Point-set triangulation

File:PointSetTriangulations.svg

A triangulation of a set of points $\mathcal{P}$ in the Euclidean space $\mathbb{R}^d$ is a simplicial complex that covers the convex hull of $\mathcal{P}$ , and whose vertices belong to $\mathcal{P}$ .{{cite book

| last1 = De Loera | first1 = Jesús A. | author-link1 = Jesús A. De Loera

| last2 = Rambau | first2 = Jörg

| last3 = Santos | first3 = Francisco | author-link3 = Francisco Santos Leal

| year = 2010

| title = Triangulations, Structures for Algorithms and Applications

| series = Algorithms and Computation in Mathematics

| volume = 25

| publisher = Springer}} In the plane (when $\mathcal{P}$ is a set of points in $\mathbb{R}^2$ ), triangulations are made up of triangles, together with their edges and vertices. Some authors require that all the points of $\mathcal{P}$ are vertices of its triangulations.{{sfn |de Berg|van Kreveld|Overmars|Schwarzkopf|2008|loc=Section 9.1}} In this case, a triangulation of a set of points $\mathcal{P}$ in the plane can alternatively be defined as a maximal set of non-crossing edges between points of $\mathcal{P}$ . 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 $\mathcal{P}$ in the plane contains the Gabriel graph, the nearest neighbor graph and the minimal spanning tree of $\mathcal{P}$ .

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).{{cite book

| last = de Berg

| first = Mark

| authorlink = Mark de Berg

|author2=Otfried Cheong |author3=Marc van Kreveld | author4-link = Mark Overmars

|author4=Mark Overmars

| title = Computational Geometry: Algorithms and Applications

| publisher = Springer-Verlag

| year = 2008

| id =

| url = http://www.cs.uu.nl/geobook/interpolation.pdf

| isbn = 978-3-540-77973-5 | author2-link = Otfried Cheong

}}

Given a set of edges that connect points of the plane, the problem to determine whether they contain a triangulation is NP-complete.{{sfn |Lloyd|1977}}

Regular triangulations

Some triangulations of a set of points $\mathcal{P}\subset\mathbb{R}^d$ can be obtained by lifting the points of $\mathcal{P}$ into $\mathbb{R}^{d+1}$ (which amounts to add a coordinate $x_{d+1}$ to each point of $\mathcal{P}$ ), by computing the convex hull of the lifted set of points, and by projecting the lower faces of this convex hull back on $\mathbb{R}^d$ . The triangulations built this way are referred to as the regular triangulations of $\mathcal{P}$ . When the points are lifted to the paraboloid of equation $x_{d+1} = x_1^2+\cdots+x_d^2$ , this construction results in the Delaunay triangulation of $\mathcal{P}$ . 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 $\mathcal{P}$ lie in the same sphere.

Combinatorics in the plane

Every triangulation of any set $\mathcal{P}$ of $n$ points in the plane has $2n - h - 2$ triangles and $3n - h - 3$ edges where $h$ is the number of points of $\mathcal{P}$ in the boundary of the convex hull of $\mathcal{P}$ . This follows from a straightforward Euler characteristic argument.{{citation

| last1 = Edelsbrunner | first1 = Herbert | author1-link = Herbert Edelsbrunner

| last2 = Tan | first2 = Tiow Seng

| last3 = Waupotitsch | first3 = Roman

| doi = 10.1137/0913058

| issue = 4

| journal = SIAM Journal on Scientific and Statistical Computing

| mr = 1166172

| pages = 994–1008

| title = An O(n² log n) time algorithm for the minmax angle triangulation

| volume = 13

| year = 1992| citeseerx = 10.1.1.66.2895 }}.

Algorithms to build triangulations in the plane

Triangle Splitting Algorithm : Find the convex hull of the point set $\mathcal{P}$ 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.Devadoss, O'Rourke Discrete and Computational Geometry. Princeton University Press, 2011, p. 60.

Incremental Algorithm : Sort the points of $\mathcal{P}$ 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 $\{p_1,..., p_k\}$ which are visible to p. Continue this process of adding one point of $\mathcal{P}$ at a time until all of $\mathcal{P}$ has been processed.Devadoss, O'Rourke Discrete and Computational Geometry. Princeton University Press, 2011, p. 62.

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.

class="wikitable" style="border:none;text-align:center;"
style="background:white;border:none;" colspan="2" \| ! style="padding:1em;" \| minimize ! style="padding:1em;" \| maximize
style="padding:1em;" \| minimum ! rowspan="2" style="padding:1em;" \| angle \| bgcolor="darkgray" \| \| $O(n \log n)$ (Delaunay triangulation)
style="padding:1em;" \| maximum \| $O(n^2 \log n)$ {{sfn \|Edelsbrunner\|Tan\|Waupotitsch\|1990}} {{sfn \|Bern\|Edelsbrunner\|Eppstein\|Mitchell\|1993}} \| bgcolor="darkgray" \|
style="padding:1em;" \| minimum ! rowspan="2" style="padding:1em;" \| area \| $O(n^2)$ {{sfn \|Chazelle\|Guibas\|Lee\|1985}} \| $O(n^2 \log n)$ {{sfn \|Vassilev\|2005}}
style="padding:1em;" \| maximum \| $O(n^2 \log n)$ {{sfn \|Vassilev\|2005}} \| bgcolor="darkgray" \|
style="padding:1em;" \| maximum ! rowspan="1" style="padding:1em;" \| degree \| NP-complete for degree of 7 {{sfn \|Jansen\|1992}} \| bgcolor="darkgray" \|
style="padding:1em;" \| maximum ! rowspan="1" style="padding:1em;" \| eccentricity \| $O(n^3)$ {{sfn \|Bern\|Edelsbrunner\|Eppstein\|Mitchell\|1993}} \| bgcolor="darkgray" \|
style="padding:1em;" \| minimum ! rowspan="3" style="padding:1em;" \| edge length \| $O(n \log n)$ (Closest pair of points problem) \| NP-complete {{sfn \|Fekete\|2012}}
style="padding:1em;" \| maximum \| $O(n^2)$ {{sfn \|Edelsbrunner\|Tan\|1991}} \| $O(n \log n)$ (using the Convex hull)
style="padding:1em;" \| sum of \| NP-hard (Minimum-weight triangulation) \| bgcolor="darkgray" \|
style="padding:1em;" \| minimum ! rowspan="1" style="padding:1em;" \| height \| bgcolor="darkgray" \| \| $O(n^2 \log n)$ {{sfn \|Bern\|Edelsbrunner\|Eppstein\|Mitchell\|1993}}
style="padding:1em;" \| maximum ! rowspan="1" style="padding:1em;" \| slope \| $O(n^3)$ {{sfn \|Bern\|Edelsbrunner\|Eppstein\|Mitchell\|1993}} \| bgcolor="darkgray" \|

class="wikitable" style="border:none;text-align:center;"

style="background:white;border:none;" colspan="2" |

! style="padding:1em;" | minimize

! style="padding:1em;" | maximize

style="padding:1em;" | minimum

! rowspan="2" style="padding:1em;" | angle

| bgcolor="darkgray" |

| $O(n \log n)$
(Delaunay triangulation)

style="padding:1em;" | maximum

| $O(n^2 \log n)$ {{sfn |Edelsbrunner|Tan|Waupotitsch|1990}} {{sfn |Bern|Edelsbrunner|Eppstein|Mitchell|1993}}

| bgcolor="darkgray" |

style="padding:1em;" | minimum

! rowspan="2" style="padding:1em;" | area

| $O(n^2)$ {{sfn |Chazelle|Guibas|Lee|1985}}

| $O(n^2 \log n)$ {{sfn |Vassilev|2005}}

style="padding:1em;" | maximum

| $O(n^2 \log n)$ {{sfn |Vassilev|2005}}

| bgcolor="darkgray" |

style="padding:1em;" | maximum

! rowspan="1" style="padding:1em;" | degree

| NP-complete
for degree of 7 {{sfn |Jansen|1992}}

| bgcolor="darkgray" |

style="padding:1em;" | maximum

! rowspan="1" style="padding:1em;" | eccentricity

| $O(n^3)$ {{sfn |Bern|Edelsbrunner|Eppstein|Mitchell|1993}}

| bgcolor="darkgray" |

style="padding:1em;" | minimum

! rowspan="3" style="padding:1em;" | edge length

| $O(n \log n)$
(Closest pair of points problem)

| NP-complete {{sfn |Fekete|2012}}

style="padding:1em;" | maximum

| $O(n^2)$ {{sfn |Edelsbrunner|Tan|1991}}

| $O(n \log n)$
(using the Convex hull)

style="padding:1em;" | sum of

| NP-hard
(Minimum-weight triangulation)

| bgcolor="darkgray" |

style="padding:1em;" | minimum

! rowspan="1" style="padding:1em;" | height

| bgcolor="darkgray" |

| $O(n^2 \log n)$ {{sfn |Bern|Edelsbrunner|Eppstein|Mitchell|1993}}

style="padding:1em;" | maximum

! rowspan="1" style="padding:1em;" | slope

| $O(n^3)$ {{sfn |Bern|Edelsbrunner|Eppstein|Mitchell|1993}}

| bgcolor="darkgray" |

Notes

References

{{citation

| last1 = Bern | first1 = M.

| last2 = Edelsbrunner | first2 = H. | author2-link = Herbert Edelsbrunner

| last3 = Eppstein | first3 = D. | author3-link = David Eppstein

| last4 = Mitchell | first4 = S.

| last5 = Tan | first5 = T. S.

| doi = 10.1007/BF02573962

| issue = 1

| journal = Discrete and Computational Geometry

| mr = 1215322

| pages = 47–65

| title = Edge insertion for optimal triangulations

| volume = 10

| year = 1993

| doi-access = free

}}

{{cite journal

| last1 = Chazelle | first1 = Bernard

| last2 = Guibas | first2 = Leo J.

| last3 = Lee | first3 = D. T.

| title = The power of geometric duality

| journal = BIT

| volume = 25

| issue = 1

| year = 1985

| issn = 0006-3835

| pages = 76–90

| url = http://www.cs.princeton.edu/~chazelle/pubs/PowerDuality.pdf

| doi = 10.1007/BF01934990

| publisher = BIT Computer Science and Numerical Mathematics

| s2cid = 122411548

}}

{{cite book

| last1 = de Berg | first1 = Mark

| last2 = van Kreveld | first2 = Marc

| last3 = Overmars | first3 = Mark

| last4 = Schwarzkopf | first4 = Otfried

| title = Computational Geometry: Algorithms and Applications

| edition = 3

| publisher = Springer-Verlag

| url = http://www.cs.uu.nl/geobook/

| year = 2008

}}

{{cite book

| last1 = O'Rourke | first1 = Joseph

| last2 = L. Devadoss | first2 = Satyan | author2-link = Satyan Devadoss

| title = Discrete and Computational Geometry

| edition = 1

| publisher = Princeton University Press

| year = 2011

| ref = dcg

}}

{{cite conference

| last1 = Edelsbrunner | first1 = Herbert

| last2 = Tan | first2 = Tiow Seng

| last3 = Waupotitsch | first3 = Roman

| year = 1990

| title = An O(n2log n) time algorithm for the MinMax angle triangulation

| conference = Proceedings of the sixth annual symposium on Computational geometry

| series = SCG '90

| pages = 44–52

| isbn = 0-89791-362-0

| doi = 10.1145/98524.98535

| publisher = ACM

| citeseerx = 10.1.1.66.2895

}}

{{cite conference

| last1 = Edelsbrunner | first1 = Herbert

| last2 = Tan | first2 = Tiow Seng

| year = 1991

| title = A quadratic time algorithm for the minmax length triangulation

| conference = 32nd Annual Symposium on Foundations of Computer Science

| pages = 414–423

| isbn = 0-8186-2445-0

| doi = 10.1109/SFCS.1991.185400

| citeseerx = 10.1.1.66.8959

}}

{{cite arXiv

| last = Fekete

| first = Sándor P.

| eprint = 1208.0202v1

| title = The Complexity of MaxMin Length Triangulation

| year = 2012

| class = cs.CG

}}

{{cite conference

| last = Jansen

| first = Klaus

| year = 1992

| title = The Complexity of the Min-max Degree Triangulation Problem

| conference = 9th European Workshop on Computational Geometry

| pages = 40–43

| url = http://tizian.cs.uni-bonn.de/EuroCG93/j-cmmdt-93.pdf

}}

{{cite conference

| last = Lloyd

| first = Errol Lynn

| conference = 18th Annual Symposium on Foundations of Computer Science (SFCS 1977)

| title = On triangulations of a set of points in the plane

| year = 1977

| pages = 228–240

| doi = 10.1109/SFCS.1977.21

| hdl = 1721.1/148916

| hdl-access = free

}}

{{cite thesis

| type = Ph.D.

| first = Tzvetalin Simeonov

| last = Vassilev

| title = Optimal Area Triangulation

| publisher = University of Saskatchewan, Saskatoon

| year = 2005

| url = http://ecommons.usask.ca/bitstream/handle/10388/etd-08232005-111957/thesisFF.pdf

| access-date = 2013-06-15

| archive-url = https://web.archive.org/web/20170813183040/https://ecommons.usask.ca/bitstream/handle/10388/etd-08232005-111957/thesisFF.pdf

| archive-date = 2017-08-13

| url-status = dead

}}

Category:Triangulation (geometry)

Category:Point (geometry)

de:Gitter (Geometrie)#Dreiecksgitter

Point-set triangulation

Regular triangulations

Combinatorics in the plane

Algorithms to build triangulations in the plane

Time complexity of various algorithms

See also

Notes

References