Geometric algorithm

Geometric algorithms (Part 6) are methods for solving problems using points and lines (and other simple geometric objects) that have only been used recently. We consider algorithms for finding the generatrix of a surface defined by a set of points, determining the intersection of geometric objects, solving problems of finding the nearest points and for performing multi-dimensional search. Many of these methods perfectly complement more elementary sorting and searching methods. [one]

Geometric algorithms include manipulating objects that are not processed at the machine language level. Therefore, the user must describe these complex objects by means of simpler data types that are directly representable in the machine. Such descriptions are called data structures. [2]

Is there a geometric algorithm for constructing degenerate groups ? [3]

To do this, you first need to find a simple way to image your own energy state, and then a simple geometric algorithm in the phase space that allows you to calculate the scalar product. This algorithm is reduced to the overlap of the areas of the phase space, representing individual quantum states. [four]

The tasks of tracing stage 2 are designed to determine the geometry of compounds, and the algorithms for solving these problems are called geometric algorithms . The main algorithms in this case are wave, ray, channel, iterative and heuristic. [five]

So is it necessary to check all the triangles defined by a set of N points to find out if a certain point lies in any of them. Graham in one of the first works specifically devoted to the development of effective geometric algorithms [Graham (1972)] showed that by performing a preliminary sorting of points, the extreme points can be found in linear time. The method he used has become a very powerful tool in the field of computational geometry. [6]

Bht polar for, then we will see that the pole on the conic is incident to its own polar. Such a transformation of points into straight lines and vice versa, called polarity, is useful when creating geometric algorithms . [7]

As indicated in sect. Although a point is considered to be a vector in Cartesian coordinates, it is desirable that the choice of a coordinate system does not affect the running time of the geometric algorithm too much. This leads to the fact that the computation model must allow the necessary transformation (Cartesian basis), and its cost per point can depend on the number of measurements, but not on the number of points to be processed. [eight]

To determine P (i) by the value of the term of equation (10.206), enclosed in square brackets, provided that the values ​​of (1 - t) m, t / (l - t) and binomial coefficients are known, eight scalar operations are required. This procedure, in essence, is the application of the Horner scheme for computing the Bezier polynomials. For small m (m5), it is more convenient to use a geometric algorithm , since its implementation requires a small number of auxiliary operations, but for large m, the Horner scheme is preferable. [9]

However, there are difficulties in analyzing the average performance. First, the input model may inaccurately characterize the input data encountered in practice, or the natural model of the input data may not exist at all. Few would object to using such input models as a randomly ordered file for a sorting algorithm or a random set of points for a geometric algorithm , and for such models you can get mathematical results that will be accurate assumptions about program performance in real-world applications. But how can you characterize the input data for a program that processes text in English. Even for sorting algorithms in certain applications, other models besides the randomly ordered data model are considered. Secondly, the analysis may require deep mathematical evidence. For example, the analysis of the average performance case for union-find algorithms is rather complicated. [ten]

Of course, we would like to say: N points are given, uniformly distributed on the plane ... Lebesgue [Kendall, Moran (1963)] 2), so we are forced to determine the specific area from which the points are selected. Fortunately, the task of calculating E (h) was given considerable attention in the literature on statistics, and the following are a number of theorems that are directly related to the analysis of some geometric algorithms . [eleven]

First of all, they were formulated not so graphically, but as geometric, their condition differed from the corresponding tasks of descriptive geometry and drawing only in that the result should be obtained without the use of drawing tools. The content of the search part of the task is to determine the line of intersection of two polyhedra. The geometric algorithm for solving this problem for students is still unknown. His search is the content of the first part of the work. [12]

The tasks of design engineering automation are divided into tasks of topological and geometric design. The formalization of problems of topological design is most simply done using graph theory. To automate the solution of layout and placement problems, combinatorial algorithms and algorithms based on mathematical programming methods are mainly used. Distribution and geometric algorithms are used to solve trace problems. [13]

It is based on the description of geometric objects in terms of properties of finite subsets. For example, a set is convex if and only if the segment defined by any pair of its points lies entirely in this set. The inadequacy of combinatorial geometry for us lies in the fact that for the majority of sets of interest to us the number of their finite subsets is infinite, and this prevents their algorithmic processing. The work of recent years in the field of geometric algorithms is aimed at getting rid of these shortcomings and the development of mathematics leading to the creation of good algorithms. [14]

Pages in the category "Lighting in three-dimensional graphics"

The 20 pages of 20 in this category are shown.