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HyperbolicTree
A hyperbolic tree (often shortened as hypertree) is an information visualization and graph drawing method inspired by hyperbolic geometry. Displaying hierarchical data as a tree suffers from visual clutter as the number of nodes per level can grow exponentially. For a simple binary tree, the maximum number of nodes at a level ''n'' is 2''n'', while the number of nodes for trees with more branching grows much more quickly. Drawing the tree as a node-link diagram thus requires exponential amounts of space to be displayed. One approach is to use a ''hyperbolic tree'', first introduced by Lamping et al. Hyperbolic trees employ hyperbolic space, which intrinsically has "more room" than Euclidean space. For instance, linearly increasing the radius of a circle in Euclidean space increases its circumference linearly, while the same circle in hyperbolic space would have its circumference increase exponentially. Exploiting this property allows laying out the tree in hyperbolic space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Visualization (graphic)
Visualization or visualisation (see spelling differences) is any technique for creating images, diagrams, or animations to communicate a message. Visualization through visual imagery has been an effective way to communicate both abstract and concrete ideas since the dawn of humanity. Examples from history include cave paintings, Egyptian hieroglyphs, Greek geometry, and Leonardo da Vinci's revolutionary methods of technical drawing for engineering and scientific purposes. Visualization today has ever-expanding applications in science, education, engineering (e.g., product visualization), interactive multimedia, medicine, etc. Typical of a visualization application is the field of computer graphics. The invention of computer graphics (and 3D computer graphics) may be the most important development in visualization since the invention of central perspective in the Renaissance period. The development of animation also helped advance visualization. Overview The use of visualiz ... [...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 menta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (data Structure)
In computer science, a tree is a widely used abstract data type that represents a hierarchical tree structure with a set of connected nodes. Each node in the tree can be connected to many children (depending on the type of tree), but must be connected to exactly one parent, except for the ''root'' node, which has no parent. These constraints mean there are no cycles or "loops" (no node can be its own ancestor), and also that each child can be treated like the root node of its own subtree, making recursion a useful technique for tree traversal. In contrast to linear data structures, many trees cannot be represented by relationships between neighboring nodes in a single straight line. Binary trees are a commonly used type, which constrain the number of children for each parent to exactly two. When the order of the children is specified, this data structure corresponds to an ordered tree in graph theory. A value or pointer to other data may be associated with every node in the tre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Disk Model
In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle. The group of orientation preserving isometries of the disk model is given by the projective special unitary group , the quotient of the special unitary group SU(1,1) by its center . Along with the Klein model and the Poincaré half-space model, it was proposed by Eugenio Beltrami who used these models to show that hyperbolic geometry was equiconsistent with Euclidean geometry. It is named after Henri Poincaré, because his rediscovery of this representation fourteen years later became better known than the original work of Beltrami. The Poincaré ball model is the similar model for ''3'' or ''n''-dimensional hyperbolic geometry in which the points of the geometry are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Model
Klein may refer to: People *Klein (surname) *Klein (musician) Places *Klein (crater), a lunar feature *Klein, Montana, United States *Klein, Texas, United States *Klein (Ohm), a river of Hesse, Germany, tributary of the Ohm *Klein River, a river in the Western Cape province of South Africa Business *Klein Bikes, a bicycle manufacturer *Klein Tools, a manufacturer *S. Klein, a department store *Klein Modellbahn, an Austrian model railway manufacturer Arts *Klein + M.B.O., an Italian musical group * Klein Award, for comic art *Yves Klein, French artist Mathematics *Klein bottle, an unusual shape in topology *Klein geometry *Klein configuration, in geometry * Klein cubic (other) *Klein graphs, in graph theory *Klein model, or Beltrami–Klein model, a model of hyperbolic geometry *Klein polyhedron, a generalization of continued fractions to higher dimensions, in the geometry of numbers *Klein surface, a dianalytic manifold of complex dimension 1 Other uses * Kleins, Line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Transformation
In geometry and complex analysis, a Möbius transformation of the complex plane is a rational function of the form f(z) = \frac of one complex variable ''z''; here the coefficients ''a'', ''b'', ''c'', ''d'' are complex numbers satisfying ''ad'' − ''bc'' ≠ 0. Geometrically, a Möbius transformation can be obtained by first performing stereographic projection from the plane to the unit two-sphere, rotating and moving the sphere to a new location and orientation in space, and then performing stereographic projection (from the new position of the sphere) to the plane. These transformations preserve angles, map every straight line to a line or circle, and map every circle to a line or circle. The Möbius transformations are the projective transformations of the complex projective line. They form a group called the Möbius group, which is the projective linear group PGL(2,C). Together with its subgroups, it has numerous applications in mathematics and physics. Möbius transfor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Geometry
In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' not on ''R'', in the plane containing both line ''R'' and point ''P'' there are at least two distinct lines through ''P'' that do not intersect ''R''. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model. When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tiling
In geometry, the binary tiling (sometimes called the Böröczky tiling) is a tiling of the hyperbolic plane, resembling a quadtree over the Poincaré half-plane model of the hyperbolic plane. It was first studied mathematically in 1974 by . However, a closely related tiling was used earlier in a 1957 print by M. C. Escher. See especially text describing ''Regular Division of the Plane VI'', pp. 112 & 114, schematic diagram, p. 116, and reproduction of the print, p. 117. Tiles In one version of the tiling, the tiles are shapes bounded by three congruent horocyclic segments (two of which are part of the same horocycle), and two line segments. All tiles are congruent. In the Poincaré half-plane model, the horocyclic segments are modeled as horizontal line segments (parallel to the boundary of the half-plane) and the line segments are modeled as vertical line segments (perpendicular to the boundary of the half-plane), giving each tile the overall shape in the model of a square or r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Visualization
Information is an abstract concept that refers to that which has the power to inform. At the most fundamental level information pertains to the interpretation of that which may be sensed. Any natural process that is not completely random, and any observable pattern in any medium can be said to convey some amount of information. Whereas digital signals and other data use discrete signs to convey information, other phenomena and artifacts such as analog signals, poems, pictures, music or other sounds, and currents convey information in a more continuous form. Information is not knowledge itself, but the meaning that may be derived from a representation through interpretation. Information is often processed iteratively: Data available at one step are processed into information to be interpreted and processed at the next step. For example, in written text each symbol or letter conveys information relevant to the word it is part of, each word conveys information relevant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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