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## Kernel

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 in the basic library 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.

### Has Models

CGAL::Cartesian<FieldNumberType>
CGAL::Homogeneous<RingNumberType>
CGAL::Simple_cartesian<FieldNumberType>
CGAL::Simple_homogeneous<RingNumberType>

### Types

Kernel::FT
| a model of FieldNumberType
Kernel::RT
| a model of RingNumberType

### Two-dimensional Kernel

### Coordinate Access

Kernel::Cartesian_const_iterator_2
| a model of Kernel::CartesianConstIterator_2

### Geometric Objects

Kernel::Point_2
| a model of Kernel::Point_2
Kernel::Vector_2
| a model of Kernel::Vector_2
Kernel::Direction_2
| a model of Kernel::Direction_2
Kernel::Line_2
| a model of Kernel::Line_2
Kernel::Ray_2
| a model of Kernel::Ray_2
Kernel::Segment_2
| a model of Kernel::Segment_2
Kernel::Triangle_2
| a model of Kernel::Triangle_2
Kernel::Iso_rectangle_2
| a model of Kernel::IsoRectangle_2
Kernel::Circle_2
| a model of Kernel::Circle_2
Kernel::Object_2
| a model of Kernel::Object_2

### Constructions

Kernel::Construct_point_2
| a model of Kernel::ConstructPoint_2
Kernel::Construct_vector_2
| a model of Kernel::ConstructVector_2
Kernel::Construct_direction_2
| a model of Kernel::ConstructDirection_2
Kernel::Construct_segment_2
| a model of Kernel::ConstructSegment_2
Kernel::Construct_line_2
| a model of Kernel::ConstructLine_2
Kernel::Construct_ray_2
| a model of Kernel::ConstructRay_2
Kernel::Construct_circle_2
| a model of Kernel::ConstructCircle_2
Kernel::Construct_triangle_2
| a model of Kernel::ConstructTriangle_2
Kernel::Construct_iso_rectangle_2
| a model of Kernel::ConstructIsoRectangle_2
Kernel::Construct_object_2
| a model of Kernel::ConstructObject_2
Kernel::Construct_scaled_vector_2
| a model of Kernel::ConstructScaledVector_2
Kernel::Construct_translated_point_2
| a model of Kernel::ConstructTranslatedPoint_2
Kernel::Construct_point_on_2
| a model of Kernel::ConstructPointOn_2
Kernel::Construct_projected_point_2
| a model of Kernel::ConstructProjectedPoint_2
Kernel::Construct_projected_xy_point_2
| a model of Kernel::ConstructProjectedXYPoint_2
Kernel::Construct_cartesian_const_iterator_2
| a model of Kernel::ConstructCartesianConstIterator_2
Kernel::Construct_vertex_2
| a model of Kernel::ConstructVertex_2
Kernel::Construct_bbox_2
| a model of Kernel::ConstructBbox_2
Kernel::Construct_perpendicular_vector_2
| a model of Kernel::ConstructPerpendicularVector_2
Kernel::Construct_perpendicular_direction_2
| a model of Kernel::ConstructPerpendicularDirection_2
Kernel::Construct_perpendicular_line_2
| a model of Kernel::ConstructPerpendicularLine_2
Kernel::Construct_max_vertex_2
| a model of Kernel::ConstructMaxVertex_2
Kernel::Construct_midpoint_2
| a model of Kernel::ConstructMidpoint_2
Kernel::Construct_min_vertex_2
| a model of Kernel::ConstructMinVertex_2
Kernel::Construct_center_2
| a model of Kernel::ConstructCenter_2
Kernel::Construct_centroid_2
| a model of Kernel::ConstructCentroid_2
Kernel::Construct_circumcenter_2
| a model of Kernel::ConstructCircumcenter_2
Kernel::Construct_bisector_2
| a model of Kernel::ConstructBisector_2
Kernel::Construct_opposite_direction_2
| a model of Kernel::ConstructOppositeDirection_2
Kernel::Construct_opposite_segment_2
| a model of Kernel::ConstructOppositeSegment_2
Kernel::Construct_opposite_ray_2
| a model of Kernel::ConstructOppositeRay_2
Kernel::Construct_opposite_line_2
| a model of Kernel::ConstructOppositeLine_2
Kernel::Construct_opposite_triangle_2
| a model of Kernel::ConstructOppositeTriangle_2
Kernel::Construct_opposite_circle_2
| a model of Kernel::ConstructOppositeCircle_2
Kernel::Construct_opposite_vector_2
| a model of Kernel::ConstructOppositeVector_2

If the result type is not determined, there is no Construct_ prefix:

Kernel::Intersect_2
| a model of Kernel::Intersect_2
Kernel::Assign_2
| a model of Kernel::Assign_2

If the result type is a number type, the prefix is Compute_:

Kernel::Compute_squared_distance_2
| a model of Kernel::ComputeSquaredDistance_2
Kernel::Compute_squared_length_2
| a model of Kernel::ComputeSquaredLength_2
Kernel::Compute_squared_radius_2
| a model of Kernel::ComputeSquaredRadius_2
Kernel::Compute_area_2
| a model of Kernel::ComputeArea_2

### Generalized Predicates

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

### Three-dimensional Kernel

### Coordinate Access

Kernel::Cartesian_const_iterator_3
| a model of Kernel::CartesianConstIterator_3

### Geometric Objects

Kernel::Point_3
| a model of Kernel::Point_3
Kernel::Vector_3
| a model of Kernel::Vector_3
Kernel::Direction_3
| a model of Kernel::Direction_3
Kernel::Iso_cuboid_3
| a model of Kernel::IsoCuboid_3
Kernel::Line_3
| a model of Kernel::Line_3
Kernel::Ray_3
| a model of Kernel::Ray_3
Kernel::Sphere_3
| a model of Kernel::Sphere_3
Kernel::Segment_3
| a model of Kernel::Segment_3
Kernel::Plane_3
| a model of Kernel::Plane_3
Kernel::Triangle_3
| a model of Kernel::Triangle_3
Kernel::Tetrahedron_3
| a model of Kernel::Tetrahedron_3
Kernel::Object_3
| a model of Kernel::Object_3

### Constructions

Kernel::Construct_point_3
| a model of Kernel::ConstructPoint_3
Kernel::Construct_vector_3
| a model of Kernel::ConstructVector_3
Kernel::Construct_direction_3
| a model of Kernel::ConstructDirection_3
Kernel::Construct_plane_3
| a model of Kernel::ConstructPlane_3
Kernel::Construct_iso_cuboid_3
| a model of Kernel::ConstructIsoCuboid_3
Kernel::Construct_line_3
| a model of Kernel::ConstructLine_3
Kernel::Construct_ray_3
| a model of Kernel::ConstructRay_3
Kernel::Construct_sphere_3
| a model of Kernel::ConstructSphere_3
Kernel::Construct_segment_3
| a model of Kernel::ConstructSegment_3
Kernel::Construct_triangle_3
| a model of Kernel::ConstructTriangle_3
Kernel::Construct_tetrahedron_3
| a model of Kernel::ConstructTetrahedron_3
Kernel::Construct_object_3
| a model of Kernel::ConstructObject_3
Kernel::Construct_scaled_vector_3
| a model of Kernel::ConstructScaledVector_3
Kernel::Construct_translated_point_3
| a model of Kernel::ConstructTranslatedPoint_3
Kernel::Construct_point_on_3
| a model of Kernel::ConstructPointOn_3
Kernel::Construct_projected_point_3
| a model of Kernel::ConstructProjectedPoint_3
Kernel::Construct_lifted_point_3
| a model of Kernel::ConstructLiftedPoint_3
Kernel::Construct_cartesian_const_iterator_3
| a model of Kernel::ConstructCartesianConstIterator_3
Kernel::Construct_vertex_3
| a model of Kernel::ConstructVertex_3
Kernel::Construct_bbox_3
| a model of Kernel::ConstructBbox_3
Kernel::Construct_supporting_plane_3
| a model of Kernel::ConstructSupportingPlane_3
Kernel::Construct_orthogonal_vector_3
| a model of Kernel::ConstructOrthogonalVector_3
Kernel::Construct_base_vector_3
| a model of Kernel::ConstructBaseVector_3
Kernel::Construct_perpendicular_plane_3
| a model of Kernel::ConstructPerpendicularPlane_3
Kernel::Construct_perpendicular_line_3
| a model of Kernel::ConstructPerpendicularLine_3
Kernel::Construct_midpoint_3
| a model of Kernel::ConstructMidpoint_3
Kernel::Construct_center_3
| a model of Kernel::ConstructCenter_3
Kernel::Construct_centroid_3
| a model of Kernel::ConstructCentroid_3
Kernel::Construct_circumcenter_3
| a model of Kernel::ConstructCircumcenter_3
Kernel::Construct_bisector_3
| a model of Kernel::ConstructBisector_3
Kernel::Construct_cross_product_vector_3
| a model of Kernel::ConstructCrossProductVector_3
Kernel::Construct_opposite_direction_3
| a model of Kernel::ConstructOppositeDirection_3
Kernel::Construct_opposite_segment_3
| a model of Kernel::ConstructOppositeSegment_3
Kernel::Construct_opposite_ray_3
| a model of Kernel::ConstructOppositeRay_3
Kernel::Construct_opposite_line_3
| a model of Kernel::ConstructOppositeLine_3
Kernel::Construct_opposite_plane_3
| a model of Kernel::ConstructOppositePlane_3
Kernel::Construct_opposite_sphere_3
| a model of Kernel::ConstructOppositeSphere_3
Kernel::Construct_opposite_vector_3
| a model of Kernel::ConstructOppositeVector_3

If the result type is not determined, there is no Construct_ prefix:

Kernel::Intersect_3
| a model of Kernel::Intersect_3
Kernel::Assign_3
| a model of Kernel::Assign_3

If the result type is a number type, the prefix is Compute_:

Kernel::Compute_area_3
| a model of Kernel::ComputeArea_3
Kernel::Compute_squared_area_3
| a model of Kernel::ComputeSquaredArea_3
Kernel::Compute_squared_distance_3
| a model of Kernel::ComputeSquaredDistance_3
Kernel::Compute_squared_length_3
| a model of Kernel::ComputeSquaredLength_3
Kernel::Compute_squared_radius_3
| a model of Kernel::ComputeSquaredRadius_3
Kernel::Compute_volume_3
| a model of Kernel::ComputeVolume_3

### Generalized Predicates

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

### d-dimensional Kernel

### Coordinate Access

Kernel::Cartesian_const_iterator_d
| a model of Kernel::CartesianConstIterator_d

### Geometric Objects

Kernel::Point_d
| a model of Kernel::Point_d

### Constructions

Kernel::Construct_Point_d
| a model of Kernel::ConstructPoint_d

### 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
| kernel.construct_vector_2_object ()

### See Also

CGAL d-dimensional Kernel manual.

Next: EuclideanRingNumberType

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The CGAL Project . Tue, December 21, 2004 .