\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 3D Envelopes

All Classes Namespaces Files Functions Variables Typedefs Enumerations Enumerator Groups Pages
List of all members
EnvelopeTraits_3 Concept Reference
Concepts

## Definition

This concept defines the minimal set of geometric predicates and operations needed to compute the envelope of a set of arbitrary surfaces in \( \mathbb{R}^3\). It refines the `ArrangementXMonotoneTraits_2` concept. In addition to the `Point_2` and `X_monotone_curve_2` types and the `Has_boundary_category` category tag listed in the base concept, it also lists the `Surface_3` and `Xy_monotone_surface_3` types, which represent arbitrary surfaces and \( xy\)-monotone surfaces, respectively, and some constructions and predicates on these types. Note however, that these operations usually involve the projection of 3D objects onto the \( xy\)-plane.

Refines:
  `ArrangementXMonotoneTraits_2`

Has Models:

  `CGAL::Env_triangle_traits_3<Kernel>`

  `CGAL::Env_sphere_traits_3<ConicTraits>`

  `CGAL::Env_plane_traits_3<Kernel>`

  `CGAL::Env_surface_data_traits_3<Traits,XyData,SData,Cnv>`

## Types
typedef unspecified_type | Surface_3
| represents an arbitrary surface in \( \mathbb{R}^3\).
typedef unspecified_type | Xy_monotone_surface_3
| represents a weakly \( xy\)-monotone surface in \( \mathbb{R}^3\).

## Functor Types
typedef unspecified_type | Make_xy_monotone_3
| provides the operator (templated by the `OutputIterator` type) : More...
typedef unspecified_type | Construct_projected_boundary_2
| provides the operator (templated by the `OutputIterator` type) : More...
typedef unspecified_type | Construct_projected_intersections_2
| provides the operator (templated by the `OutputIterator` type) : More...
typedef unspecified_type | Compare_z_at_xy_3
| provides the operators : More...
typedef unspecified_type | Compare_z_at_xy_above_3
| provides the operator : More...
typedef unspecified_type | Compare_z_at_xy_below_3
| provides the operator : More...

## Creation
| EnvelopeTraits_3 ()
| default constructor.
| EnvelopeTraits_3 (EnvelopeTraits_3 other)
| copy constructor.
EnvelopeTraits_3 | operator= (other)
| assignment operator.

## Accessing Functor Objects
Make_xy_monotone_3 | make_xy_monotone_3_object ()
Construct_projected_boundary_2 | construct_projected_boundary_2_object ()
Construct_projected_intersections_2 | construct_projected_intersections_2_object ()
Compare_z_at_xy_3 | compare_z_at_xy_3_object ()
Compare_z_at_xy_above_3 | compare_z_at_xy_above_3_object ()
Compare_z_at_xy_below_3 | compare_z_at_xy_below_3_object ()

## Member Typedef Documentation

typedef unspecified_type EnvelopeTraits_3::Compare_z_at_xy_3

provides the operators :

- `Comparison_result operator() (Point_2 p, Xy_monotone_surface_3 s1, Xy_monotone_surface_3 s2)`
  which determines the relative \( z\)-order of the two given \( xy\)-monotone surfaces at the \( xy\)-coordinates of the point `p`, with the precondition that both surfaces are defined over `p`. Namely, it returns the comparison result of \( s_1(p)\) and \( s_2(p)\).
- `Comparison_result operator() (X_monotone_curve_2 c, Xy_monotone_surface_3 s1, Xy_monotone_surface_3 s2)`
  which determines the relative \( z\)-order of the two given \( xy\)-monotone surfaces over the interior of a given \( x\)-monotone curve \( c\), with the precondition that \( c\) is fully contained in the \( xy\)-definition range of both \( s_1\) and \( s_2\), and that the surfaces do not intersect over \( c\). The functor should therefore return the comparison result of \( s_1(p')\) and \( s_2(p')\) for some point \( p'\) in the interior of \( c\).
- `Comparison_result operator() (Xy_monotone_surface_3 s1, Xy_monotone_surface_3 s2)`
  which determines the relative \( z\)-order of the two given unbounded \( xy\)-monotone surfaces, which are defined over the entire \( xy\)-plane and have no boundary, with the precondition that the surfaces do not intersect at all. The functor should therefore return the comparison result of \( s_1(p)\) and \( s_2(p)\) for some planar point \( p \in\mathbb{R}^2\). This operator is required iff the category tag `Has_boundary_category` is defined as `Tag_true`.

typedef unspecified_type EnvelopeTraits_3::Compare_z_at_xy_above_3

provides the operator :

- `Comparison_result operator() (X_monotone_curve_2 c, Xy_monotone_surface_3 s1, Xy_monotone_surface_3 s2)`
  which determines the relative \( z\)-order of the two given \( xy\)-monotone surfaces immediately above their projected intersection curve \( c\) (a planar point \( p\) is above an \( x\)-monotone curve \( c\) if it is in the \( x\)-range of \( c\), and lies to its left when the curve is traversed from its \( xy\)-lexicographically smaller endpoint to its larger endpoint). We have the precondition that both surfaces are defined "above" \( c\), and their relative \( z\)-order is the same for some small enough neighborhood of points above \( c\).

typedef unspecified_type EnvelopeTraits_3::Compare_z_at_xy_below_3

provides the operator :

- `Comparison_result operator() (X_monotone_curve_2 c, Xy_monotone_surface_3 s1, Xy_monotone_surface_3 s2)`
  which determines the relative \( z\)-order of the two given \( xy\)-monotone surfaces immediately below their projected intersection curve \( c\) (a planar point \( p\) is below an \( x\)-monotone curve \( c\) if it is in the \( x\)-range of \( c\), and lies to its right when the curve is traversed from its \( xy\)-lexicographically smaller endpoint to its larger endpoint). We have the precondition that both surfaces are defined "below" \( c\), and their relative \( z\)-order is the same for some small enough neighborhood of points below \( c\).

typedef unspecified_type EnvelopeTraits_3::Construct_projected_boundary_2

provides the operator (templated by the `OutputIterator` type) :

- `OutputIterator operator() (Xy_monotone_surface_3 s, OutputIterator oi)`
  which computes all planar \( x\)-monotone curves and possibly isolated planar points that form the projection of the boundary of the given \( xy\)-monotone surface \( s\) onto the \( xy\)-plane, and inserts them into the output iterator. The value-type of `OutputIterator` is `Object`, where `Object` wraps either a `Point_2`, or a `pair<X_monotone_curve_2, Oriented_side>`. In the former case, the object represents an isolated point of the projected boundary. In the latter, more general, case the object represents an \( x\)-monotone boundary curve along with an enumeration value which is either `ON_NEGATIVE_SIDE` or `ON_POSITIVE_SIDE`, indicating whether whether the projection of the surface onto the \( xy\)-plane lies below or above this \( x\)-monotone curve, respectively. In degenerate case, namely when the surface itself is vertical, and its projection onto the plane is \( 1\)-dimensional, the `Oriented_side` value is `ON_ORIENTED_BOUNDARY`. The operator returns a past-the-end iterator for the output sequence.

typedef unspecified_type EnvelopeTraits_3::Construct_projected_intersections_2

provides the operator (templated by the `OutputIterator` type) :

- `OutputIterator operator() (Xy_monotone_surface_3 s1, Xy_monotone_surface_3 s2, OutputIterator oi)`
  which computes the projection of the intersections of the \( xy\)-monotone surfaces `s1` and `s2` onto the \( xy\)-plane, and inserts them into the output iterator. The value-type of `OutputIterator` is `Object`, where each `Object` either wraps a `pair<X_monotone_curve_2,Multiplicity>` instance, which represents a projected intersection curve with its multiplicity (in case the multiplicity is undefined or not known, it should be set to \( 0\)) or an `Point_2` instance, representing the projected image of a degenerate intersection (the projection of an isolated intersection point, or of a vertical intersection curve). The operator returns a past-the-end iterator for the output sequence.

typedef unspecified_type EnvelopeTraits_3::Make_xy_monotone_3

provides the operator (templated by the `OutputIterator` type) :

- `OutputIterator operator() (Surface_3 S, bool is_lower, OutputIterator oi)`
  which subdivides the given surface `S` into \( xy\)-monotone parts and inserts them into the output iterator. The value of `is_lower` indicates whether we compute the lower or the upper envelope, so that \( xy\)-monotone surfaces that are irrelevant to the lower-envelope (resp. upper-envelope) computation may be discarded. The value-type of `OutputIterator` is `Xy_monotone_surface_3`. The operator returns a past-the-end iterator for the output sequence.