1.
Geometry
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Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

2.
Tessellation
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A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellations can be generalized to higher dimensions and a variety of geometries, a periodic tiling has a repeating pattern. The patterns formed by periodic tilings can be categorized into 17 wallpaper groups, a tiling that lacks a repeating pattern is called non-periodic. An aperiodic tiling uses a set of tile shapes that cannot form a repeating pattern. In the geometry of higher dimensions, a space-filling or honeycomb is called a tessellation of space. A real physical tessellation is a made of materials such as cemented ceramic squares or hexagons. Such tilings may be decorative patterns, or may have such as providing durable and water-resistant pavement. Historically, tessellations were used in Ancient Rome and in Islamic art such as in the decorative geometric tiling of the Alhambra palace, in the twentieth century, the work of M. C. Escher often made use of tessellations, both in ordinary Euclidean geometry and in geometry, for artistic effect. Tessellations are sometimes employed for decorative effect in quilting, Tessellations form a class of patterns in nature, for example in the arrays of hexagonal cells found in honeycombs. Tessellations were used by the Sumerians in building wall decorations formed by patterns of clay tiles, decorative mosaic tilings made of small squared blocks called tesserae were widely employed in classical antiquity, sometimes displaying geometric patterns. In 1619 Johannes Kepler made a documented study of tessellations. He wrote about regular and semiregular tessellations in his Harmonices Mundi, he was possibly the first to explore and to explain the structures of honeycomb. Some two hundred years later in 1891, the Russian crystallographer Yevgraf Fyodorov proved that every periodic tiling of the features one of seventeen different groups of isometries. Fyodorovs work marked the beginning of the mathematical study of tessellations. Other prominent contributors include Shubnikov and Belov, and Heinrich Heesch, in Latin, tessella is a small cubical piece of clay, stone or glass used to make mosaics. The word tessella means small square and it corresponds to the everyday term tiling, which refers to applications of tessellations, often made of glazed clay. Tessellation or tiling in two dimensions is a topic in geometry that studies how shapes, known as tiles, can be arranged to fill a plane without any gaps, according to a given set of rules

3.
Euclidean space
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In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria, the term Euclidean distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions, classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. Geometric constructions are used to define rational numbers. It means that points of the space are specified with collections of real numbers and this approach brings the tools of algebra and calculus to bear on questions of geometry and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions. From the modern viewpoint, there is only one Euclidean space of each dimension. With Cartesian coordinates it is modelled by the coordinate space of the same dimension. In one dimension, this is the line, in two dimensions, it is the Cartesian plane, and in higher dimensions it is a coordinate space with three or more real number coordinates. One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance, for example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that point is shifted in the same direction. The other is rotation about a point in the plane. In order to all of this mathematically precise, the theory must clearly define the notions of distance, angle, translation. Even when used in theories, Euclidean space is an abstraction detached from actual physical locations, specific reference frames, measurement instruments. The standard way to such space, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. The reason for working with vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner. Once the Euclidean plane has been described in language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulae, and calculations are not made any more difficult by the presence of more dimensions. Intuitively, the distinction says merely that there is no choice of where the origin should go in the space

4.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces

5.
Trilinear interpolation
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Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of a point within the local axial rectangular prism linearly. For an arbitrary, unstructured mesh, other methods of interpolation must be used, if all the elements are tetrahedra. Trilinear interpolation is used in numerical analysis, data analysis. Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension D =1, and bilinear interpolation, the order of accuracy is 1 for all these interpolation schemes, and it requires D =8 adjacent pre-defined values surrounding the interpolation point. This gives us a value for the point. The above operations can be visualized as follows, First we find the eight corners of a cube that surround our point of interest and these corners have the values C000, C100, C010, C110, C001, C101, C011, C111. Next, we perform linear interpolation between C000 and C100 to find C00, C001 and C101 to find C01, C011 and C111 to find C11, now we do interpolation between C00 and C10 to find C0, C01 and C11 to find C1. Contains a very clever and simple method to find trilinear interpolation that is based on binary logic and can be extended to any dimension

6.
Bilinear interpolation
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In mathematics, bilinear interpolation is an extension of linear interpolation for interpolating functions of two variables on a rectilinear 2D grid. The key idea is to linear interpolation first in one direction. Although each step is linear in the values and in the position. Suppose that we want to find the value of the function f at the point. It is assumed that we know the value of f at the four points Q11 =, Q12 =, Q21 = and we first do linear interpolation in the x-direction. Or equivalently, in operations, f ≈. Contrary to what the name suggests, the interpolant is not linear. The interpolant is linear along lines parallel to either the x or the y direction, along any other straight line, the interpolant is quadratic. The result of interpolation is independent of which axis is interpolated first. If we had first performed the linear interpolation in the y-direction and then in the x-direction, the obvious extension of bilinear interpolation to three dimensions is called trilinear interpolation. In computer vision and image processing, bilinear interpolation is one of the basic resampling techniques, in texture mapping, it is also known as bilinear filtering or bilinear texture mapping, and it can be used to produce a reasonably realistic image. An algorithm is used to map a screen pixel location to a point on the texture map. A weighted average of the attributes of the four surrounding texels is computed and applied to the screen pixel and this process is repeated for each pixel forming the object being textured. When an image needs to be scaled up, each pixel of the image needs to be moved in a certain direction based on the scale constant. However, when scaling up an image by a scale factor. In this case, those holes should be assigned appropriate RGB or grayscale values so that the image does not have non-valued pixels. Bilinear interpolation can be used where perfect image transformation with pixel matching is impossible, so one can calculate. Bilinear interpolation considers the closest 2x2 neighborhood of pixel values surrounding the unknown pixels computed location

7.
Plane (geometry)
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In mathematics, a plane is a flat, two-dimensional surface that extends infinitely far. A plane is the analogue of a point, a line. When working exclusively in two-dimensional Euclidean space, the article is used, so. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory and graphing are performed in a space, or in other words. Euclid set forth the first great landmark of mathematical thought, a treatment of geometry. He selected a small core of undefined terms and postulates which he used to prove various geometrical statements. Although the plane in its sense is not directly given a definition anywhere in the Elements. In his work Euclid never makes use of numbers to measure length, angle, in this way the Euclidean plane is not quite the same as the Cartesian plane. This section is concerned with planes embedded in three dimensions, specifically, in R3. In a Euclidean space of any number of dimensions, a plane is determined by any of the following. A line and a point not on that line, a line is either parallel to a plane, intersects it at a single point, or is contained in the plane. Two distinct lines perpendicular to the plane must be parallel to each other. Two distinct planes perpendicular to the line must be parallel to each other. Specifically, let r0 be the vector of some point P0 =. The plane determined by the point P0 and the vector n consists of those points P, with position vector r, such that the vector drawn from P0 to P is perpendicular to n. Recalling that two vectors are perpendicular if and only if their dot product is zero, it follows that the plane can be described as the set of all points r such that n ⋅ =0. Expanded this becomes a + b + c =0, which is the form of the equation of a plane. This is just a linear equation a x + b y + c z + d =0 and this familiar equation for a plane is called the general form of the equation of the plane

8.
Reservoir simulation
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Reservoir simulation is an area of reservoir engineering in which computer models are used to predict the flow of fluids through porous media. Reservoir simulation models are used by oil and gas companies in the development of new fields, also, models are used in developed fields where production forecasts are needed to help make investment decisions. As building and maintaining a robust, reliable model of a field is often time-consuming and expensive, improvements in simulation software have lowered the time to develop a model. Also, models can be run on personal rather than more expensive workstations. For ongoing reservoir management, models may help in improved oil recovery by hydraulic fracturing, highly deviated or horizontal wells can also be represented. Specialized software may be used in the design of hydraulic fracturing, also, future improvement in oil recovery with pressure maintenance by re-injection of produced gas or by water injection into an aquifer can be evaluated. Water flooding resulting in the displacement of oil is commonly evaluated using reservoir simulation. The application of enhanced oil recovery processes requires that the field possesses the characteristics to make application successful. Model studies can assist in this evaluation, EOR processes include miscible displacement by natural gas, CO2, or nitrogen and chemical flooding. Special features in simulation software is needed to represent these processes, in some miscible applications, the smearing of the flood front, also called numerical dispersion, may be a problem. Reservoir simulation is used extensively to identify opportunities to increase oil production in oil deposits. Oil recovery is improved by lowering the oil viscosity by injecting steam or hot water, typical processes are steam soaks and steam flooding. A recent application of reservoir simulation is the modeling of coalbed methane production and this application requires a specialized CBM simulator. Traditional finite difference simulators dominate both theoretical and practical work in reservoir simulation, conventional FD simulation is underpinned by three physical concepts, conservation of mass, isothermal fluid phase behavior, and the Darcy approximation of fluid flow through porous media. Thermal simulators add conservation of energy to this list, allowing temperatures to change within the reservoir, 2-D approximations are also used in various conceptual models, such as cross-sections and 2-D radial grid models. Theoretically, finite difference models permit discretization of the reservoir using both structured and more complex unstructured grids to accurately represent the geometry of the reservoir, local grid refinements are also a feature provided by many simulators to more accurately represent the near wellbore multi-phase flow effects. This refined meshing near wellbores is extremely important when analyzing issues such as water, Other types of simulators include finite element and streamline. Representation of faults and their transmissibilities are advanced features provided in many simulators, in these models, inter-cell flow transmissibilities must be computed for non-adjacent layers outside of conventional neighbor-to-neighbor connections