Polygonal chain
Introduction
A polygonal chain (also known as a polyline) is a connected series of line segments in a given order. It is a fundamental concept in computational geometry and computer graphics, often used to represent shapes, paths, and boundaries. Polygonal chains can be open or closed, depending on whether the first and last vertices are connected.
Definition and Properties
A polygonal chain is defined by a sequence of points \( P_1, P_2, \ldots, P_n \) in the Euclidean plane. Each pair of consecutive points \( (P_i, P_{i+1}) \) forms a line segment. The chain is open if \( P_1 \neq P_n \) and closed if \( P_1 = P_n \).
Types of Polygonal Chains
- **Simple Polygonal Chain**: A chain that does not intersect itself.
- **Self-Intersecting Polygonal Chain**: A chain that intersects itself at one or more points.
- **Closed Polygonal Chain**: A chain where the first and last points coincide, forming a loop.
- **Open Polygonal Chain**: A chain where the first and last points do not coincide.
Applications
Polygonal chains are used in various fields such as computer graphics, geographic information systems (GIS), robotics, and computer-aided design (CAD). They are fundamental in representing and manipulating shapes, paths, and boundaries.
Computer Graphics
In computer graphics, polygonal chains are used to model the edges of polygons and polyhedra. They are essential in rendering algorithms, mesh generation, and vector graphics.
Geographic Information Systems (GIS)
In GIS, polygonal chains represent geographical features such as roads, rivers, and boundaries. They are used in spatial analysis, map rendering, and geospatial data processing.
Robotics
In robotics, polygonal chains are used in path planning and obstacle avoidance. They help in defining the workspace and movement paths of robotic arms and mobile robots.
Computer-Aided Design (CAD)
In CAD, polygonal chains are used to define the edges of 2D and 3D models. They are crucial in solid modeling, surface modeling, and wireframe modeling.
Mathematical Representation
A polygonal chain can be represented mathematically as a sequence of vectors. Each vector corresponds to a line segment between two consecutive points. The length of the chain is the sum of the lengths of these vectors.
Length of a Polygonal Chain
The length \( L \) of a polygonal chain defined by points \( P_1, P_2, \ldots, P_n \) is given by:
\[ L = \sum_{i=1}^{n-1} \|P_{i+1} - P_i\| \]
where \( \|P_{i+1} - P_i\| \) denotes the Euclidean distance between points \( P_i \) and \( P_{i+1} \).
Curvature of a Polygonal Chain
The curvature of a polygonal chain at a vertex \( P_i \) is defined by the angle between the vectors \( P_{i-1}P_i \) and \( P_iP_{i+1} \). This angle is called the turning angle.
Algorithms and Computational Aspects
Several algorithms are used to process and analyze polygonal chains. These include algorithms for convex hull construction, polygon triangulation, and shortest path computation.
Convex Hull Construction
The convex hull of a set of points is the smallest convex polygon that contains all the points. Algorithms such as Graham's scan and Jarvis's march are used to compute the convex hull of a polygonal chain.
Polygon Triangulation
Polygon triangulation involves dividing a polygonal chain into non-overlapping triangles. This is useful in finite element analysis, computer graphics, and geometric modeling.
Shortest Path Computation
Algorithms such as Dijkstra's algorithm and A* search algorithm are used to find the shortest path between points on a polygonal chain. These algorithms are essential in pathfinding and network routing.