From ecf83cd96aa85b267eaad71dee5164b0a8249068 Mon Sep 17 00:00:00 2001 From: Orestis Malaspinas <orestis.malaspinas@hesge.ch> Date: Mon, 17 Feb 2020 18:22:43 +0100 Subject: [PATCH] ajout delaunay --- delaunay.md | 159 ++++++++++++ figs/Bowyer-Watson_0.png | Bin 0 -> 63926 bytes figs/Bowyer-Watson_1.png | Bin 0 -> 90205 bytes figs/Bowyer-Watson_2.png | Bin 0 -> 104605 bytes figs/Bowyer-Watson_3.png | Bin 0 -> 105853 bytes figs/Bowyer-Watson_4.png | Bin 0 -> 110111 bytes figs/Bowyer-Watson_6.png | Bin 0 -> 61952 bytes figs/PointSetTriangulations.svg | 421 ++++++++++++++++++++++++++++++++ 8 files changed, 580 insertions(+) create mode 100644 delaunay.md create mode 100644 figs/Bowyer-Watson_0.png create mode 100644 figs/Bowyer-Watson_1.png create mode 100644 figs/Bowyer-Watson_2.png create mode 100644 figs/Bowyer-Watson_3.png create mode 100644 figs/Bowyer-Watson_4.png create mode 100644 figs/Bowyer-Watson_6.png create mode 100644 figs/PointSetTriangulations.svg diff --git a/delaunay.md b/delaunay.md new file mode 100644 index 0000000..9143430 --- /dev/null +++ b/delaunay.md @@ -0,0 +1,159 @@ +% Triangulation de Delaunay +% Travail pratique de programmation déquentielle +% 19 février 2020 + +# La triangulation de Delaunay + +## Généralités + +* Soit un ensemble de $N$ points $\mathcal{P}=\{p_i\}_{i=1}^N$. +* Une triangulation est un ensemble de triangle dont les sommets sont formés des points de $\mathcal{P}$ et couvrant leur envelope convexe. + +).](figs/PointSetTriangulations.svg){#fig:tri width=100%} + +## La triangulation de Delaunay + +* La triangulation de Delaunay (TD) d'un ensemble de points $\mathcal{P}$ est telle qu'aucun point de $\mathcal{P}$ ne se trouve dans le cercle circonscrit d'un triangle de la TD. +* Les points sont tous sur la surface de chaque cercle circonscrit des triangles de la TD. +* Elle maximise le plus petit angle de tous les triangles (on essaie d'éviter les triangle avec les angles très petits). +* Il existe plusieurs algorithmes pour construire cette triangulation: *flip algorithm*, *incremental edge flip*, *Bowyer--Watson**, ... + +## L'algorithme de Bowyer--Watson + +* Dans cet algorithme on commence par construire un **super-triangle** contenant tous les points de $\mathcal{P}$. +* On ajoute ensuite les points itérativement: + * Chaque fois qu'on ajoute un point, $\vec p$, on retire tous les triangles dont le triangle circonscrit contient le point $\vec p$. + * On se retrouve avec avec un "trou" qui contient $\vec p$. + * Toutes les arrêtes du polygones sont reliées à $\vec p$ pour former les nouveau triangles à ajouter à la triangulation. +* On retire tous les triangles qui contiennent les points du super-triangle. + + +## Exemple d'ajouts (1/6) (voir [wikipedia](https://bit.ly/2vJs5p2)) + +{#fig:tri width=100%} + +## Exemple d'ajouts (2/6) (voir [wikipedia](https://bit.ly/2vJs5p2)) + +{#fig:tri width=100%} + +## Exemple d'ajouts (3/6) (voir [wikipedia](https://bit.ly/2vJs5p2)) + +{#fig:tri width=100%} + +## Exemple d'ajouts (4/6) (voir [wikipedia](https://bit.ly/2vJs5p2)) + +{#fig:tri width=100%} + +## Exemple d'ajouts (5/6) (voir [wikipedia](https://bit.ly/2vJs5p2)) + +{#fig:tri width=100%} + +## Exemple d'ajouts (6/6) (voir [wikipedia](https://bit.ly/2vJs5p2)) + +{#fig:tri width=100%} + +<!-- ## Pseudo-code de l'algorithme (1/2) + +\scriptsize + +```C +void bowyer_watson(point *points, int num_p, vec_triangle *t, int *num_t) { + ajout_super_triangle(vec_triangle, num_t); // ajout super triangle + + pour chaque p dans points { + mauvais_triangles = vecteur_vide(); + pour chaque t dans vec_triangle { + si p est dans le triangle circonscrit de t { + vecteur_push(mauvais_triangles, t); + } + } + + vec_arretes = vecteur_vide(); + pour chaque t dans mauvais_triangles { + pour chaque arrete dans t { + si l'arrete n'est partagée avec aucun triangle de mauvais_triangles { + vecteur_push(vec_arretes, arrete); + } + } + } + + pour chaque t dans mauvais_triangles { + vec_remove(vec_triangles, t); + } + + pour chaque arrete dans vec_arretes { + t = form_triangle_from_edge_and_point(arrete, p); + vector_push(vec_triangles, t); + } + + pour chaque t dans vec_triangles { + if t contient un point du super triangle { + vec_remove(vec_triangles, t); + } + } + } +} +``` --> + +## Pseudo-code de l'algorithme (1/2) + +\scriptsize + +```C +void bowyer_watson(point *points, int num_p, vec_triangle *t, int *num_t) { + ajout_super_triangle(vec_triangle, num_t); // ajout super triangle + + pour chaque p dans points { + mauvais_triangles = vecteur_vide(); + // on détermine les mauvais triangles + pour chaque t dans vec_triangle { + si p est dans le triangle circonscrit de t { + vecteur_push(mauvais_triangles, t); + } + } + vec_arretes = vecteur_vide(); + // stockage du polynogone entourant le "trou" + pour chaque t dans mauvais_triangles { + pour chaque arrete dans t { + si l'arrete n'est partagée avec aucun triangle de mauvais_triangles { + vecteur_push(vec_arretes, arrete); + } + } + } + // on retire les mauvais triangles + pour chaque t dans mauvais_triangles { + vec_remove(vec_triangles, t); + } +} +``` + +## Pseudo-code de l'algorithme (2/2) + +\scriptsize + +```C + // on forme les nouveaux triangles et on les ajoute + pour chaque arrete dans vec_arretes { + t = form_triangle_from_edge_and_point(arrete, p); + vector_push(vec_triangles, t); + } + + // on retire les points du super triangle + pour chaque t dans vec_triangles { + if t contient un point du super triangle { + vec_remove(vec_triangles, t); + } + } + } +} +``` + + +## Appartenance au cercle circonscrit + +* Une partie fondamentale de l'algorithme réside dans la capacité à déterminer si un point est dans le cercle circonscrit. +* On peut déterminer si un point $D$ est dans le triangle circonscrit délimité par les points $A$, $B$, et $C$ (qui sont ordonnés dans le sens inverse des aiguilles d'une montre), si +$$ +\begin{vmatrix}A_{x}-D_{x}&A_{y}-D_{y}&(A_{x}-D_{x})^{2}+(A_{y}-D_{y})^{2}\\B_{x}-D_{x}&B_{y}-D_{y}&(B_{x}-D_{x})^{2}+(B_{y}-D_{y})^{2}\\C_{x}-D_{x}&C_{y}-D_{y}&(C_{x}-D_{x})^{2}+(C_{y}-D_{y})^{2}\end{vmatrix}>0, +$$ +où $|.|$ signifie le calcul du déterminant. \ No newline at end of file diff --git a/figs/Bowyer-Watson_0.png b/figs/Bowyer-Watson_0.png new file mode 100644 index 0000000000000000000000000000000000000000..2050bf90cb6e85cbaf9eaa4d12cc6e1721031665 GIT binary patch literal 63926 zcmeFZby!qg`!+nZl7fWN(nv{ngVG=^jdZuf&`3*-gmi;+cMFozl0zfi-SF<={>A;g z&;9=S{qr4<haBka*=y}}#d%)Wx%S?nN(%2VPzg~%AP|PMl!P(}gjfy&A$)#<417{y z(|Zm4LUa<7R(%5e@qA(&0{s7}z0?OM5QqR9_5;^Xb0Q1^k%OcqL{;5VcIVw=bzD*( z9?aBS&K4%-M^bb6JIW;xMX|o$zWqQ|i5Mf=OKhlPKupY$^rkYPnBrL?D75>PHG=-o zOA-4aIv#xlLZu&yhzaj~J{KD!h>dOK@VAuJ+}+pDv&x$JIlIp9dkYOU<TJSJEZmf~ zHl%BxZ4ckgNpU3)Qb<FnfWLvN98*YAkN=bg8~IW{eww4i-u~A|%HLla{Pjip-{|GA z6Qm)ymdKia9Yy6tVEpT7U=m#XUq{6TL9%}x%{2YbaR0I0V^sVnQ2vt@|3Q@h&*1ab 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