ajout delaunay

ecf83cd9 · orestis.malaspin · 7684dc2c · ecf83cd9 · ecf83cd9 · ecf83cd9
Commit ecf83cd9 authored 5 years ago by orestis.malaspin
--- a/delaunay.md
+++ b/delaunay.md
+% 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.
+
+![Deux exemples de triangulation (source: [wikipedia](https://bit.ly/2P2XCJ9)).](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))
+
+![Ajout du premier point dans le super-triangle](figs/Bowyer-Watson_0.png){#fig:tri width=100%}
+
+## Exemple d'ajouts (2/6) (voir [wikipedia](https://bit.ly/2vJs5p2))
+
+![Ajout du premier point dans le super-triangle](figs/Bowyer-Watson_1.png){#fig:tri width=100%}
+
+## Exemple d'ajouts (3/6) (voir [wikipedia](https://bit.ly/2vJs5p2))
+
+![Ajout du premier point dans le super-triangle](figs/Bowyer-Watson_2.png){#fig:tri width=100%}
+
+## Exemple d'ajouts (4/6) (voir [wikipedia](https://bit.ly/2vJs5p2))
+
+![Ajout du premier point dans le super-triangle](figs/Bowyer-Watson_3.png){#fig:tri width=100%}
+
+## Exemple d'ajouts (5/6) (voir [wikipedia](https://bit.ly/2vJs5p2))
+
+![Ajout du premier point dans le super-triangle](figs/Bowyer-Watson_4.png){#fig:tri width=100%}
+
+## Exemple d'ajouts (6/6) (voir [wikipedia](https://bit.ly/2vJs5p2))
+
+![Ajout du premier point dans le super-triangle](figs/Bowyer-Watson_6.png){#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
--- a/figs/Bowyer-Watson_0.png
+++ b/figs/Bowyer-Watson_0.png
--- a/figs/Bowyer-Watson_1.png
+++ b/figs/Bowyer-Watson_1.png
--- a/figs/Bowyer-Watson_2.png
+++ b/figs/Bowyer-Watson_2.png
--- a/figs/Bowyer-Watson_3.png
+++ b/figs/Bowyer-Watson_3.png
--- a/figs/Bowyer-Watson_4.png
+++ b/figs/Bowyer-Watson_4.png
--- a/figs/Bowyer-Watson_6.png
+++ b/figs/Bowyer-Watson_6.png
--- a/figs/PointSetTriangulations.svg
+++ b/figs/PointSetTriangulations.svg
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