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Straight Skeleton
In geometry, a straight skeleton is a method of representing a polygon by a topological skeleton. It is similar in some ways to the medial axis but differs in that the skeleton is composed of straight line segments, while the medial axis of a polygon may involve parabolic curves. However, both are homotopy-equivalent to the underlying polygon.. Straight skeletons were first defined for simple polygons by ,. and generalized to planar straight-line graphs (PSLG) by . In their interpretation as projection of roof surfaces, they are already extensively discussed by . Definition The straight skeleton of a polygon is defined by a continuous shrinking process in which the edges of the polygon are moved inwards parallel to themselves at a constant speed. As the edges move in this way, the vertices where pairs of edges meet also move, at speeds that depend on the angle of the vertex. If one of these moving vertices collides with a nonadjacent edge, the polygon is split in two by the coll ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fold-and-cut Theorem
The fold-and-cut theorem states that any shape with straight sides can be cut from a single (idealized) sheet of paper by folding it flat and making a single straight complete cut. Such shapes include polygons, which may be concave, shapes with holes, and collections of such shapes (i.e. the regions need not be connected space, connected). The corresponding problem that the theorem solves is known as the fold-and-cut problem, which asks what shapes can be obtained by the so-called fold-and-cut method. A particular instance of the problem, which asks how a particular shape can be obtained by the fold-and-cut method, is known as ''a'' fold-and-cut problem. History The earliest known description of a fold-and-cut problem appears in ''Wakoku Chiyekurabe'' (Mathematical Contests), a book that was published in 1721 by Kan Chu Sen in Japan. An 1873 article in ''Harper's New Monthly Magazine'' describes how Betsy Ross may have proposed that stars on the American flag have five poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a Bounded set, bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its projective duality, dual problem of intersecting Half-space (geome ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Polygon
In geometry, a convex polygon is a polygon that is the boundary of a convex set. This means that the line segment between two points of the polygon is contained in the union of the interior and the boundary of the polygon. In particular, it is a simple polygon (not self-intersecting). Equivalently, a polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. Strictly convex polygon A convex polygon is ''strictly'' convex if no line contains more than two vertices of the polygon. In a convex polygon, all interior angles are less than ''or equal'' to 180 degrees, while in a strictly convex polygon all interior angles are strictly less than 180 degrees. Properties The following properties of a simple polygon are all equivalent to convexity: *Every internal angle is less than or equal to 180 degrees. *Every point on every line segment between two points inside or on the boundary of the polygon remains inside or on the bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degree (graph Theory)
In graph theory, the degree (or valency) of a vertex of a graph is the number of edges that are incident to the vertex; in a multigraph, a loop contributes 2 to a vertex's degree, for the two ends of the edge. The degree of a vertex v is denoted \deg(v) or \deg v. The maximum degree of a graph G is denoted by \Delta(G), and is the maximum of G's vertices' degrees. The minimum degree of a graph is denoted by \delta(G), and is the minimum of G's vertices' degrees. In the multigraph shown on the right, the maximum degree is 5 and the minimum degree is 0. In a regular graph, every vertex has the same degree, and so we can speak of ''the'' degree of the graph. A complete graph (denoted K_n, where n is the number of vertices in the graph) is a special kind of regular graph where all vertices have the maximum possible degree, n-1. In a signed graph, the number of positive edges connected to the vertex v is called positive deg(v) and the number of connected negative edges is enti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geographic Information System
A geographic information system (GIS) consists of integrated computer hardware and Geographic information system software, software that store, manage, Spatial analysis, analyze, edit, output, and Cartographic design, visualize Geographic data and information, geographic data. Much of this often happens within a spatial database; however, this is not essential to meet the definition of a GIS. In a broader sense, one may consider such a system also to include human users and support staff, procedures and workflows, the Geographic Information Science and Technology Body of Knowledge, body of knowledge of relevant concepts and methods, and institutional organizations. The uncounted plural, ''geographic information systems'', also abbreviated GIS, is the most common term for the industry and profession concerned with these systems. The academic discipline that studies these systems and their underlying geographic principles, may also be abbreviated as GIS, but the unambiguous GIScie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Color Gradient
In color science, a color gradient (also known as a color ramp or a color progression) specifies a range of position-dependent colors, usually used to fill a region. In assigning colors to a set of values, a gradient is a continuous colormap, a type of color scheme. In computer graphics, the term swatch has come to mean a palette of active colors. RAL K5 Fächer RGB.jpg, RAL CLASSIC K5 color fan Guidapantonestandard.JPG, Pantone color guide Nuancier Pantone.jpg, cards of Pantone base colors and blends HKS-K-Farbfaecher.jpg, HKS (colour system), HKS colour fan Definitions * Color gradient is a set of colors arranged in a linear order ( ordered) * A continuous colormap is a curve through a colorspace 3D RGB profile of the Linear Gray Continous color gradient.png, gray 3D RGB profile of cubehelix color gradient.png, cubehelix 0 3d 60 75 v.png, HSV rainbow 3D RGB profile of the Smooth Cool Warm diverging color gradient by Kenneth Moreland.png, diverging Strict definition A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Miter Joint
A miter joint (mitre in British English) is a joint made by cutting each of two parts to be joined, across the main surface, usually at a 45° angle, to form a corner, usually to form a 90° angle, though it can comprise any angle greater than 0 degrees. It is called beveling when the angled cut is done on the side, although the resulting joint is still a miter joint. For woodworking, a disadvantage of a miter joint is its weakness, but it can be strengthened with a spline (a thin wafer of wood inserted into a slot, usually arranged with the long grain of the spline across the short grain of the frame timber). There are two common variations of a splined miter joint, one where the spline is long and runs the length of the mating surfaces and another where the spline is perpendicular to the joined edges. Common applications include picture frames, pipes, and molding. Non-perpendicular joints For miter joints occurring at angles other than 90°, for materials of the same ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel (curve)
A parallel of a curve is the envelope of a family of congruent circles centered on the curve. It generalises the concept of '' parallel (straight) lines''. It can also be defined as a curve whose points are at a constant '' normal distance'' from a given curve. These two definitions are not entirely equivalent as the latter assumes smoothness, whereas the former does not. In computer-aided design the preferred term for a parallel curve is offset curve. (In other geometric contexts, the term offset can also refer to translation.) Offset curves are important, for example, in numerically controlled machining, where they describe, for example, the shape of the cut made by a round cutting tool of a two-axis machine. The shape of the cut is offset from the trajectory of the cutter by a constant distance in the direction normal to the cutter trajectory at every point. In the area of 2D computer graphics known as vector graphics, the (approximate) computation of parallel curves i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Drawing
Graph drawing is an area of mathematics and computer science combining methods from geometric graph theory and information visualization to derive two-dimensional depictions of graph (discrete mathematics), graphs arising from applications such as social network analysis, cartography, linguistics, and bioinformatics. A drawing of a graph or network diagram is a pictorial representation of the vertex (graph theory), vertices and edge (graph theory), edges of a graph. This drawing should not be confused with the graph itself: very different layouts can correspond to the same graph., p. 6. In the abstract, all that matters is which pairs of vertices are connected by edges. In the concrete, however, the arrangement of these vertices and edges within a drawing affects its understandability, usability, fabrication cost, and aesthetics. The problem gets worse if the graph changes over time by adding and deleting edges (dynamic graph drawing) and the goal is to preserve the user's men ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symposium On Computational Geometry
The International Symposium on Computational Geometry (SoCG) is an academic conference in computational geometry. Today its acronym is pronounced "sausage." It was founded in 1985, with the program committee consisting of David Dobkin, Joseph O'Rourke, Franco Preparata, and Godfried Toussaint; O'Rourke was the conference chair. The symposium was originally sponsored by the SIGACT and SIGGRAPH Special Interest Groups of the Association for Computing Machinery (ACM). It dissociated from the ACM in 2014, motivated by the difficulties of organizing ACM conferences outside the United States and by the possibility of turning to an open-access system of publication. Since 2015 the conference proceedings have been published by the Leibniz International Proceedings in Informatics Dagstuhl is a computer science research center in Germany, located in and named after a district of the town of Wadern, Merzig-Wadern, Saarland. Location Following the model of the mathematical center at Mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
