Triangulating the Real Projective Plane MACIS 07

Transcription

1 Tiangulating the Real Pojective Plane Midul Aanjaneya Moniue Teillaud MACIS 07

2 The Real Pojective Plane V () R P = R {0} / if = λ fo λ R {0} P the shee model

3 The Real Pojective Plane P = R {0} / if = λ fo λ R {0} not oientable segment [,]?

4 Tiangulation (abstact) simlicial comlex = set K and collection of S of (abstact) simlices = subsets of K such that Fo all v K, {v} S = vetex of K If τ σ S, then τ S.

5 Tiangulation (abstact) simlicial comlex = set K and collection of S of (abstact) simlices = subsets of K such that Fo all v K, {v} S = vetex of K If τ σ S, then τ S. Tiangulation of a toological sace X = simlicial comlex K such that σ K σ is homeomohic to X.

6 Tiangulations of P Studied mainly fom a gah-theoetic esective. P admits exactly two ieducible tiangulations. [Banette, 98]

7 Ou Poblem Comute a tiangulation of P whose vetices ae oints of a given inut set P = {,,..., n }

8 Ou Poblem Comute a tiangulation of P whose vetices ae oints of a given inut set P = {,,..., n } Famous oblem in R d, many famous algoithms use the oientation of R d c do not extend to P a in-tiangle test b

9 Ou Poblem Comute a tiangulation of P whose vetices ae oints of a given inut set P = {,,..., n } Famous oblem in R d, many famous algoithms use the oientation of R d V() V() V() do not extend to P

10 Ou Poblem Comute a tiangulation of P whose vetices ae oints of a given inut set P = {,,..., n } Famous oblem in R d, many famous algoithms use the oientation of R d V() V() V() do not extend to P

11 Ou Poblem Comute a tiangulation of P whose vetices ae oints of a given inut set P = {,,..., n } Famous oblem in R d, many famous algoithms use the oientation of R d V() V() V() do not extend to P

12 Ou Poblem Comute a tiangulation of P whose vetices ae oints of a given inut set P = {,,..., n } Famous oblem in R d, many famous algoithms use the oientation of R d V() V() V() do not extend to P

13 Ou Poblem - some emaks Obvious aoach: D convex hull of {, } on the shee V() V() V() not a tiangulation

14 Ou Poblem - some emaks Obvious aoach: D convex hull of {, } on the shee V() V() V() not a tiangulation Oiented ojective lane two half-shees. Two coies of each oint. Two indeendent tiangulations, n oints. [Stolfi]

15 Ou algoithm Basics: the in-tiangle test Algoithm to comute a tiangulation of P diectly in P in two main stes initialization of the tiangulation sufficient condition fo existence insetion of oints

16 in-tiangle test No oientation in P, but notion of inteio/exteio well-defined: P with a cell (toologically euivalent to a disk) cut out is toologically euivalent to a Möbius band.

17 in-tiangle test The inteio of a tiangle of P can be unambiguously defined/checked using a distinguishing lane s P lies in tiangle (,, ) V() V() V() iff P (,, ) v(s) R lies in cone (V (), V (), V (s)) not cut by the lane P(,, ) sum of signs of deteminants

18 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7}

19 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, 7

20 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, b : 5, 7

21 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, b : 5, c : 45, 7

22 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, b : 5, c : 45, d : 46, 7

23 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, b : 5, c : 45, d : 46, e : 747, 7

24 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, b : 5, c : 45, d : 46, e : 747, f : 77 7

25 Comuting an initial tiangulation Idea: use one of the known minimal tiangulations of P and its incidence stuctue. P = {,,, 4, 5, 6, 7} a : 6, b : 5, c : 45, d : 46, e : 747, f : 77 incidences: a, d, f, a, b, e, b, d, f, 4c, 4d, 4e, 5b, 5c, 6a, 6d, 7e, 7f

26 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea

27 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

28 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

29 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

30 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

31 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

32 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

33 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation 4

34 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation Associate a distinguishing lane to each tiangle Relace the fake oints by oints of P

35 Comuting an initial tiangulation Take 4 oints in P, no of which ae collinea Add fake oints and fom the minimal tiangulation Associate a distinguishing lane to each tiangle Relace the fake oints by oints of P Sufficient condition: at least 6 oints of P ae in geneal osition (no oints in these 6 ae collinea) O(n )

36 Adding futhe oints Comute a tiangulation T n stating fom T init = T 7 in a dynamic way: Fo each new i Find the tiangle of T i containing i, Cut it into tiangles T i. in a static way: Fo each tiangle of T init, find the tiangle of T init containing, Comute each of the small tiangulations using an algoithm in R. O(n ) O(n log n) thanks to anonymous efeee

37 Conclusion Fist tiangulation algoithm comuting in P. Easy to code in CGAL. Futhe wok Delaunay tiangulation Hieachical data stuctues and andomized incemental constuction?...