Point-set triangulation explained

A triangulation of a set of points

l{P}

in the Euclidean space

Rd

is a simplicial complex that covers the convex hull of

l{P}

, and whose vertices belong to

l{P}

.[1] In the plane (when

l{P}

is a set of points in

R2

), triangulations are made up of triangles, together with their edges and vertices. Some authors require that all the points of

l{P}

are vertices of its triangulations. In this case, a triangulation of a set of points

l{P}

in the plane can alternatively be defined as a maximal set of non-crossing edges between points of

l{P}

. In the plane, triangulations are special cases of planar straight-line graphs.

A particularly interesting kind of triangulations are the Delaunay triangulations. They are the geometric duals of Voronoi diagrams. The Delaunay triangulation of a set of points

l{P}

in the plane contains the Gabriel graph, the nearest neighbor graph and the minimal spanning tree of

l{P}

.

Triangulations have a number of applications, and there is an interest to find the "good" triangulations of a given point set under some criteria as, for instance minimum-weight triangulations. Sometimes it is desirable to have a triangulation with special properties, e.g., in which all triangles have large angles (long and narrow ("splinter") triangles are avoided).[2]

Given a set of edges that connect points of the plane, the problem to determine whether they contain a triangulation is NP-complete.

Regular triangulations

Some triangulations of a set of points

l{P}\subsetRd

can be obtained by lifting the points of

l{P}

into

Rd+1

(which amounts to add a coordinate

xd+1

to each point of

l{P}

), by computing the convex hull of the lifted set of points, and by projecting the lower faces of this convex hull back on

Rd

. The triangulations built this way are referred to as the regular triangulations of

l{P}

. When the points are lifted to the paraboloid of equation

xd+1=

2
x
d
, this construction results in the Delaunay triangulation of

l{P}

. Note that, in order for this construction to provide a triangulation, the lower convex hull of the lifted set of points needs to be simplicial. In the case of Delaunay triangulations, this amounts to require that no

d+2

points of

l{P}

lie in the same sphere.

Combinatorics in the plane

Every triangulation of any set

l{P}

of

n

points in the plane has

2n-h-2

triangles and

3n-h-3

edges where

h

is the number of points of

l{P}

in the boundary of the convex hull of

l{P}

. This follows from a straightforward Euler characteristic argument.[3]

Algorithms to build triangulations in the plane

Triangle Splitting Algorithm : Find the convex hull of the point set

l{P}

and triangulate this hull as a polygon. Choose an interior point and draw edges to the three vertices of the triangle that contains it. Continue this process until all interior points are exhausted.[4]

Incremental Algorithm : Sort the points of

l{P}

according to x-coordinates. The first three points determine a triangle. Consider the next point

p

in the ordered set and connect it with all previously considered points

\{p1,...,pk\}

which are visible to p. Continue this process of adding one point of

l{P}

at a time until all of

l{P}

has been processed.[5]

Time complexity of various algorithms

The following table reports time complexity results for the construction of triangulations of point sets in the plane, under different optimality criteria, where

n

is the number of points.
minimizemaximize
minimumangle

O(nlogn)


(Delaunay triangulation)
maximum

O(n2logn)

minimumarea

O(n2)

O(n2logn)

maximum

O(n2logn)

maximumdegreeNP-complete
for degree of 7
maximumeccentricity

O(n3)

minimumedge length

O(nlogn)


(Closest pair of points problem)
maximum

O(n2)

O(nlogn)


(using the Convex hull)
sum ofNP-hard
(Minimum-weight triangulation)
minimumheight

O(n2logn)

maximumslope

O(n3)

See also

References

Notes and References

  1. Book: De Loera . Jesús A. . Jesús A. De Loera . Rambau . Jörg . Santos . Francisco . Francisco Santos Leal . 2010 . Triangulations, Structures for Algorithms and Applications . Algorithms and Computation in Mathematics . 25 . Springer.
  2. Book: de Berg , Mark . Mark de Berg

    . Mark de Berg . Otfried Cheong . Marc van Kreveld . Mark Overmars . Mark Overmars . Computational Geometry: Algorithms and Applications . Springer-Verlag . 2008 . 978-3-540-77973-5 . Otfried Cheong .

  3. .
  4. Devadoss, O'Rourke Discrete and Computational Geometry. Princeton University Press, 2011, p. 60.
  5. Devadoss, O'Rourke Discrete and Computational Geometry. Princeton University Press, 2011, p. 62.

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