1.
Two-dimensional space
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In physics and mathematics, two-dimensional space is a geometric model of the planar projection of the physical universe. The two dimensions are commonly called length and width, both directions lie in the same plane. A sequence of n numbers can be understood as a location in n-dimensional space. When n =2, the set of all locations is called two-dimensional space or bi-dimensional space. Each reference line is called an axis or just axis of the system. The coordinates can also be defined as the positions of the projections of the point onto the two axes, expressed as signed distances from the origin. The idea of system was developed in 1637 in writings by Descartes and independently by Pierre de Fermat. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Later, the plane was thought of as a field, where any two points could be multiplied and, except for 0, divided and this was known as the complex plane. The complex plane is called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand, although they were first described by Norwegian-Danish land surveyor, Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. In mathematics, analytic geometry describes every point in space by means of two coordinates. Two perpendicular coordinate axes are given which cross each other at the origin and they are usually labeled x and y. Another widely used system is the polar coordinate system, which specifies a point in terms of its distance from the origin. In two dimensions, there are infinitely many polytopes, the polygons, the first few regular ones are shown below, The Schläfli symbol represents a regular p-gon. The regular henagon and regular digon can be considered degenerate regular polygons and they can exist nondegenerately in non-Euclidean spaces like on a 2-sphere or a 2-torus. There exist infinitely many non-convex regular polytopes in two dimensions, whose Schläfli symbols consist of rational numbers and they are called star polygons and share the same vertex arrangements of the convex regular polygons

2.
Cartesian plane
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Each reference line is called a coordinate axis or just axis of the system, and the point where they meet is its origin, usually at ordered pair. The coordinates can also be defined as the positions of the projections of the point onto the two axis, expressed as signed distances from the origin. One can use the principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes. In general, n Cartesian coordinates specify the point in an n-dimensional Euclidean space for any dimension n and these coordinates are equal, up to sign, to distances from the point to n mutually perpendicular hyperplanes. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes can be described by Cartesian equations, algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, a familiar example is the concept of the graph of a function. Cartesian coordinates are also tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering. They are the most common system used in computer graphics, computer-aided geometric design. Nicole Oresme, a French cleric and friend of the Dauphin of the 14th Century, used similar to Cartesian coordinates well before the time of Descartes. The adjective Cartesian refers to the French mathematician and philosopher René Descartes who published this idea in 1637 and it was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery. Both authors used a single axis in their treatments and have a length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes La Géométrie was translated into Latin in 1649 by Frans van Schooten and these commentators introduced several concepts while trying to clarify the ideas contained in Descartes work. Many other coordinate systems have developed since Descartes, such as the polar coordinates for the plane. The development of the Cartesian coordinate system would play a role in the development of the Calculus by Isaac Newton. The two-coordinate description of the plane was later generalized into the concept of vector spaces. Choosing a Cartesian coordinate system for a one-dimensional space – that is, for a straight line—involves choosing a point O of the line, a unit of length, and an orientation for the line. An orientation chooses which of the two half-lines determined by O is the positive, and which is negative, we say that the line is oriented from the negative half towards the positive half

3.
Sheet metal
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Sheet metal is metal formed by an industrial process into thin, flat pieces. It is one of the forms used in metalworking and it can be cut. Countless everyday objects are fabricated from sheet metal, thicknesses can vary significantly, extremely thin thicknesses are considered foil or leaf, and pieces thicker than 6 mm are considered plate. Sheet metal is available in flat pieces or coiled strips, the coils are formed by running a continuous sheet of metal through a roll slitter. The thickness of metal is in the USA commonly specified by a traditional. The larger the number, the thinner the metal. Commonly used steel sheet metal ranges from 30 gauge to about 7 gauge, in the rest of the world, the sheet metal thickness is given in millimeters. There are many different metals that can be made into metal, such as aluminium, brass, copper, steel, tin, nickel. Sheet metal of iron and other materials with high permeability, also known as laminated steel cores, has applications in transformers. Historically, an important use of metal was in plate armor worn by cavalry. Sheet metal workers are known as tin bashers, a name derived from the hammering of panel seams when installing tin roofs. Grade 304 is the most common of the three grades and it offers good corrosion resistance while maintaining formability and weldability. Available finishes are #2B, #3, and #4, grade 303 is not available in sheet form. Grade 316 possesses more corrosion resistance and strength at elevated temperatures than 304 and it is commonly used for pumps, valves, chemical equipment, and marine applications. Available finishes are #2B, #3, and #4, grade 410 is a heat treatable stainless steel, but it has a lower corrosion resistance than the other grades. It is commonly used in cutlery, the only available finish is dull. Grade 430 is popular grade, low cost alternative to series 300s grades, used when high corrosion resistance is not a primary criteria. Common grade for appliance products, often with a brushed finish, aluminum is also a popular metal used in sheet metal due to its flexibility, wide range of options, cost effectiveness, and other properties

4.
Image
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Images may be two-dimensional, such as a photograph or screen display, or three-dimensional, such as a statue or hologram. They may be captured by optical devices – such as cameras, mirrors, lenses, telescopes, microscopes, etc. and natural objects and phenomena, such as the human eye or water. The word image is used in the broader sense of any two-dimensional figure such as a map, a graph. A volatile image is one that only for a short period of time. This may be a reflection of an object by a mirror, a fixed image, also called a hard copy, is one that has been recorded on a material object, such as paper or textile by photography or any other digital process. A mental image exists in a mind, as something one remembers or imagines. The subject of an image need not be real, it may be a concept, such as a graph, function. For example, Sigmund Freud claimed to have dreamed purely in aural-images of dialogs, a still image is a single static image, as distinguished from a kinetic image. This phrase is used in photography, visual media and the industry to emphasize that one is not talking about movies. A film still is a taken on the set of a movie or television program during production. In literature, imagery is a picture which appeals to the senses. It can both be figurative and literal, a moving image is typically a movie or video, including digital video. It could also be an animated display such as a zoetrope, library of Congress – Format Descriptions for Still Images Image Processing – Online Open Research Group Legal Issues Regarding Images Image Copyright Case

5.
Diagram
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A diagram is a symbolic representation of information according to some visualization technique. Diagrams have been used since ancient times, but became prevalent during the Enlightenment. Sometimes, the uses a three-dimensional visualization which is then projected onto a two-dimensional surface. The word graph is used as a synonym for diagram. Specific kind of display, This is the genre that shows qualitative data with shapes that are connected by lines, arrows. In science the term is used in both ways, on the other hand, Lowe defined diagrams as specifically abstract graphic portrayals of the subject matter they represent. Or in Halls words diagrams are simplified figures, caricatures in a way and these simplified figures are often based on a set of rules. The basic shape according to White can be characterized in terms of elegance, clarity, ease, pattern, simplicity, elegance is basically determined by whether or not the diagram is the simplest and most fitting solution to a problem. g. Many of these types of diagrams are generated using diagramming software such as Visio. Chart Diagrammatic reasoning Diagrammatology List of graphical methods Mathematical diagram Plot commons, michael Anderson, Peter Cheng, Volker Haarslev. Theory and Application of Diagrams, First International Conference, Diagrams 2000, edinburgh, Scotland, UK, September 1–3,2000. Garcia, M The Diagrams of Architecture

6.
Logo
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A logo is a graphic mark, emblem, or symbol commonly used by commercial enterprises, organizations, and even individuals to aid and promote instant public recognition. There are purely graphic emblems, symbols, icons and logos, in the days of hot metal typesetting, a logotype was one word cast as a single piece of type. By extension, the term was used for a uniquely set. At the level of communication and in common usage, a companys logo is today often synonymous with its trademark or brand. The arts were expanding in purpose—from expression and decoration of an artistic, storytelling nature, to a differentiation of brands, consultancies and trades-groups in the commercial arts were growing and organizing, by 1890, the US had 700 lithographic printing firms employing more than 8,000 people. Artistic credit tended to be assigned to the company, as opposed to the individual artists who usually performed less important jobs. Playful children’s books, authoritative newspapers, and conversational periodicals developed their own visual and editorial styles for unique, as printing costs decreased, literacy rates increased, and visual styles changed, the Victorian decorative arts led to an expansion of typographic styles and methods of representing businesses. A renewal of interest in craftsmanship and quality also provided the artists and companies with a greater interest in credit, leading to the creation of unique logos and marks. By the 1950s, Modernism had shed its roots as an artistic movement in Europe to become an international, commercialized movement with adherents in the United States. Modernist-inspired logos proved successful in the era of mass visual communication ushered in by television, improvements in printing technology, the current era of logo design began in the 1870s with the first abstract logo, the Bass red triangle. As of 2014, many corporations, products, brands, services, agencies, as a result, only a few of the thousands of ideograms in circulation are recognizable without a name. Ideograms and symbols may be effective than written names, especially for logos translated into many alphabets in increasingly globalized markets. For instance, a written in Arabic script might have little resonance in most European markets. By contrast, ideograms keep the general nature of a product in both markets. In non-profit areas, the Red Cross exemplifies a well-known emblem that does not need an accompanying name, the red cross and red crescent are among the best-recognized symbols in the world. National Red Cross and Red Crescent Societies and their Federation as well as the International Committee of the Red Cross include these symbols in their logos, branding can aim to facilitate cross-language marketing. Consumers and potential consumers can identify the Coca-Cola name written in different alphabets because of the standard color, the text was written in Spencerian Script, which was a popular writing style when the Coca Cola Logo was being designed. Since a logo is the visual entity signifying an organization, logo design is an important area of graphic design, a logo is the central element of a complex identification system that must be functionally extended to all communications of an organization

7.
Glyph
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In typography, a glyph /ˈɡlɪf/ is an elemental symbol within an agreed set of symbols, intended to represent a readable character for the purposes of writing. In Turkish, however, it is not a glyph because that language has two versions of the letter i, with and without a dot. In Japanese syllabaries, a number of the characters are made up of more than one separate mark, however, in some cases, additional marks fulfill the role of diacritics, to differentiate distinct characters. In general, a diacritic is a glyph, even if it is contiguous with the rest of the character, two or more glyphs which have the same significance, whether used interchangeably or chosen depending on context, are called allographs of each other. The term has been used in English since 1727, borrowed from glyphe, from the Greek γλυφή, glyphē, carving, and the verb γλύφειν, glýphein, to hollow out, engrave, carve. The word glyph first came to widespread European attention with the engravings, in archaeology, a glyph is a carved or inscribed symbol. It may be a pictogram or ideogram, or part of a system such as a syllable. In 1897 Dana Evans discovered glyphs written on rocks in the Colorado Desert and these ancient characters have been called the most enlightening discovery in Native American History in the 19th Century. In typography, a glyph has a different definition, it is the specific shape, design. The same is true in computing, in computing as well as typography, the term character refers to a grapheme or grapheme-like unit of text, as found in natural language writing systems. The range of glyphs required increases correspondingly, in summary, in typography and computing, a glyph is a graphical unit. In graphonomics, the glyph is used for a noncharacter. Most typographic glyphs originate from the characters of a typeface, in the mobile text input technologies, Glyph is a family of text input methods based on the decomposition of letters into basic shapes. In role-playing games, the glyph is sometimes used alongside the word rune in describing magical drawings or etchings. Runes often refer to placing the image on an object or person to empower it, whereas the magic in a glyph lies dormant and is only triggered when the glyph is read or approached

8.
Typeface
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In metal typesetting, a font is a particular size, weight and style of a typeface. Each font was a set of type, one piece for each glyph. In modern usage, with the advent of digital typography, font is frequently synonymous with typeface, in particular, the use of vector or outline fonts means that different sizes of a typeface can be dynamically generated from one design. The word font derives from Middle French fonte melted, a casting, the term refers to the process of casting metal type at a type foundry. In a manual printing house the word font would refer to a set of metal type that would be used to typeset an entire page. Unlike a digital typeface it would not include a definition of each character. A font when bought new would often be sold as 12pt 14A 34a, meaning that it would be a size 12-point font containing 14 uppercase As, given the name upper and lowercase because of which case the metal type was located in, otherwise known as majuscule and minuscule. The rest of the characters would be provided in quantities appropriate for the distribution of letters in that language. Some metal type characters required in typesetting, such as dashes, spaces and line-height spacers, were not part of a specific font, line spacing is still often called leading, because the strips used for line spacing were made of lead. In the 1880s–90s, hot lead typesetting was invented, in which type was cast as it was set, either piece by piece or in entire lines of type at one time. In European alphabetic scripts, i. e. Latin, Cyrillic and Greek, the main properties are the stroke width, called weight, the style or angle. The regular or standard font is sometimes labeled roman, both to distinguish it from bold or thin and from italic or oblique. The keyword for the default, regular case is often omitted for variants and never repeated, otherwise it would be Bulmer regular italic, Bulmer bold regular, Roman can also refer to the language coverage of a font, acting as a shorthand for Western European. Different fonts of the same typeface may be used in the work for various degrees of readability and emphasis. The weight of a font is the thickness of the character outlines relative to their height. A typeface may come in fonts of many weights, from ultra-light to extra-bold or black, four to six weights are not unusual, many typefaces for office, web and non-professional use come with just a normal and a bold weight which are linked together. If no bold weight is provided, many renderers support faking a bolder font by rendering the outline a second time at an offset, the base weight differs among typefaces, that means one normal font may appear bolder than some other normal font. For example, fonts intended to be used in posters are often quite bold by default while fonts for long runs of text are rather light, therefore, weight designations in font names may differ in regard to the actual absolute stroke weight or density of glyphs in the font

9.
2D computer graphics
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2D computer graphics is the computer-based generation of digital images—mostly from two-dimensional models and by techniques specific to them. The word may stand for the branch of science that comprises such techniques. This representation is more flexible since it can be rendered at different resolutions to suit different output devices. For these reasons, documents and illustrations are often stored or transmitted as 2D graphic files, 2D computer graphics started in the 1950s, based on vector graphics devices. These were largely supplanted by raster-based devices in the following decades, the PostScript language and the X Window System protocol were landmark developments in the field. 2D graphics models may combine geometric models, digital images, text to be typeset, mathematical functions and equations and these components can be modified and manipulated by two-dimensional geometric transformations such as translation, rotation, scaling. In object-oriented graphics, the image is described indirectly by an object endowed with a self-rendering method—a procedure which assigns colors to the pixels by an arbitrary algorithm. Complex models can be built by combining simpler objects, in the paradigms of object-oriented programming, in Euclidean geometry, a translation moves every point a constant distance in a specified direction. A translation can be described as a motion, other rigid motions include rotations and reflections. A translation can also be interpreted as the addition of a constant vector to every point, a translation operator is an operator T δ such that T δ f = f. If v is a vector, then the translation Tv will work as Tv = p + v. If T is a translation, then the image of a subset A under the function T is the translate of A by T, the translate of A by Tv is often written A + v. In a Euclidean space, any translation is an isometry, the set of all translations forms the translation group T, which is isomorphic to the space itself, and a normal subgroup of Euclidean group E. The quotient group of E by T is isomorphic to the orthogonal group O, E / T ≅ O, thus we write the 3-dimensional vector w = using 4 homogeneous coordinates as w =. Similarly, the product of matrices is given by adding the vectors. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In linear algebra, a matrix is a matrix that is used to perform a rotation in Euclidean space. R = rotates points in the xy-Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system

10.
Three-dimensional space
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Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space, when n =3, the set of all such locations is called three-dimensional Euclidean space. It is commonly represented by the symbol ℝ3 and this serves as a three-parameter model of the physical universe in which all known matter exists. However, this space is one example of a large variety of spaces in three dimensions called 3-manifolds. Furthermore, in case, these three values can be labeled by any combination of three chosen from the terms width, height, depth, and breadth. In mathematics, analytic geometry describes every point in space by means of three coordinates. Three coordinate axes are given, each perpendicular to the two at the origin, the point at which they cross. They are usually labeled x, y, and z, below are images of the above-mentioned systems. Two distinct points determine a line. Three distinct points are either collinear or determine a unique plane, four distinct points can either be collinear, coplanar or determine the entire space. Two distinct lines can intersect, be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a plane, so skew lines are lines that do not meet. Two distinct planes can either meet in a line or are parallel. Three distinct planes, no pair of which are parallel, can meet in a common line. In the last case, the three lines of intersection of each pair of planes are mutually parallel, a line can lie in a given plane, intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane that are parallel to the given line, a hyperplane is a subspace of one dimension less than the dimension of the full space. The hyperplanes of a space are the two-dimensional subspaces, that is

11.
Decal
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A decal or transfer is a plastic, cloth, paper or ceramic substrate that has printed on it a pattern or image that can be moved to another surface upon contact, usually with the aid of heat or water. The word is short for decalcomania, which is the English version of the French word décalcomanie, a decal is composed of the following layers from top to bottom, A paper or film face-stock makes up the top layer of the labelstock. The printing is done on the side of the facestock. An adhesive layer is applied to the bottom of the face stock, a silicone or release coating layer is applied to the upper side of the backing material. A paper or film liner provides the bottom layer of the labelstock, an RFID circuit can be included in the paper or film face stock. Different variations of decals include, water-slide or water-dip, and vinyl peel-and-stick, a water-slide decal is screen-printed on a layer of water-soluble adhesive on a water-resistant paper, that must first be dipped in water prior to its application. Upon contact with water, the glue is loosened and the decal can be removed from its backing, overlong exposure, however, dissolves the glue completely causing the decal to fail to adhere. A peel-and-stick decal is actually not a decal as described above, but a vinyl sticker with adhesive backing, the sign industry calls these peel-and-stick vinyl stickers vinyl-cut-decals. Mass-production of vinyl decals starts with large rolls of vinyl sheet, vinyl is fed through a plotter or large-format printer/cutter which prints the desired image and cuts out the desired shapes. Designs are typically created using specialized software and sent to the machines electronically. After the patterns are cut, excess vinyl on the sheet is removed in a process called weeding, finally, a paper pre-mask can be applied to the top of the vinyl design allowing easy application of multiple letters and shapes. Decals are commonly used on hot rod automobiles and plastic models and they are also used on guitars as a way of personalizing them. Government agencies use decals on vehicles for identification and these decals are referred to as fleet markings and are required by law on all fire and law enforcement vehicles in the US. Most fleet markings are created from reflective vinyl with an adhesive backing that is applied in a peel-and-stick manner

12.
Car
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A car is a wheeled, self-powered motor vehicle used for transportation and a product of the automotive industry. The year 1886 is regarded as the year of the modern car. In that year, German inventor Karl Benz built the Benz Patent-Motorwagen, cars did not become widely available until the early 20th century. One of the first cars that was accessible to the masses was the 1908 Model T, an American car manufactured by the Ford Motor Company. Cars were rapidly adopted in the United States of America, where they replaced animal-drawn carriages and carts, cars are equipped with controls used for driving, parking, passenger comfort and safety, and controlling a variety of lights. Over the decades, additional features and controls have been added to vehicles, examples include rear reversing cameras, air conditioning, navigation systems, and in car entertainment. Most cars in use in the 2010s are propelled by a combustion engine. Both fuels cause air pollution and are blamed for contributing to climate change. Vehicles using alternative fuels such as ethanol flexible-fuel vehicles and natural gas vehicles are also gaining popularity in some countries, electric cars, which were invented early in the history of the car, began to become commercially available in 2008. There are costs and benefits to car use, the costs of car usage include the cost of, acquiring the vehicle, interest payments, repairs and auto maintenance, fuel, depreciation, driving time, parking fees, taxes, and insurance. The costs to society of car use include, maintaining roads, land use, road congestion, air pollution, public health, health care, road traffic accidents are the largest cause of injury-related deaths worldwide. The benefits may include transportation, mobility, independence. The ability for humans to move flexibly from place to place has far-reaching implications for the nature of societies and it was estimated in 2010 that the number of cars had risen to over 1 billion vehicles, up from the 500 million of 1986. The numbers are increasing rapidly, especially in China, India, the word car is believed to originate from the Latin word carrus or carrum, or the Middle English word carre. In turn, these originated from the Gaulish word karros, the Gaulish language was a branch of the Brythoic language which also used the word Karr, the Brythonig language evolved into Welsh where Car llusg and car rhyfel still survive. It originally referred to any wheeled vehicle, such as a cart, carriage. Motor car is attested from 1895, and is the formal name for cars in British English. Autocar is a variant that is attested from 1895

13.
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

14.
Boundary representation
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In solid modeling and computer-aided design, boundary representation—often abbreviated as B-rep or BREP—is a method for representing shapes using the limits. A solid is represented as a collection of connected surface elements, Boundary representation models are composed of two parts, topology and geometry. The main topological items are, faces, edges and vertices, a face is a bounded portion of a surface, an edge is a bounded piece of a curve and a vertex lies at a point. Other elements are the shell, the loop and loop-edge links which are used to create the edge circuits, the edges are like the edges of a table, bounding a surface portion. In addition to the Boolean operations, B-rep has extrusion, chamfer, blending, drafting, shelling, tweaking, the basic method for BREP was developed independently in the early 1970s by both Ian C. Braid in Cambridge and Bruce G. Baumgart at Stanford, braid continued his work with the research solid modeller BUILD which was the forerunner of many research and commercial solid modelling systems. Braid worked on the commercial systems ROMULUS, the forerunner of Parasolid, Parasolid and ACIS are the basis for many of todays commercial CAD systems. In Finland, Martti Mäntylä produced a solid modelling system called GWB, in the USA Eastman and Weiler were also working on Boundary Representation and in Japan Professor Fumihiko Kimura and his team at Tokyo University also produced their own B-rep modelling system. Initially CSG was used by several commercial systems because it was easier to implement, the advent of reliable commercial B-rep kernel systems like Parasolid and ACIS, mentioned above, has led to widespread adoption of B-rep for CAD. Boundary representation is essentially a local representation connecting faces, edges and vertices, an extension of this was to group sub-elements of the shape into logical units called geometric features, or simply features. Pioneering work was done by Kyprianou in Cambridge also using the BUILD system and continued and extended by Jared, features are the basis of many other developments, allowing high-level geometric reasoning about shape for comparison, process-planning, manufacturing, etc. Boundary representation has also extended to allow special, non-solid model types called non-manifold models. An important sub-class of non-manifold models are sheet objects which are used to represent thin-plate objects, in the world of data-exchange, STEP, the Standard for the Exchange of Product Model data also defines some data models for boundary representations. The common generic topological and geometric models are defined in ISO 10303-42 Geometric and topological representation

15.
Boolean operations on polygons
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Boolean operations on polygons are a set of Boolean operations operating on one or more sets of polygons in computer graphics. These sets of operations are used in computer graphics, CAD. Using bitmaps in modeling polygon shapes has many drawbacks, one of the drawbacks is that the memory usage can be very large, since the resolution of polygons is proportional to the number of bits used to represent polygons. The higher the resolution is desired, the more the number of bits is required, modern implementations for Boolean operations on polygons tend to use plane sweep algorithms. A list of papers using plane sweep algorithms for Boolean operations on polygons can be found in References below, Boolean operations on convex polygons and monotone polygons of the same direction may be performed in linear time. Software Michael Leonov has compiled a comparison of polygon clippers, angus Johnson has also compared three clipping libraries. SINED GmbH has compared performance and memory utilization of three polygon clippers, a comparison of 5 clipping libraries at rogue-modron. blogspot. com A commercial library for 3D Boolean operations, sgCore C++/C# library. The comp. graphics. algorithms FAQ, solutions to problems with 2D. Matthias Kramms gfxpoly, a free C library for 2D polygons, klaas Holwerdas Boolean, a C++ library for 2D polygons. David Kennisons Polypack, a FORTRAN library based on the Vatti algorithm, klamer Schuttes Clippoly, a polygon clipper written in C++. Michael Leonovs poly_Boolean, a C++ library, which extends the Schutte algorithm. Angus Johnsons Clipper, an open-source freeware library thats based on the Vatti algorithm, geoLib, a commercial library available in C++ and C#. Alan Murtas GPC, General Polygon Clipper library, polygonLib, C++ and COM libraries for 2D polygons

16.
Two-dimensional graph
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A two-dimensional graph is a set of points in two-dimensional space. If the points are real and if Cartesian coordinates are used, with real variables on the axes, each point in the graph depicts the values of two real variables. Alternatively, each point in a graph may depict the value of a complex variable. The function could be a function or a transcendental function. For example, the graph of the polynomial f = x 3 −9 x is. If this set is plotted on a Cartesian plane, the result is a curve, in some cases the relation between two real variables cannot be written in the form y = f. In other words, it is not a function, nevertheless, the set of all points given by the relation is still a two-dimensional graph, as in the accompanying graph of the circle 2 +2 =1. An image of a curve is also a two-dimensional graph. In some contexts it is useful to graph two or more together in the same diagram. An example is the supply and demand graph commonly used in economics, two-dimensional geometric shapes are sets of points bounded by line segments or curves, so a shape can also be constructed by graphs of several equations of its boundary. Polygons are the shapes that are bounded by line segments. These can be visualized by using two-dimensional graphs, graphs of two polygons, a parallelogram and a right triangle, are shown here along with the graph of a circle. Graph Three-dimensional graph List of two-dimensional geometric shapes Analytic geometry Cartesian coordinate system Euclidean space Coordinate system Dimension