unspecified_type Construct_triangle_3
 a model of Kernel::ConstructTriangle_3
 
typedef unspecified_type Construct_tetrahedron_3
 a model of Kernel::ConstructTetrahedron_3
 
typedef unspecified_type Construct_circle_3
 a model of Kernel::ConstructCircle_3
 
typedef unspecified_type Construct_object_3
 a model of Kernel::ConstructObject_3
 
typedef unspecified_type Construct_scaled_vector_3
 a model of Kernel::ConstructScaledVector_3
 
typedef unspecified_type Construct_translated_point_3
 a model of Kernel::ConstructTranslatedPoint_3
 
typedef unspecified_type Construct_point_on_3
 a model of Kernel::ConstructPointOn_3
 
typedef unspecified_type Construct_projected_point_3
 a model of Kernel::ConstructProjectedPoint_3
 
typedef unspecified_type Construct_lifted_point_3
 a model of Kernel::ConstructLiftedPoint_3
 
typedef unspecified_type Construct_cartesian_const_iterator_3
 a model of Kernel::ConstructCartesianConstIterator_3
 
typedef unspecified_type Construct_vertex_3
 a model of Kernel::ConstructVertex_3
 
typedef unspecified_type Construct_source_3
 a model of Kernel::ConstructSource_3
 
typedef unspecified_type Construct_target_3
 a model of Kernel::ConstructTarget_3
 
typedef unspecified_type Construct_max_vertex_3
 a model of Kernel::ConstructMaxVertex_3
 
typedef unspecified_type Construct_min_vertex_3
 a model of Kernel::ConstructMinVertex_3
 
typedef unspecified_type Construct_bbox_3
 a model of Kernel::ConstructBbox_3
 
typedef unspecified_type Construct_supporting_plane_3
 a model of Kernel::ConstructSupportingPlane_3
 
typedef unspecified_type Construct_orthogonal_vector_3
 a model of Kernel::ConstructOrthogonalVector_3
 
typedef unspecified_type Construct_base_vector_3
 a model of Kernel::ConstructBaseVector_3
 
typedef unspecified_type Construct_difference_of_vectors_3
 a model of Kernel::ConstructDifferenceOfVectors_3
 
typedef unspecified_type Construct_sum_of_vectors_3
 a model of Kernel::ConstructSumOfVectors_3
 
typedef unspecified_type Construct_divided_vector_3
 a model of Kernel::ConstructDividedVector_3
 
typedef unspecified_type Construct_normal_3
 a model of Kernel::ConstructNormal_3
 
typedef unspecified_type Construct_perpendicular_plane_3
 a model of Kernel::ConstructPerpendicularPlane_3
 
typedef unspecified_type Construct_radical_plane_3
 a model of Kernel::ConstructRadicalPlane_3
 
typedef unspecified_type Construct_perpendicular_line_3
 a model of Kernel::ConstructPerpendicularLine_3
 
typedef unspecified_type Construct_midpoint_3
 a model of Kernel::ConstructMidpoint_3
 
typedef unspecified_type Construct_center_3
 a model of Kernel::ConstructCenter_3
 
typedef unspecified_type Construct_centroid_3
 a model of Kernel::ConstructCentroid_3
 
typedef unspecified_type Construct_circumcenter_3
 a model of Kernel::ConstructCircumcenter_3
 
typedef unspecified_type Construct_weighted_circumcenter_3
 a model of Kernel::ConstructWeightedCircumcenter_3
 
typedef unspecified_type Construct_bisector_3
 a model of Kernel::ConstructBisector_3
 
typedef unspecified_type Construct_cross_product_vector_3
 a model of Kernel::ConstructCrossProductVector_3
 
typedef unspecified_type Construct_barycenter_3
 a model of Kernel::ConstructBarycenter_3
 
typedef unspecified_type Construct_second_point_3
 a model of Kernel::ConstructSecondPoint_3 More...
 
typedef unspecified_type Construct_opposite_direction_3
 a model of Kernel::ConstructOppositeDirection_3
 
typedef unspecified_type Construct_opposite_segment_3
 a model of Kernel::ConstructOppositeSegment_3
 
typedef unspecified_type Construct_opposite_ray_3
 a model of Kernel::ConstructOppositeRay_3
 
typedef unspecified_type Construct_opposite_line_3
 a model of Kernel::ConstructOppositeLine_3
 
typedef unspecified_type Construct_opposite_plane_3
 a model of Kernel::ConstructOppositePlane_3
 
typedef unspecified_type Construct_opposite_sphere_3
 a model of Kernel::ConstructOppositeSphere_3
 
typedef unspecified_type Construct_opposite_vector_3
 a model of Kernel::ConstructOppositeVector_3
 
typedef unspecified_type Intersect_3
 a model of Kernel::Intersect_3
 
typedef unspecified_type Assign_3
 a model of Kernel::Assign_3
 
typedef unspecified_type Compute_weight_3
 a model of Kernel::ComputeWeight_3
 
typedef unspecified_type Compute_x_3
 a model of Kernel::ComputeX_3
 
typedef unspecified_type Compute_y_3
 a model of Kernel::ComputeY_3
 
typedef unspecified_type Compute_z_3
 a model of Kernel::ComputeZ_3
 
typedef unspecified_type Compute_a_3
 a model of Kernel::ComputeA_3
 
typedef unspecified_type Compute_b_3
 a model of Kernel::ComputeB_3
 
typedef unspecified_type Compute_c_3
 a model of Kernel::ComputeC_3
 
typedef unspecified_type Compute_d_3
 a model of Kernel::ComputeD_3
 
typedef unspecified_type Compute_hx_3
 a model of Kernel::ComputeHx_3
 
typedef unspecified_type Compute_hy_3
 a model of Kernel::ComputeHy_3
 
typedef unspecified_type Compute_hz_3
 a model of Kernel::ComputeHz_3
 
typedef unspecified_type Compute_hw_3
 a model of Kernel::ComputeHw_3
 
typedef unspecified_type Compute_L_infinity_distance_3
 a model of Kernel::ComputeLInfinityDistance_3
 
typedef unspecified_type Compute_dx_3
 a model of Kernel::ComputeDx_3
 
typedef unspecified_type Compute_dy_3
 a model of Kernel::ComputeDy_3
 
typedef unspecified_type Compute_dz_3
 a model of Kernel::ComputeDz_3
 
typedef unspecified_type Compute_xmax_3
 a model of Kernel::ComputeXmax_3
 
typedef unspecified_type Compute_xmin_3
 a model of Kernel::ComputeXmin_3
 
typedef unspecified_type Compute_ymax_3
 a model of Kernel::ComputeYmax_3
 
typedef unspecified_type Compute_ymin_3
 a model of Kernel::ComputeYmin_3
 
typedef unspecified_type Compute_zmax_3
 a model of Kernel::ComputeZmax_3
 
typedef unspecified_type Compute_zmin_3
 a model of Kernel::ComputeZmin_3
 
typedef unspecified_type Compute_area_3
 a model of Kernel::ComputeArea_3
 
typedef unspecified_type Compute_squared_area_3
 a model of Kernel::ComputeSquaredArea_3
 
typedef unspecified_type Compute_area_divided_by_pi_3
 a model of Kernel::ComputeAreaDividedByPi_3
 
typedef unspecified_type Compute_approximate_area_3
 a model of Kernel::ComputeApproximateArea_3
 
typedef unspecified_type Compute_approximate_dihedral_angle_3
 a model of Kernel::ComputeApproximateDihedralAngle_3
 
typedef unspecified_type Compute_determinant_3
 a model of Kernel::ComputeDeterminant_3
 
typedef unspecified_type Compute_scalar_product_3
 a model of Kernel::ComputeScalarProduct_3
 
typedef unspecified_type Compute_squared_distance_3
 a model of Kernel::ComputeSquaredDistance_3
 
typedef unspecified_type Compute_squared_length_3
 a model of Kernel::ComputeSquaredLength_3
 
typedef unspecified_type Compute_squared_length_divided_by_pi_square_3
 a model of Kernel::ComputeSquaredLengthDividedByPiSquare_3
 
typedef unspecified_type Compute_approximate_squared_length_3
 a model of Kernel::ComputeApproximateSquaredLength_3
 
typedef unspecified_type Compute_squared_radius_3
 a model of Kernel::ComputeSquaredRadius_3
 
typedef unspecified_type Compute_volume_3
 a model of Kernel::ComputeVolume_3
 
typedef unspecified_type Compute_power_product_3
 a model of Kernel::ComputePowerProduct_3
 
typedef unspecified_type Compute_squared_radius_smallest_orthogonal_sphere_3
 a model of Kernel::ComputeSquaredRadiusSmallestOrthogonalSphere_3
 

Three-dimensional Generalized Predicates

typedef unspecified_type Angle_3
 a model of Kernel::Angle_3
 
typedef unspecified_type Equal_3
 a model of Kernel::Equal_3
 
typedef unspecified_type Equal_x_3
 a model of Kernel::EqualX_3
 
typedef unspecified_type Equal_y_3
 a model of Kernel::EqualY_3
 
typedef unspecified_type Equal_z_3
 a model of Kernel::EqualZ_3
 
typedef unspecified_type Equal_xy_3
 a model of Kernel::EqualXY_3
 
typedef unspecified_type Less_x_3
 a model of Kernel::LessX_3
 
typedef unspecified_type Less_y_3
 a model of Kernel::LessY_3
 
typedef unspecified_type Less_z_3
 a model of Kernel::LessZ_3
 
typedef unspecified_type Less_xy_3
 a model of Kernel::LessXY_3
 
typedef unspecified_type Less_xyz_3
 a model of Kernel::LessXYZ_3
 
typedef unspecified_type Compare_weighted_squared_radius_3
 a model of Kernel::CompareWeightedSquaredRadius_3
 
typedef unspecified_type Compare_x_3
 a model of Kernel::CompareX_3
 
typedef unspecified_type Compare_y_3
 a model of Kernel::CompareY_3
 
typedef unspecified_type Compare_z_3
 a model of Kernel::CompareZ_3
 
typedef unspecified_type Compare_xy_3
 a model of Kernel::CompareXY_3
 
typedef unspecified_type Compare_xyz_3
 a model of Kernel::CompareXYZ_3
 
typedef unspecified_type Compare_slope_3
 a model of Kernel::CompareSlope_3
 
typedef unspecified_type Compare_squared_distance_3
 a model of Kernel::CompareSquaredDistance_3
 
typedef unspecified_type Less_signed_distance_to_plane_3
 a model of Kernel::LessSignedDistanceToPlane_3
 
typedef unspecified_type Less_distance_to_point_3
 a model of Kernel::LessDistanceToPoint_3
 
typedef unspecified_type Compare_dihedral_angle_3
 a model of Kernel::CompareDihedralAngle_3
 
typedef unspecified_type Compare_distance_3
 a model of Kernel::CompareDistance_3
 
typedef unspecified_type Compare_power_distance_3
 a model of Kernel::ComparePowerDistance_3
 
typedef unspecified_type Compare_power_distance_to_power_sphere_3
 a model of Kernel::ComparePowerDistanceToPowerSphere_3
 
typedef unspecified_type Collinear_3
 a model of Kernel::Collinear_3
 
typedef unspecified_type Coplanar_3
 a model of Kernel::Coplanar_3
 
typedef unspecified_type Orientation_3
 a model of Kernel::Orientation_3
 
typedef unspecified_type Coplanar_orientation_3
 a model of Kernel::CoplanarOrientation_3
 
typedef unspecified_type Coplanar_side_of_bounded_circle_3
 a model of Kernel::CoplanarSideOfBoundedCircle_3
 
typedef unspecified_type Side_of_oriented_sphere_3
 a model of Kernel::SideOfOrientedSphere_3
 
typedef unspecified_type Side_of_bounded_sphere_3
 a model of Kernel::SideOfBoundedSphere_3
 
typedef unspecified_type Power_side_of_oriented_power_sphere_3
 a model of Kernel::PowerSideOfOrientedPowerSphere_3
 
typedef unspecified_type Power_side_of_bounded_power_sphere_3
 a model of Kernel::PowerSideOfBoundedPowerSphere_3
 
typedef unspecified_type Is_degenerate_3
 a model of Kernel::IsDegenerate_3
 
typedef unspecified_type Has_on_3
 a model of Kernel::HasOn_3
 
typedef unspecified_type Has_on_bounded_side_3
 a model of Kernel::HasOnBoundedSide_3
 
typedef unspecified_type Has_on_unbounded_side_3
 a model of Kernel::HasOnUnboundedSide_3
 
typedef unspecified_type Has_on_boundary_3
 a model of Kernel::HasOnBoundary_3
 
typedef unspecified_type Has_on_positive_side_3
 a model of Kernel::HasOnPositiveSide_3
 
typedef unspecified_type Has_on_negative_side_3
 a model of Kernel::HasOnNegativeSide_3
 
typedef unspecified_type Oriented_side_3
 a model of Kernel::OrientedSide_3
 
typedef unspecified_type Bounded_side_3
 a model of Kernel::BoundedSide_3
 
typedef unspecified_type Are_parallel_3
 a model of Kernel::AreParallel_3
 
typedef unspecified_type Are_ordered_along_line_3
 a model of Kernel::AreOrderedAlongLine_3
 
typedef unspecified_type Are_strictly_ordered_along_line_3
 a model of Kernel::AreStrictlyOrderedAlongLine_3
 
typedef unspecified_type Collinear_are_ordered_along_line_3
 a model of Kernel::CollinearAreOrderedAlongLine_3
 
typedef unspecified_type Collinear_are_strictly_ordered_along_line_3
 a model of Kernel::CollinearAreStrictlyOrderedAlongLine_3
 
typedef unspecified_type Do_intersect_3
 a model of Kernel::DoIntersect_3
 

Operations

For each of the function objects above, there must exist a member function that requires no arguments and returns an instance of that function object.

The name of the member function is the uncapitalized name of the type returned with the suffix _object appended. For example, for the function object Kernel::Construct_vector_2 the following member function must exist:

Kernel::Construct_vector_2 construct_vector_2_object () const
 

Member Typedef Documentation

Member Data Documentation

const bool Kernel::Has_filtered_predicates
static

A Boolean value indicating whether the predicates are filtered (as in CGAL::Filtered_kernel).

This helps propagating such decisions to traits classes which are built on top of a kernel, so that they can decide to filter their own predicates or not.