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.

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.