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Implicit Kd-tree
An implicit ''k''-d tree is a ''k''-d tree defined implicitly above a rectilinear grid. Its split planes' positions and orientations are not given explicitly but implicitly by some recursive splitting-function defined on the hyperrectangles belonging to the tree's nodes. Each inner node's split plane is positioned on a grid plane of the underlying grid, partitioning the node's grid into two subgrids. Nomenclature and references The terms " min/max ''k''-d tree" and "implicit ''k''-d tree" are sometimes mixed up. This is because the first publication using the term "implicit ''k''-d tree" did actually use explicit min/max ''k''-d trees but referred to them as "implicit ''k''-d trees" to indicate that they may be used to ray trace implicitly given iso surfaces. Nevertheless, this publication used also slim ''k''-d trees which are a subset of the implicit ''k''-d trees with the restriction that they can only be built over integer hyperrectangles with sidelengths that are powers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-d Tree
In computer science, a ''k''-d tree (short for ''k-dimensional tree'') is a space-partitioning data structure for organizing points in a ''k''-dimensional space. K-dimensional is that which concerns exactly k orthogonal axes or a space of any number of dimensions. ''k''-d trees are a useful data structure for several applications, such as: * Searches involving a multidimensional search key (e.g. range searches and nearest neighbor searches) & * Creating point clouds. ''k''-d trees are a special case of binary space partitioning trees. Description The ''k''-d tree is a binary tree in which ''every'' node is a ''k''-dimensional point. Every non-leaf node can be thought of as implicitly generating a splitting hyperplane that divides the space into two parts, known as half-spaces. Points to the left of this hyperplane are represented by the left subtree of that node and points to the right of the hyperplane are represented by the right subtree. The hyperplane direction is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rectilinear Grid
A regular grid is a tessellation of ''n''-dimensional Euclidean space by congruent parallelotopes (e.g. bricks). Its opposite is irregular grid. Grids of this type appear on graph paper and may be used in finite element analysis, finite volume methods, finite difference methods, and in general for discretization of parameter spaces. Since the derivatives of field variables can be conveniently expressed as finite differences, structured grids mainly appear in finite difference methods. Unstructured grids offer more flexibility than structured grids and hence are very useful in finite element and finite volume methods. Each cell in the grid can be addressed by index (i, j) in two dimensions or (i, j, k) in three dimensions, and each vertex has coordinates (i\cdot dx, j\cdot dy) in 2D or (i\cdot dx, j\cdot dy, k\cdot dz) in 3D for some real numbers ''dx'', ''dy'', and ''dz'' representing the grid spacing. Related grids A Cartesian grid is a special case where the elements are un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plane (mathematics)
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so ''the'' Euclidean plane refers to the whole space. Several notions of a plane may be defined. The Euclidean plane follows Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ..., and in particular the parallel postulate. A projective plane may be constructed by adding "points at infinity" where two otherwise parallel lines would intersect, so that every pair of lines intersects in exactly one point. The elliptic plane may be further defined by adding a metr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (geometry)
In geometry, the orientation, attitude, bearing, direction, or angular position of an object – such as a Line (geometry), line, plane (geometry), plane or rigid body – is part of the description of how it is placed in the Euclidean space, space it occupies. More specifically, it refers to the imaginary rotation that is needed to move the object from a reference placement to its current placement. A rotation may not be enough to reach the current placement, in which case it may be necessary to add an imaginary translation (geometry), translation to change the object's position (geometry), position (or linear position). The position and orientation together fully describe how the object is placed in space. The above-mentioned imaginary rotation and translation may be thought to occur in any order, as the orientation of an object does not change when it translates, and its position does not change when it rotates. Euler's rotation theorem shows that in three dimensions any orien ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recursion
Recursion occurs when the definition of a concept or process depends on a simpler or previous version of itself. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function (mathematics), function being defined is applied within its own definition. While this apparently defines an infinite number of instances (function values), it is often done in such a way that no infinite loop or infinite chain of references can occur. A process that exhibits recursion is ''recursive''. Video feedback displays recursive images, as does an infinity mirror. Formal definitions In mathematics and computer science, a class of objects or methods exhibits recursive behavior when it can be defined by two properties: * A simple ''base case'' (or cases) — a terminating scenario that does not use recursion to produce an answer * A ''recursive step'' — a set of rules that reduce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperrectangle
In geometry, a hyperrectangle (also called a box, hyperbox, k-cell or orthotopeCoxeter, 1973), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions. A necessary and sufficient condition is that it is Congruence (geometry), congruent to the Cartesian product of finite interval (mathematics), intervals. This means that a k-dimensional rectangular solid has each of its edges equal to one of the closed intervals used in the definition. Every k-cell is compact (mathematics), compact. If all of the edges are equal length, it is a ''hypercube''. A hyperrectangle is a special case of a parallelohedron#Related shapes, parallelotope. Formal definition For every integer i from 1 to k, let a_i and b_i be real numbers such that a_i < b_i. The set of all points in whose coordinates satisfy the inequalities is a -cell. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Node (computer Science)
A node is a basic unit of a data structure, such as a linked list or Tree (data structure), tree data structure. Nodes contain data and also may link to other nodes. Links between nodes are often implemented by Pointer (computer programming), pointers. Nodes and trees Nodes are often arranged into tree structures. A node represents the information contained in a single data structure. These nodes may contain a value or condition, or possibly serve as another independent data structure. Nodes are represented by a single parent node. The highest point on a tree structure is called a root node, which does not have a parent node, but serves as the parent or 'grandparent' of all of the nodes below it in the tree. The height of a node is determined by the total number of edges on the path from that node to the furthest leaf node, and the height of the tree is equal to the height of the root node. Node depth is determined by the distance between that particular node and the root node. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Min Max Kd Tree
A min/max ''k''d-tree is a ''k''-d tree with two scalar values—a minimum and a maximum—assigned to its nodes. The minimum/maximum of an inner node is equal to the minimum/maximum of its children's minima/maxima. Construction Min/max ''k''d-trees may be constructed recursively. Starting with the root node, the splitting plane orientation and position is evaluated. Then the children's splitting planes and min/max values are evaluated recursively. The min/max value of the current node is simply the minimum/maximum of its children's minima/maxima. Properties The min/max ''k''d-tree has—besides the properties of an ''k''d-tree—the special property that an inner node's min/max values coincide each with a min/max value of either one child. This allows to discard the storage of min/max values at the leaf nodes by storing two bits at inner nodes, assigning min/max values to the children: Each inner node's min/max values will be known in advance, where the root node's min/max values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Tree
In computer science, a binary tree is a tree data structure in which each node has at most two children, referred to as the ''left child'' and the ''right child''. That is, it is a ''k''-ary tree with . A recursive definition using set theory is that a binary tree is a triple , where ''L'' and ''R'' are binary trees or the empty set and ''S'' is a singleton (a single–element set) containing the root. From a graph theory perspective, binary trees as defined here are arborescences. A binary tree may thus be also called a bifurcating arborescence, a term which appears in some early programming books before the modern computer science terminology prevailed. It is also possible to interpret a binary tree as an undirected, rather than directed graph, in which case a binary tree is an ordered, rooted tree. Some authors use rooted binary tree instead of ''binary tree'' to emphasize the fact that the tree is rooted, but as defined above, a binary tree is always rooted. In ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ray Casting
Ray casting is the methodological basis for 3D CAD/CAM solid modeling and image rendering. It is essentially the same as ray tracing (graphics), ray tracing for computer graphics where virtual light rays are "cast" or "traced" on their path from the focal point of a camera through each pixel in the camera sensor to determine what is visible along the ray in the 3D scene. The term "Ray Casting" was introduced by Scott Roth while at the General Motors Research Labs from 1978–1980. His paper, "Ray Casting for Modeling Solids", describes modeled solid objects by combining primitive solids, such as blocks and cylinders, using the set operators union (+), intersection (&), and difference (−). The general idea of using these binary operators for solid modeling is largely due to Voelcker and Requicha's geometric modelling group at the University of Rochester. See ''solid modeling'' for a broad overview of solid modeling methods. Before ray casting (and ray tracing), computer graphics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isosurface
An isosurface is a three-dimensional analog of an isoline. It is a surface that represents points of a constant value (e.g. pressure, temperature, velocity, density) within a volume of space; in other words, it is a level set of a continuous function whose domain is 3-space. The term ''isoline'' is also sometimes used for domains of more than 3 dimensions. Applications Isosurfaces are normally displayed using computer graphics, and are used as data visualization methods in computational fluid dynamics (CFD), allowing engineers to study features of a fluid flow (gas or liquid) around objects, such as aircraft wings. An isosurface may represent an individual shock wave in supersonic flight, or several isosurfaces may be generated showing a sequence of pressure values in the air flowing around a wing. Isosurfaces tend to be a popular form of visualization for volume datasets since they can be rendered by a simple polygonal model, which can be drawn on the screen very qui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
