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The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [5] [6] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [7] or they can be used to build a pointer based quadtree.
Order theory is a branch of mathematics that studies various kinds of objects (often binary relations) that capture the intuitive notion of ordering, providing a framework for saying when one thing is "less than" or "precedes" another. An alphabetical list of many notions of order theory can be found in the order theory glossary.
A subset of a vector space is called a cone if for all real >,.A cone is called pointed if it contains the origin. A cone is convex if and only if +. The intersection of any non-empty family of cones (resp. convex cones) is again a cone (resp. convex cone); the same is true of the union of an increasing (under set inclusion) family of cones (resp. convex cones).
The induced graph of an ordered graph is obtained by adding some edges to an ordering graph, using the method outlined below. The induced width of an ordered graph is the width of its induced graph. [2] Given an ordered graph, its induced graph is another ordered graph obtained by joining some pairs of nodes that are both parents of another node.
Many different types of lattice have been studied; see map of lattices for a list. Partially ordered sets (or posets), orderings in which some pairs are comparable and others might not be; Preorders, a generalization of partial orders allowing ties (represented as equivalences and distinct from incomparabilities)
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
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In the theory of cluster analysis, the nearest-neighbor chain algorithm is an algorithm that can speed up several methods for agglomerative hierarchical clustering.These are methods that take a collection of points as input, and create a hierarchy of clusters of points by repeatedly merging pairs of smaller clusters to form larger clusters.