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	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_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_max_vertex_2
	a model of Kernel::ConstructMaxVertex_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_min_vertex_2
	a model of Kernel::ConstructMinVertex_2
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_bisector_2
	a model of Kernel::ConstructBisector_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_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_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_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_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	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	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_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_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_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_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_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

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_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_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	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	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

typedef unspecified_type Kernel::Construct_second_point_2

a model of Kernel::ConstructSecondPoint_2

typedef unspecified_type Kernel::Construct_second_point_3

a model of Kernel::ConstructSecondPoint_3

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.