 CGAL 5.3 - 2D Envelopes
EnvelopeDiagram_1 Concept Reference

## 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.

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.