Version 5.0 The LEDA User Manual

15.2 Point Sets and Delaunay Triangulations (
POINT SET )
1. Definition
There are three instantiations of POINT SET : point set (floating point kernel),
rat point set (rational kernel) and real point set (real kernel). The respective header file
name corresponds to the type name (with “.h” appended).
An instance T of data type POINT SET is a planar embedded bidirected graph (map)
representing the Delaunay Triangulation of its vertex set. The position of a vertex v is
given by T.pos(v) and we use S = {T.pos(v) | v ∈ T } to denote the underlying point set.
Each face of T (except for the outer face) is a triangle whose circumscribing circle does
not contain any point of S in its interior. For every edge e, the sequence
e, T.face cycle succ(e), T.face cycle succ(T.face cycle succ(e)), . . .
traces the boundary of the face to the left of e. The edges of the outer face of T form the
convex hull of S; the trace of the convex hull is clockwise. The subgraph obtained from
T by removing all diagonals of co-circular quadrilaterals is called the Delaunay Diagram
of S.
POINT SET provides all constant graph operations, e.g., T.reversal(e) returns the reversal
of edge e, T.all edges( ) returns the list of all edges of T , and forall edges(e, T )
iterates over all edges of T . In addition, POINT SET provides operations for inserting
and deleting points, point location, nearest neighbor searches, and navigation in both the
triangulation and the diagram.
POINT SET s are essentially objects of type GRAPH , where the node information
is the position of the node and the edge information is irrelevant. For a graph G
of type GRAPH the function Is Delaunay(G) tests whether G is a Delaunay
triangulation of its vertices.
The data type POINT SET is illustrated by the point set demo in the LEDA demo directory.
Be aware that the nearest neighbor queries for a point (not for a node) and the range
search queries for circles, triangles, and rectangles are non-const operations and modify
the underlying graph. The set of nodes and edges is not changed; however, it is not
guaranteed that the underlying Delaunay triangulation is unchanged.
#include < LEDA/geo/generic/POINT SET.h >
2. Creation
POINT SET T ; creates an empty POINT SET T .