Discrete geometry

Exploring further into discrete geometry reveals its influence on tessellations and tilings, which are methods of covering a plane using shapes without gaps or overlaps. These concepts are not only mathematically interesting but also have practical applications in art, design, and architecture. Tessellations can efficiently use space and materials, showcasing the blend of mathematical theory and practical utility.

Going deeper into everyday applications, consider the planning and design of urban areas. Urban planners utilise geometrical shapes and principles to design cities that are easy to navigate, ensuring that roads, buildings, and public spaces are organised in a logical and accessible manner. This meticulous planning involves discrete geometry at every step, from the layout of city blocks to the design of public transportation systems.

Diving deeper into the computer science application, algorithms for pathfinding in networks or graphs are heavily grounded in discrete geometry. These algorithms, like Dijkstra's or A*, are critical for network routing, AI movement in games, and solving mazes. Here's a simple representation of Dijkstra's algorithm in Python:

def dijkstra(graph, start):
    shortest_path = {}
    unvisited = set(graph)
    while unvisited:
        node = min(unvisited, key=lambda node: shortest_path.get(node,float('inf')))
        unvisited.remove(node)
        for neighbour, distance in graph[node].items():
            path = shortest_path.get(node,float('inf')) + distance
            if path < shortest_path.get(neighbour, float('inf')):
                shortest_path[neighbour] = path
    return shortest_path

Exploring the concept of convex hulls offers an exciting excursion into applied discrete geometry. A convex hull of a set of points is the smallest convex set containing all the points. This concept has critical applications in computational geometry, such as designing algorithms for pattern recognition, data analysis, and image processing.For instance, in computer vision, determining the convex hull of a set of points can help in identifying the outer boundary of an object within an image, aiding in object recognition and classification.

Delving deeper, computational geometry's reliance on discrete geometry is evident in algorithms for 3D object recognition. The ability to precisely define and manipulate geometric shapes within a computer requires the discrete representation of these shapes, illustrating the symbiotic relationship between these areas.For instance, 3D object recognition algorithms often rely on discretely approximating the shapes of objects using polygons or polyhedra, allowing computers to efficiently process and analyse their geometric properties for recognition tasks.

Exploring further, the impact of discrete geometry on modern technology is vividly seen in the realm of robotics, particularly in motion planning and spatial reasoning. Robots operate in environments that are discretised into navigable and non-navigable spaces, allowing them to plot paths and make decisions based on geometric algorithms. This is crucial in autonomous vehicles, where discrete geometry algorithms process sensor data to avoid obstacles and navigate efficiently.For example, pathfinding algorithms utilising discrete geometry, such as A* or Dijkstra's, empower autonomous vehicles to compute the most efficient route between two points, considering real-world constraints. Here's a simple representation of the A* algorithm in Python:

def a_star_algorithm(start, stop):
    # Open set is initially empty
    # Closed set starts with the initial position
    # Assume some functions defined elsewhere: heuristic(), neighbours(), move_cost()
    open_set = set(start)
    closed_set = set()
    g = {}
    parents = {}
    g[start] = 0
    
    while len(open_set) > 0:
        n = None
        for v in open_set:
            if n == None or g[v] + heuristic(v) < g[n] + heuristic(n):
                n = v;
        if n == None:
            print('Path does not exist!')
            return None
        if n == stop:
            reconst_path = []
            
            while parents[n] != n:
                reconst_path.append(n)
                n = parents[n]
            reconst_path.append(start)
            reconst_path.reverse()
            return reconst_path
        
        for (m, weight) in neighbours(n):
            if m not in open_set and m not in closed_set:
                open_set.add(m)
                parents[m] = n
                g[m] = g[n] + weight
            else:
                if g[m] > g[n] + weight:
                    g[m] = g[n] + weight
                    parents[m] = n
                    
                    if m in closed_set:
                        closed_set.remove(m)
                        open_set.add(m)
        open_set.remove(n)
        closed_set.add(n)
    print('Path does not exist!')
    return None