In geometry, triangulation refers to the process of dividing a geometric shape, such as a polygon, into triangles. This is often done to simplify calculations, especially in fields like computer graphics, spatial analysis, and geographic information systems (GIS). **Key points about triangulation in geometry:** 1. **Purpose:** Triangulation allows for easier computation of areas, volumes, and various properties of complex shapes since triangles are the simplest polygons.
An antiprism is a type of polyhedron that consists of two parallel polygonal bases connected by a band of triangular faces. It can be described as a generalized version of a prism, where instead of the bases being congruent and aligned, the bases are offset from each other and connected by equilateral triangles. Key features of an antiprism include: 1. **Polygonal Bases**: The two bases are identical polygons, such as triangles, squares, or pentagons.
An Apollonian network is a type of geometric network that is constructed using a recursive process based on the properties of triangular tiling. It begins with a single triangle, which is then subdivided into smaller triangles recursively. The network has a rich structure and exhibits fractal characteristics, making it interesting in the study of complex networks.
The Bowyer-Watson algorithm is a computational geometry algorithm used to incrementally construct a Delaunay triangulation of a set of points in a two-dimensional space. A Delaunay triangulation maximizes the minimum angle of the triangles formed, avoiding skinny triangles and ensuring better numerical stability for applications such as mesh generation and interpolation.
Constrained Delaunay triangulation (CDT) is a type of triangulation for a planar point set that respects certain constraints, particularly the inclusion of specified edges (or line segments) in the triangulation. This is an extension of the standard Delaunay triangulation, which is defined without any constraints.
Delaunay refinement is a computational geometry technique primarily used in the context of mesh generation. It aims to create a mesh composed of triangles (or tetrahedra in 3D) that satisfies certain optimality criteria, such as minimizing the maximum angle of the triangles (maximizing the minimum angle), and ensuring that the mesh conforms to specified geometric constraints of the underlying domain.
Delaunay triangulation is a geometric method for dividing a set of points into triangles such that no point is inside the circumcircle of any triangle in the triangulation. This property maximizes the minimum angle of the triangles, which helps avoid skinny triangles and is particularly useful in computational geometry and various applications including computer graphics, geographical information systems (GIS), and numerical simulations.
Fan triangulation is a method used in computational geometry, particularly in the field of computer graphics and geographic information systems. The process involves breaking down a polygon (usually a simple polygon) into a set of triangles, which can be more easily processed in various applications such as rendering or spatial analysis. The distinguishing feature of fan triangulation is that it typically starts from a single vertex (the "fan" vertex) and connects it to all other vertices of the polygon, forming a series of triangles.
Kinetic triangulation is a concept from computational geometry that deals with the dynamic problem of maintaining the properties of a triangulation of a set of points in motion. Specifically, it refers to the process of efficiently updating the triangulation structure as the points in the plane change their positions over time.
Minimum-weight triangulation (MWT) refers to the problem of dividing a simple polygon into triangles in such a way that the total weight of the edges used in the triangulation is minimized. The "weight" of an edge can be defined in various ways depending on the context, but it commonly relates to the length of the edge in geometric scenarios.
A **nonobtuse mesh** is a type of geometric mesh used primarily in finite element methods and computational geometry. In this context, a mesh is a collection of vertices, edges, and faces that defines a geometric shape or domain over which computations are performed. The term "nonobtuse" refers to the angles formed by the elements (usually triangles or tetrahedra) in the mesh.
Pitteway triangulation is a method used in mathematics and computer graphics for the triangulation of polyhedral surfaces, which involves breaking down a complex surface into simpler triangular components. This technique is particularly useful in computer graphics for rendering 3D models, as it simplifies the geometry and allows for easier manipulation and computation. The method typically involves defining a set of points on the surface and then systematically creating triangles that connect these points, ensuring that the entire surface is covered without overlaps or gaps.
Point-set triangulation is a computational geometry concept that involves subdividing a set of points into a collection of triangles, typically in a two-dimensional space. This method is essential for various applications in computer graphics, geographic information systems (GIS), finite element analysis, and mesh generation. In point-set triangulation, the key objectives are: 1. **Covering the Point Set**: The triangulation should cover all the points in the given set.
Polygon triangulation is the process of dividing a polygon into triangles, which are simpler geometric shapes. This is useful in various fields such as computer graphics, geographical information systems (GIS), and computational geometry because triangles are easier to work with for tasks like rendering, mesh generation, and mathematical computations.
Quasi-triangulation refers to a type of planar division that is similar to triangulation, but instead of dividing a region into triangles, it divides the region into a more generalized subdivision, which may include other polygonal shapes. This concept is relevant in computational geometry, where the goal is often to break down a complex shape into simpler components for analysis, representation, or processing.
Rotation distance, also known as **tree rotation distance**, is a concept from computational biology and bioinformatics that quantifies the minimum number of rotation operations required to transform one binary tree into another. A binary tree can be defined as a tree structure where each node has at most two children referred to as the left and right child. A rotation operation involves changing the structure of the tree without altering its nodes.
A simplicial complex is a mathematical structure used in algebraic topology and combinatorial mathematics to study spaces and their properties. It is a way of building up a geometric object from simpler building blocks called simplices. ### Definition of a Simplicial Complex A simplicial complex \( K \) is a set of simplices that satisfies two conditions: 1. **Non-emptiness**: The empty set is in \( K \).
A triangle mesh is a type of geometric representation commonly used in computer graphics, 3D modeling, and computational geometry. It consists of a collection of triangular faces that define a 3D shape or surface. Each triangle is typically defined by three vertices, which are points in 3D space, and the edges connecting these vertices.
A Triangulated Irregular Network (TIN) is a method used in geographic information systems (GIS) and computer graphics to represent a surface. It consists of a collection of triangles that are formed by connecting a set of irregularly spaced points (also known as vertices or nodes) in a way that creates a continuous representation of a surface, such as terrain elevation.
In topology, triangulation refers to the process of dividing a topological space into simpler pieces called simplices, specifically triangles (in two dimensions), tetrahedra (in three dimensions), or their higher-dimensional analogues. This technique is often employed in the study of geometric structures and algebraic topology.
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