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CGAL 4.3 - 2D Envelopes

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EnvelopeDiagram_1 Concept Reference
Concepts

## Definition

This concept defines the representation of an envelope diagram of a set of planar curve. The envelope diagram is a subdivision of the \( x\)-axis into 0-dimensional cells (vertices) and 1-dimensional cells (edges), such that the identity of the curves that induce the lower envelope (or the upper envelope) over each cell is fixed.

A vertex in an envelope diagram is therefore associated with a point on the envelope, and corresponds to either a curve endpoint or to an intersection point of two (or more) curves. Therefore each vertex is associated with a set of \( x\)-monotone curves that induce the envelope over this point. Each vertex is incident to two edges, one lying to its left and the other to its right.

An edge in the envelope diagram represents a continuous portion of the \( x\)-axis, and is associated with a (possibly empty) set of curves that induce the envelope over this portion of the \( x\)-axis. An edge may be bounded by two vertices, one to its left and the other to its right. However, the diagram contains two unbounded edges, its leftmost edge, representing the interval \( (-\infty, x_l)\), and its rightmost edge, representing the interval \( (x_r, \infty)\), where \( x_l\) and \( x_r\) are the \( x\)-coodinates of the leftmost and the rightmost vertices in the diagram, respectively. Note that a diagram may contain no vertices at all, in which case it comprises a single edge.

Note that any model of the `EnvelopeDiagram_1` concept must define a geometric traits class, which in turn defines the `Point_2` and `X_monotone_curve_2` types defined with the diagram features.

See Also
  `EnvelopeDiagramVertex`
  `EnvelopeDiagramEdge`

## Types
typedef unspecified_type | Traits_2
| the geometric traits class.
typedef Traits_2::Point_2 | Point_2
| the point type.
typedef Traits_2::X_monotone_curve_2 | X_monotone_curve_2
| the \( x\)-monotone curve type.
typedef unspecified_type | Size
| the size type (convertible to `size_t`).
typedef unspecified_type | Curve_const_iterator
| an iterator for the \( x\)-monotone curves that induce a diagram feature, with value type `X_monotone_curve_2`.
typedef unspecified_type | Vertex
| the vertex type, a model of the concept `EnvelopeDiagramVertex`.
typedef unspecified_type | Edge
| the edge type, a model of the concept `EnvelopeDiagramEdge`.
typedef unspecified_type | Vertex_handle
| a handle to a diagram vertex.
typedef unspecified_type | Vertex_const_handle
| a non-mutable handle to a diagram vertex.
typedef unspecified_type | Edge_handle
| a handle to a diagram edge.
typedef unspecified_type | Edge_const_handle
| a non-mutable handle to a diagram edge.

## Creation
| EnvelopeDiagram_1 ()
| constructs an empty diagram containing one unbounded edge, which corresponds to the entire plane and has no \( x\)-monotone curves that are associated with it.
| Envelope_diagram_1 (const Self &other)
| copy constructor.

## Access Functions
Edge_const_handle | leftmost () const
| returns the leftmost edge of the diagram (a non-const version is also available).
Edge_const_handle | rightmost () const
| returns the rightmost edge of the diagram (a non-const version is also available).

## Modifiers
void | set_leftmost (Edge_const_handle e)
| sets the leftmost edge of the diagram to be `e`.
void | set_rightmost (Edge_const_handle e)
| sets the rightmost edge of the diagram to be `e`.
Vertex_handle | new_vertex (const Point_2 &p)
| creates a new diagram vertex, associated with the point `p`.
Edge_handle | new_edge ()
| creates a new diagram edge.
void | delete_vertex (Vertex_handle v)
| deletes the given vertex `v`.
void | delete_edge (Edge_handle e)
| deletes the given edge `e`.