Update the README with triangulation and convex hull

README.md

@@ -1,65 +1,96 @@
 Übergeekism
 ===========
 
-This is an attempt at using as many as possible **cool** computer science stuff to produce a single image.
+This is an attempt at using as many as possible **cool** computer science stuff
+to produce a single image.
 
-Algorithms may not be implemented in the most efficient manner, as the aim is to have elegant and simple code for
+Algorithms may not be implemented in the most efficient manner, as the aim is to
-educationnal purpose.
+have elegant and simple code for educational purpose.
 
 Until now, the following algorithms/data structure/concepts are used:
 - the (logo) turtle,
 - Lindenmayer systems,
 - Penrose tiling,
-- Travelling salesman problem,
+- travelling salesman problem,
 - ant colony algorithm,
 - A\* shortest path,
+- Delaunay triangulation,
+- Bowyer-Watson algorithm,
+- convex hull,
+- Chan's algorithm,
 - graph (adjacency list, adjacency matrix),
 - hash table.
 
 The current code is written in Python.
 
 
-Penrose tiling
+Penrose graph
---------------
+-------------
 
-The main shape visible on the image is a Penrose tiling (type P3), which is a **non-periodic tiling** with an absurd level
+The main shape visible on the image is a Penrose tiling (type P3), which is a
-of coolness.
+**non-periodic tiling** with an absurd level of coolness.
 
-The edges are **recursively** built with a **Lindenmayer system**. Yes, it is capable of building a Penrose tiling if you
+The edges are **recursively** built with a **Lindenmayer system**. Yes, it is
-know which grammar to use. Yes, this is insanely cool.
+capable of building a Penrose tiling if you know which grammar to use. Yes, this
+is insanely cool.
 
-The Lindenamyer system works by drawing edges one after another, we thus use a (LOGO) **turtle** to draw them.
+The Lindenamyer system works by drawing edges one after another, we thus use a
+(LOGO) **turtle** to draw them.
 
-Because the L-system grammar is not very efficient to build the tiling, we insert edges in a data structure that
+Because the L-system grammar is not very efficient to build the tiling, we
-contains an unordered collection of unique element: a **hash table**.
+insert edges in a data structure that contains an unordered collection of unique
+element: a **hash table**.
 
 
 Travelling Salesman Problem
 ---------------------------
 
-The Penrose tiling defines a **graph**, which connects a set of vertices with a set of edges. We can consider the vertices
+The Penrose tiling segments defines a **graph**, which connects a set of
-as cities and edges as roads between them.
+vertices with a set of edges. We can consider the vertices as cities and edges
+as roads between them.
 
-Now we want to find the shortest possible route that visits each city exactly once and returns to the origin city. This
+Now we want to find the shortest possible route that visits each city exactly
-is the **Travelling Salesman Problem**. We use an **Ant Colony Optimization** algorithm to (try) to solve it.
+once and returns to the origin city. This is the **Travelling Salesman
+Problem**. We use an **Ant Colony Optimization** algorithm to (try) to solve it.
 
-Because each city is not connected to every other cities, we need to find the shortest path between two cities. This is
+Because each city is not connected to every other cities, we need to find the
-done with the help of the **A-star** algorithm.
+shortest path between two cities. This is done with the help of the **A-star**
+algorithm.
 
-The ant colony algorithm output a path that connect every cities, which is drawn on the image, but it also stores a
+The ant colony algorithm output a path that connect every cities, which is drawn
-so-called pheromones matrix, which can be drawn as edges with variable transparency/width.
+on the image, but it also stores a so-called pheromones matrix, which can be
+drawn as edges with variable transparency/width.
+
+
+Penrose tiling
+--------------
+
+Because the L-system draws the Penrose tiling segments by segments, we need to
+compute how each segment is related to the diamonds to rebuild the tiling
+corresponding to all those edges.
+
+Fortunately, computing a **Delaunay triangulation** of the Penrose vertices
+brings back the Penrose graph (how cool is that!?) and stores plain shapes
+(triangles) instead of unordered segments. This is done thanks to the
+**Bowyer-Watson algorithm**.
+
+But this triangulation contains edges that link the set of exterior vertices,
+which are not in the Penrose graph. This is solved by computing the **convex
+hull**, with the **Chan's algorithm**, and removing the triangles that contains
+those edges from the triangulation.
 
 
 TODO
 ----
 
 More coolness:
-- Compute the **Delaunay triangulation** from the Penrose graph vertices,
+- Compute the **Voronoï diagram** from the triangulation,
-- Compute the **Voronoï diagram** (Bowyer-Watson or Fortune's algorithm?) from the triangulation,
 - Remove Voronoï edges that intersects with the Penrose graph,
 - **quad trees** may be useful somewhere to query neighbors points?,
 - The center of remaining segments is the center of the Penrose tiles,
 - Build back the neighborhood of those tiles from the Voronoï diagram,
+- Draw the neighborhood with **splines** across the center of diamonds
+  segments,
 - Run a **cellular automata** on this Penrose tiling,
 - Draw a **planner** on it.