1.
Parametric equation
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In mathematics, parametric equations define a group of quantities as functions of one or more independent variables called parameters. For example, the equations x = cos t y = sin t form a representation of the unit circle. Parametric equations are used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a parameter is often labeled t, however. Parameterizations are non-unique, more than one set of equations can specify the same curve. In kinematics, objects paths through space are described as parametric curves. Used in this way, the set of equations for the objects coordinates collectively constitute a vector-valued function for position. Such parametric curves can then be integrated and differentiated termwise, thus, if a particles position is described parametrically as r = then its velocity can be found as v = r ′ = and its acceleration as a = r ″ =. Another important use of equations is in the field of computer-aided design. For example, consider the three representations, all of which are commonly used to describe planar curves. These problems can be addressed by rewriting the non-parametric equations in parametric form, numerous problems in integer geometry can be solved using parametric equations. A classical such solution is Euclids parametrization of right triangles such that the lengths of their sides a, b, by multiplying a, b and c by an arbitrary positive integer, one gets a parametrization of all right triangles whose three sides have integer lengths. Converting a set of equations to a single equation involves eliminating the variable t from the simultaneous equations x = x, y = y. If one of these equations can be solved for t, the expression obtained can be substituted into the equation to obtain an equation involving x and y only. If the parametrization is given by rational functions x = p r, y = q r, where p, q, r are set-wise coprime polynomials, in some cases there is no single equation in closed form that is equivalent to the parametric equations. The simplest equation for a parabola, y = x 2 can be parameterized by using a free parameter t, and setting x = t, y = t 2 f o r − ∞ < t < ∞. More generally, any given by an explicit equation y = f can be parameterized by using a free parameter t. A more sophisticated example is the following, consider the unit circle which is described by the ordinary equation x 2 + y 2 =1
2.
Computer graphics
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Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with help from specialized hardware and software. It is a vast and recent area in computer science, the phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes referred to as CGI. The overall methodology depends heavily on the sciences of geometry, optics. Computer graphics is responsible for displaying art and image data effectively and meaningfully to the user and it is also used for processing image data received from the physical world. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, the term computer graphics has been used a broad sense to describe almost everything on computers that is not text or sound. Such imagery is found in and on television, newspapers, weather reports, a well-constructed graph can present complex statistics in a form that is easier to understand and interpret. In the media such graphs are used to illustrate papers, reports, thesis, many tools have been developed to visualize data. Computer generated imagery can be categorized into different types, two dimensional, three dimensional, and animated graphics. As technology has improved, 3D computer graphics have become more common, Computer graphics has emerged as a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Screens could display art since the Lumiere brothers use of mattes to create effects for the earliest films dating from 1895. New kinds of displays were needed to process the wealth of information resulting from such projects, early projects like the Whirlwind and SAGE Projects introduced the CRT as a viable display and interaction interface and introduced the light pen as an input device. Douglas T. Ross of the Whirlwind SAGE system performed an experiment in 1954 in which a small program he wrote captured the movement of his finger. Electronics pioneer Hewlett-Packard went public in 1957 after incorporating the decade prior, and established ties with Stanford University through its founders. This began the transformation of the southern San Francisco Bay Area into the worlds leading computer technology hub - now known as Silicon Valley. The field of computer graphics developed with the emergence of computer graphics hardware, further advances in computing led to greater advancements in interactive computer graphics. In 1959, the TX-2 computer was developed at MITs Lincoln Laboratory, the TX-2 integrated a number of new man-machine interfaces
3.
Dimension
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In physics and mathematics, the dimension of a mathematical space is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one only one coordinate is needed to specify a point on it – for example. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces, in classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a space but not the one that was found necessary to describe electromagnetism. The four dimensions of spacetime consist of events that are not absolutely defined spatially and temporally, Minkowski space first approximates the universe without gravity, the pseudo-Riemannian manifolds of general relativity describe spacetime with matter and gravity. Ten dimensions are used to string theory, and the state-space of quantum mechanics is an infinite-dimensional function space. The concept of dimension is not restricted to physical objects, high-dimensional spaces frequently occur in mathematics and the sciences. They may be parameter spaces or configuration spaces such as in Lagrangian or Hamiltonian mechanics, in mathematics, the dimension of an object is an intrinsic property independent of the space in which the object is embedded. This intrinsic notion of dimension is one of the ways the mathematical notion of dimension differs from its common usages. The dimension of Euclidean n-space En is n, when trying to generalize to other types of spaces, one is faced with the question what makes En n-dimensional. One answer is that to cover a ball in En by small balls of radius ε. This observation leads to the definition of the Minkowski dimension and its more sophisticated variant, the Hausdorff dimension, for example, the boundary of a ball in En looks locally like En-1 and this leads to the notion of the inductive dimension. While these notions agree on En, they turn out to be different when one looks at more general spaces, a tesseract is an example of a four-dimensional object. The rest of this section some of the more important mathematical definitions of the dimensions. A complex number has a real part x and an imaginary part y, a single complex coordinate system may be applied to an object having two real dimensions. For example, an ordinary two-dimensional spherical surface, when given a complex metric, complex dimensions appear in the study of complex manifolds and algebraic varieties. The dimension of a space is the number of vectors in any basis for the space. This notion of dimension is referred to as the Hamel dimension or algebraic dimension to distinguish it from other notions of dimension
4.
Vector graphics
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Vector graphics is the use of polygons to represent images in computer graphics. Vector graphics are based on vectors, which lead through locations called control points or nodes, one of the first uses of vector graphic displays was the US SAGE air defense system. Vector graphics systems were retired from U. S. en route air traffic control in 1999. Vector graphics were used on the TX-2 at the MIT Lincoln Laboratory by computer graphics pioneer Ivan Sutherland to run his program Sketchpad in 1963. Subsequent vector graphics systems, most of which iterated through dynamically modifiable stored lists of drawing instructions, include the IBM2250, Imlac PDS-1, storage scope displays, such as the Tektronix 4014, could display vector images but not modify them without first erasing the display. In computer typography, modern outline fonts describe printable characters by cubic or quadratic mathematical curves with control points, nevertheless, bitmap fonts are still in use. Processing outline character data in sophisticated fashion to create satisfactory bitmaps for rendering is called hinting, although the term implies suggestion, the process is deterministic, and done by executable code, essentially a special-purpose computer language. While automatic hinting is possible, results can be inferior to that done by experts, although a typical plot might easily require a few thousand paper motions, back and forth, the paper doesnt slip. In a tiny roll-fed plotter made by Alps in Japan, teeth on thin sprockets indented the paper near its edges on the first pass, some Hewlett-Packard pen plotters had two-axis pen carriers and stationery paper. However, the moving-paper H-P plotters had grit wheels which, on the first pass, indented the paper surface, present-day vector graphic files such as engineering drawings are typically printed as bitmaps, after vector-to-raster conversion. The term vector graphics is used today in the context of two-dimensional computer graphics. It is one of several modes an artist can use to create an image on a raster display, Vector graphics can be uploaded to online databases for other designers to download and manipulate, speeding up the creative process. Other modes include text, multimedia, and 3D rendering, virtually all modern 3D rendering is done using extensions of 2D vector graphics techniques. Plotters used in technical drawing still draw vectors directly to paper, the World Wide Web Consortium standard for vector graphics is Scalable Vector Graphics. The standard is complex and has been slow to be established at least in part owing to commercial interests. Many web browsers now have support for rendering SVG data. In recent years, SVG has become a significant format that is independent of the resolution of the rendering device. SVG files are essentially printable text that describes both straight and curved paths, as well as other attributes, wikipedia prefers SVG for images such as simple maps, line illustrations, coats of arms, and flags, which generally are not like photographs or other continuous-tone images
5.
Raster graphics
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Raster images are stored in image files with varying formats. A raster is technically characterized by the width and height of the image in pixels, the printing and prepress industries know raster graphics as contones. The opposite to contones is line work, usually implemented as vector graphics in digital systems, the word raster has its origins in the Latin rastrum, which is derived from radere. It originates from the scan of cathode ray tube video monitors. By association, it can refer to a rectangular grid of pixels. The word rastrum is now used to refer to a device for drawing musical staff lines, most modern computers have bitmapped displays, where each on-screen pixel directly corresponds to a small number of bits in memory. The screen is refreshed simply by scanning through pixels and coloring them according to set of bits. The refresh procedure, being speed critical, is implemented by dedicated circuitry. Most computer images are stored in raster graphics formats or compressed variations, including GIF, JPEG, and PNG, three-dimensional voxel raster graphics are employed in video games and are also used in medical imaging such as MRI scanners. GIS programs commonly use rasters that encode geographic data in the values as well as the pixel locations. Raster graphics are resolution dependent, meaning they cannot scale up to a resolution without loss of apparent quality. This property contrasts with the capabilities of graphics, which easily scale up to the quality of the device rendering them. Raster graphics deal more practically than vector graphics with photographs and photo-realistic images, typically, a resolution of 150 to 300 PPI works well for 4-color process printing. However, for printing technologies that perform color mixing through dithering rather than through overprinting, printer DPI and image PPI have a different meaning. Thus, for instance, printing an image at 250 PPI may actually require a printer setting of 1200 DPI, when an image is rendered in a raster-based image editor, the image is composed of millions of pixels. At its core, an image editor works by manipulating each individual pixel. Most pixel-based image editors work using the RGB color model, and this article is based on material taken from the Free On-line Dictionary of Computing prior to 1 November 2008 and incorporated under the relicensing terms of the GFDL, version 1.3 or later
6.
Animation
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Animation is the process of making the illusion of motion and the illusion of change by means of the rapid display of a sequence of images that minimally differ from each other. The illusion—as in motion pictures in general—is thought to rely on the phi phenomenon, animators are artists who specialize in the creation of animation. Animation can be recorded with either analogue media, a book, motion picture film, video tape, digital media, including formats with animated GIF, Flash animation. To display animation, a camera, computer, or projector are used along with new technologies that are produced. Animation creation methods include the traditional animation creation method and those involving stop motion animation of two and three-dimensional objects, paper cutouts, puppets and clay figures, Images are displayed in a rapid succession, usually 24,25,30, or 60 frames per second. Computer animation processes generating animated images with the general term computer-generated imagery, 3D animation uses computer graphics, while 2D animation is used for stylistic, low bandwidth and faster real-time renderings. An earthen goblet discovered at the site of the 5, 200-year-old Shahr-e Sūkhté in southeastern Iran, the artifact bears five sequential images depicting a Persian Desert Ibex jumping up to eat the leaves of a tree. They may, of course, refer to Chinese shadow puppets, in the 19th century, the phenakistoscope, zoetrope and praxinoscope were introduced. A thaumatrope is a toy with a small disk with different pictures on each side. The phenakistoscope was invented simultaneously by Belgian Joseph Plateau and Austrian Simon von Stampfer in 1831, the phenakistoscope consists of a disk with a series of images, drawn on radi evenly space around the center of the disk. John Barnes Linnett patented the first flip book in 1868 as the kineograph, the first animated projection was created in France, by Charles-Émile Reynaud, who was a French science teacher. Reynaud created the Praxinoscope in 1877 and the Théâtre Optique in December 1888, on 28 October 1892, he projected the first animation in public, Pauvre Pierrot, at the Musée Grévin in Paris. This film is notable as the first known instance of film perforations being used. His films were not photographed, they were drawn directly onto the transparent strip, in 1900, more than 500,000 people had attended these screenings. Stuart Blackton, who, because of that, is considered the father of American animation, in Europe, the French artist, Émile Cohl, created the first animated film using what came to be known as traditional animation creation methods - the 1908 Fantasmagorie. The film largely consisted of a figure moving about and encountering all manner of morphing objects. There were also sections of live action in which the hands would enter the scene. The film was created by drawing each frame on paper and then shooting each frame onto negative film, the author of the first puppet-animated film was the Russian-born director Wladyslaw Starewicz, known as Ladislas Starevich
7.
User interface
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The user interface, in the industrial design field of human–computer interaction, is the space where interactions between humans and machines occur. Examples of this concept of user interfaces include the interactive aspects of computer operating systems, hand tools, heavy machinery operator controls. The design considerations applicable when creating user interfaces are related to or involve such disciplines as ergonomics and psychology. Generally, the goal of user interface design is to produce a user interface makes it easy, efficient. This generally means that the needs to provide minimal input to achieve the desired output. Other terms for user interface are man–machine interface and when the machine in question is a computer human–computer interface, the user interface or human–machine interface is the part of the machine that handles the human–machine interaction. Membrane switches, rubber keypads and touchscreens are examples of the part of the Human Machine Interface which we can see. In complex systems, the interface is typically computerized. The term human–computer interface refers to this kind of system, in the context of computing the term typically extends as well to the software dedicated to control the physical elements used for human-computer interaction. The engineering of the interfaces is enhanced by considering ergonomics. The corresponding disciplines are human factors engineering and usability engineering, which is part of systems engineering, tools used for incorporating human factors in the interface design are developed based on knowledge of computer science, such as computer graphics, operating systems, programming languages. Nowadays, we use the graphical user interface for human–machine interface on computers. There is a difference between a user interface and an interface or a human–machine interface. A human-machine interface is typically local to one machine or piece of equipment, an operator interface is the interface method by which multiple equipment that are linked by a host control system is accessed or controlled. The system may expose several user interfaces to serve different kinds of users, for example, a computerized library database might provide two user interfaces, one for library patrons and the other for library personnel. The user interface of a system, a vehicle or an industrial installation is sometimes referred to as the human–machine interface. HMI is a modification of the original term MMI, in practice, the abbreviation MMI is still frequently used although some may claim that MMI stands for something different now. Another abbreviation is HCI, but is commonly used for human–computer interaction
8.
Graphical user interface
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GUIs were introduced in reaction to the perceived steep learning curve of command-line interfaces, which require commands to be typed on a computer keyboard. The actions in a GUI are usually performed through direct manipulation of the graphical elements, beyond computers, GUIs are used in many handheld mobile devices such as MP3 players, portable media players, gaming devices, smartphones and smaller household, office and industrial controls. Designing the visual composition and temporal behavior of a GUI is an important part of application programming in the area of human–computer interaction. Its goal is to enhance the efficiency and ease of use for the logical design of a stored program. Methods of user-centered design are used to ensure that the language introduced in the design is well-tailored to the tasks. The visible graphical interface features of an application are sometimes referred to as chrome or GUI, typically, users interact with information by manipulating visual widgets that allow for interactions appropriate to the kind of data they hold. The widgets of an interface are selected to support the actions necessary to achieve the goals of users. A model–view–controller allows a structure in which the interface is independent from and indirectly linked to application functions. This allows users to select or design a different skin at will, good user interface design relates to users more, and to system architecture less. Large widgets, such as windows, usually provide a frame or container for the main presentation content such as a web page, smaller ones usually act as a user-input tool. A GUI may be designed for the requirements of a market as application-specific graphical user interfaces. By the 1990s, cell phones and handheld game systems also employed application specific touchscreen GUIs, newer automobiles use GUIs in their navigation systems and multimedia centers, or navigation multimedia center combinations. Sample graphical desktop environments A GUI uses a combination of technologies and devices to provide a platform that users can interact with, a series of elements conforming a visual language have evolved to represent information stored in computers. This makes it easier for people with few computer skills to work with, the most common combination of such elements in GUIs is the windows, icons, menus, pointer paradigm, especially in personal computers. The WIMP style of interaction uses a virtual device to represent the position of a pointing device, most often a mouse. Available commands are compiled together in menus, and actions are performed making gestures with the pointing device, a window manager facilitates the interactions between windows, applications, and the windowing system. The windowing system handles hardware devices such as pointing devices, graphics hardware, window managers and other software combine to simulate the desktop environment with varying degrees of realism. Smaller mobile devices such as personal assistants and smartphones typically use the WIMP elements with different unifying metaphors, due to constraints in space
9.
France
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France, officially the French Republic, is a country with territory in western Europe and several overseas regions and territories. The European, or metropolitan, area of France extends from the Mediterranean Sea to the English Channel and the North Sea, Overseas France include French Guiana on the South American continent and several island territories in the Atlantic, Pacific and Indian oceans. France spans 643,801 square kilometres and had a population of almost 67 million people as of January 2017. It is a unitary republic with the capital in Paris. Other major urban centres include Marseille, Lyon, Lille, Nice, Toulouse, during the Iron Age, what is now metropolitan France was inhabited by the Gauls, a Celtic people. The area was annexed in 51 BC by Rome, which held Gaul until 486, France emerged as a major European power in the Late Middle Ages, with its victory in the Hundred Years War strengthening state-building and political centralisation. During the Renaissance, French culture flourished and a colonial empire was established. The 16th century was dominated by civil wars between Catholics and Protestants. France became Europes dominant cultural, political, and military power under Louis XIV, in the 19th century Napoleon took power and established the First French Empire, whose subsequent Napoleonic Wars shaped the course of continental Europe. Following the collapse of the Empire, France endured a succession of governments culminating with the establishment of the French Third Republic in 1870. Following liberation in 1944, a Fourth Republic was established and later dissolved in the course of the Algerian War, the Fifth Republic, led by Charles de Gaulle, was formed in 1958 and remains to this day. Algeria and nearly all the colonies became independent in the 1960s with minimal controversy and typically retained close economic. France has long been a centre of art, science. It hosts Europes fourth-largest number of cultural UNESCO World Heritage Sites and receives around 83 million foreign tourists annually, France is a developed country with the worlds sixth-largest economy by nominal GDP and ninth-largest by purchasing power parity. In terms of household wealth, it ranks fourth in the world. France performs well in international rankings of education, health care, life expectancy, France remains a great power in the world, being one of the five permanent members of the United Nations Security Council with the power to veto and an official nuclear-weapon state. It is a member state of the European Union and the Eurozone. It is also a member of the Group of 7, North Atlantic Treaty Organization, Organisation for Economic Co-operation and Development, the World Trade Organization, originally applied to the whole Frankish Empire, the name France comes from the Latin Francia, or country of the Franks
10.
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
11.
Renault
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Groupe Renault is a French multinational automobile manufacturer established in 1899. The company produces a range of cars and vans, and in the past has manufactured trucks, tractors, tanks, buses/coaches, the Renault–Nissan Alliance is the fourth-largest automotive group. Renault has a 43. 4% controlling stake in Nissan of Japan, a 37% indirectly-owned stake in AvtoVAZ of Russia, Renault also owns subsidiaries RCI Banque, Renault Retail Group and Motrio. Renault has various joint ventures, including Oyak-Renault, Renault Pars, Carlos Ghosn is the current chairman and CEO. The French government owns a 19. 73% share of Renault as of April 2015, Renault Trucks, previously known as Renault Véhicules Industriels, has been part of Volvo Trucks since 2001. Renault Agriculture became 100% owned by German agricultural equipment manufacturer CLAAS in 2008, together Renault and Nissan invested €4 billion in eight electric vehicles over three to four years beginning in 2011. Renault is known for its role in sport, particularly rallying, Formula 1. Its early work on mathematical curve modeling for car bodies is important in the history of computer graphics, the Renault corporation was founded in 1899 as Société Renault Frères by Louis Renault and his brothers Marcel and Fernand. While Louis handled design and production, Marcel and Fernand managed the business, the first Renault car, the Renault Voiturette 1CV, was sold to a friend of Louis father after giving him a test ride on 24 December 1898. In 1903, Renault began to manufacture its own engines, until then it had purchased them from De Dion-Bouton, the first major volume sale came in 1905 when Société des Automobiles de Place bought Renault AG1 cars to establish a fleet of taxis. These vehicles were used by the French military to transport troops during World War I which earned them the nickname Taxi de la Marne. By 1907, a significant percentage London and Paris taxis had been built by Renault, Renault was also the best-selling foreign brand in New York in 1907 and 1908. In 1908 the company produced 3,575 units, becoming the countrys largest car manufacturer, the brothers recognised the value of publicity that participation in motor racing could generate for their vehicles. Renault made itself known through succeeding in the first city-to-city races held in Switzerland, both Louis and Marcel raced company vehicles, but Marcel was killed in an accident during the 1903 Paris-Madrid race. Although Louis never raced again, his company remained very involved, Louis took full control of the company as the only remaining brother in 1906 when Fernand retired for health reasons. Fernand died in 1909 and Louis became the owner, renaming the company Société des Automobiles Renault. Renault fostered its reputation for innovation from very early on, at the time, cars were luxury items. The price of the smallest Renaults at the time were ₣3000 francs, in 1905 the company introduced mass-production techniques and Taylorism in 1913
12.
De Casteljau's algorithm
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In the mathematical field of numerical analysis, De Casteljaus algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau. De Casteljaus algorithm can also be used to split a single Bézier curve into two Bézier curves at an arbitrary parameter value, although the algorithm is slower for most architectures when compared with the direct approach, it is more numerically stable. A Bézier curve B can be written in Bernstein form as follows B = ∑ i =0 n β i b i, n, the result B is given by, B = β0. When evaluating a Bézier curve of degree n in 3-dimensional space with n+1 control points Pi B = ∑ i =0 n P i b i, n, t ∈ with P i, =, the geometric interpretation of De Casteljaus algorithm is straightforward. Consider a Bézier curve with control points P0, connecting the consecutive points we create the control polygon of the curve. Subdivide now each line segment of this polygon with the ratio t and this way you arrive at the new polygon having one fewer segment. Repeat the process until you arrive at the single point - this is the point of the corresponding to the parameter t. This algorithm not only evaluates the curve at t, but splits the curve into two pieces at t, and provides the equations of the two sub-curves in Bézier form, the interpretation given above is valid for a nonrational Bézier curve. The algorithm then proceeds as usual, interpolating in R4, the resulting four-dimensional points may be projected back into three-space with a perspective divide. In general, operations on a curve are equivalent to operations on a nonrational curve in a projective space. This representation as the control points and weights is often convenient when evaluating rational curves. Bézier curves De Boors algorithm Horner scheme to evaluate polynomials in monomial form Clenshaw algorithm to evaluate polynomials in Chebyshev form Farin, Gerald & Hansford, natic, MA, A K Peters, Ltd
13.
Numerical stability
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In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context, one is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra the principal concern is instabilities caused by proximity to singularities of various kinds, some numerical algorithms may damp out the small fluctuations in the input data, others might magnify such errors. Calculations that can be not to magnify approximation errors are called numerically stable. One of the tasks of numerical analysis is to try to select algorithms which are robust – that is to say. Typically, an algorithm involves an approximate method, and in some cases one could prove that the algorithm would approach the solution in some limit. There are different ways to formalize the concept of stability, the following definitions of forward, backward, and mixed stability are often used in numerical linear algebra. Consider the problem to be solved by the algorithm as a function f mapping the data x to the solution y. The result of the algorithm, say y*, will deviate from the true solution y. The main causes of error are round-off error and truncation error, the forward error of the algorithm is the difference between the result and the solution, in this case, Δy = y* − y. The backward error is the smallest Δx such that f = y*, in other words, the forward and backward error are related by the condition number, the forward error is at most as big in magnitude as the condition number multiplied by the magnitude of the backward error. In many cases, it is natural to consider the relative error | Δ x | | x | instead of the absolute error Δx. The algorithm is said to be stable if the backward error is small for all inputs x. Of course, small is a term and its definition will depend on the context. Often, we want the error to be of the order as, or perhaps only a few orders of magnitude bigger than. The usual definition of numerical stability uses a general concept, called mixed stability, which combines the forward error. An algorithm is stable in this if it solves a nearby problem approximately. Hence, a stable algorithm is always stable
14.
Adobe Illustrator
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Adobe Illustrator is a vector graphics editor developed and marketed by Adobe Systems. The latest version, Illustrator CC2017, is the twenty-first generation in the product line, Adobe Illustrator was first developed for the Apple Macintosh in December 1986 as a commercialization of Adobes in-house font development software and PostScript file format. Adobe Illustrator is the product of Adobe Photoshop. Early magazine advertisements referred to the product as the Adobe Illustrator, Illustrator 88, the product name for version 1.7, was released in 1988 and introduced many new tools and features. BYTE in 1989 listed Illustrator 88 as among the Distinction winners of the BYTE Awards, although during its first decade Adobe developed Illustrator primarily for Macintosh, it sporadically supported other platforms. In the early 1990s, Adobe released versions of Illustrator for NeXT, Silicon Graphics, and Sun Solaris platforms, the first version of Illustrator for Windows, version 2.0, was released in early 1989 and flopped. Version 4 was, however, the first version of Illustrator to support editing in preview mode, version 6 was the last truly Macintosh version of Illustrator. The interface changed radically with the version to make it more Windows-friendly. The changes remained until CS6 when some small steps were taken to restore the app to a slightly more Mac-like interface. With the introduction of Illustrator 7 in 1997, Adobe made critical changes in the interface with regard to path editing. Illustrator also began to support TrueType, effectively ending the font wars between PostScript Type 1 and TrueType, like Photoshop, Illustrator also began supporting plug-ins, greatly and quickly extending its abilities. With true user interface parity between Macintosh and Windows versions starting with 7.0, designers could finally standardize on Illustrator, corel did port CorelDRAW6.0 to the Macintosh in late 1996, but it was received as too little, too late. Designers tended to prefer Illustrator, Drawcord, or Free Hand based on which software they learned first, as an example, there are capabilities in Freehand still not available in Illustrator. Famously, Aldus did a comparison matrix between its own Freehand, Illustrator and Draw, and Draws one win was that it came with three different clip art views of the human pancreas, Adobe bought Aldus in 1994 for PageMaker. As a result, Macromedia acquired FreeHand in 1995 from its developer, Altsys. Using the Adobe SVG Viewer, introduced in 2000, allowed users to view SVG images in most major browsers until it was discontinued in 2009, native support for SVG was not complete in all major browsers until Internet Explorer 9 in 2011. Version 9 included a feature, similar to that within Adobes discontinued product Streamline. Illustrator CS was the first version to include 3-dimensional capabilities allowing users to extrude or revolve shapes to create simple 3D objects, Illustrator CS2 was available for both the Mac OS X and Microsoft Windows operating systems
15.
Convex hull
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In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or in a Euclidean space is the smallest convex set that contains X. With the latter definition, convex hulls may be extended from Euclidean spaces to arbitrary real vector spaces, they may also be generalized further, to oriented matroids. The algorithmic problem of finding the convex hull of a set of points in the plane or other low-dimensional Euclidean spaces is one of the fundamental problems of computational geometry. A set of points is defined to be if it contains the line segments connecting each pair of its points. The convex hull of a given set X may be defined as The minimal convex set containing X The intersection of all convex sets containing X The set of all combinations of points in X. The union of all simplices with vertices in X and it is not obvious that the first definition makes sense, why should there exist a unique minimal convex set containing X, for every X. Thus, it is exactly the unique minimal convex set containing X. Each convex set containing X must contain all convex combinations of points in X, in fact, according to Carathéodorys theorem, if X is a subset of an N-dimensional vector space, convex combinations of at most N +1 points are sufficient in the definition above. If the convex hull of X is a set, then it is the intersection of all closed half-spaces containing X. The hyperplane separation theorem proves that in case, each point not in the convex hull can be separated from the convex hull by a half-space. However, there exist convex sets, and convex hulls of sets, more abstractly, the convex-hull operator Conv has the characteristic properties of a closure operator, It is extensive, meaning that the convex hull of every set X is a superset of X. It is non-decreasing, meaning that, for two sets X and Y with X ⊆ Y, the convex hull of X is a subset of the convex hull of Y. It is idempotent, meaning that for every X, the hull of the convex hull of X is the same as the convex hull of X. The convex hull of a point set S is the set of all convex combinations of its points. For each choice of coefficients, the convex combination is a point in the convex hull. Expressing this as a formula, the convex hull is the set. The convex hull of a point set S ⊊ R n forms a convex polygon when n =2. Each point x i in S that is not in the hull of the other points is called a vertex of Conv . In fact, every convex polytope in R n is the hull of its vertices
16.
Affine transformation
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In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation, an affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. If X and Y are affine spaces, then every affine transformation f, X → Y is of the form x ↦ M x + b, unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear, all Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, another way to deal with affine transformations systematically is to select a point as the origin, then, any affine transformation is equivalent to a linear transformation followed by a translation. An affine map f, A → B between two spaces is a map on the points that acts linearly on the vectors. In symbols, f determines a linear transformation φ such that and we can interpret this definition in a few other ways, as follows. If an origin O ∈ A is chosen, and B denotes its image f ∈ B, the conclusion is that, intuitively, f consists of a translation and a linear map. In other words, f preserves barycenters, as shown above, an affine map is the composition of two functions, a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. If A is a matrix, = is equivalent to the following y → = A x → + b →, the above-mentioned augmented matrix is called an affine transformation matrix, or projective transformation matrix. This representation exhibits the set of all affine transformations as the semidirect product of K n and G L. This is a group under the operation of composition of functions, ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate 1 to every vector, one considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the coordinate is 1. Thus the origin of the space can be found at. A translation within the space by means of a linear transformation of the higher-dimensional space is then possible
17.
Translation (geometry)
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In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same amount in a given direction. In Euclidean geometry a transformation is a correspondence between two sets of points or a mapping from one plane to another. )A translation can be described as a rigid motion. 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. 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, a translation is an affine transformation with no fixed points. Matrix multiplications always have the origin as a fixed point, similarly, the product of translation matrices is given by adding the vectors, T u T v = T u + v. Because addition of vectors is commutative, multiplication of matrices is therefore also commutative. In physics, translation is movement that changes the position of an object, for example, according to Whittaker, A translation is the operation changing the positions of all points of an object according to the formula → where is the same vector for each point of the object. When considering spacetime, a change of time coordinate is considered to be a translation, for example, the Galilean group and the Poincaré group include translations with respect to time
18.
Rotation
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A rotation is a circular movement of an object around a center of rotation. A three-dimensional object always rotates around a line called a rotation axis. If the axis passes through the center of mass, the body is said to rotate upon itself. A rotation about a point, e. g. the Earth about the Sun, is called a revolution or orbital revolution. The axis is called a pole, mathematically, a rotation is a rigid body movement which, unlike a translation, keeps a point fixed. This definition applies to rotations within both two and three dimensions All rigid body movements are rotations, translations, or combinations of the two, a rotation is simply a progressive radial orientation to a common point. That common point lies within the axis of that motion, the axis is 90 degrees perpendicular to the plane of the motion. If the axis of the rotation lies external of the body in question then the body is said to orbit, there is no fundamental difference between a “rotation” and an “orbit” and or spin. The key distinction is simply where the axis of the rotation lies and this distinction can be demonstrated for both “rigid” and “non rigid” bodies. If a rotation around a point or axis is followed by a rotation around the same point/axis. The reverse of a rotation is also a rotation, thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis and that is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the rotations are known as yaw, pitch. This terminology is used in computer graphics. In astronomy, rotation is an observed phenomenon. Stars, planets and similar bodies all spin around on their axes, the rotation rate of planets in the solar system was first measured by tracking visual features. Stellar rotation is measured through Doppler shift or by tracking active surface features and this rotation induces a centrifugal acceleration in the reference frame of the Earth which slightly counteracts the effect of gravity the closer one is to the equator
19.
Quadratic function
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A univariate quadratic function has the form f = a x 2 + b x + c, a ≠0 in the single variable x. The graph of a quadratic function is a parabola whose axis of symmetry is parallel to the y-axis. If the quadratic function is set equal to zero, then the result is a quadratic equation, the solutions to the univariate equation are called the roots of the univariate function. In general there can be a large number of variables, in which case the resulting surface is called a quadric. The adjective quadratic comes from the Latin word quadrātum, a term like x2 is called a square in algebra because it is the area of a square with side x. In general, a prefix indicates the number 4. Quadratum is the Latin word for square, the coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring. When using the quadratic polynomial, authors sometimes mean having degree exactly 2. If the degree is less than 2, this may be called a degenerate case, usually the context will establish which of the two is meant. Sometimes the word order is used with the meaning of degree, a quadratic polynomial may involve a single variable x, or multiple variables such as x, y, and z. Any single-variable quadratic polynomial may be written as a x 2 + b x + c, where x is the variable, and a, b, and c represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation a x 2 + b x + c =0, each quadratic polynomial has an associated quadratic function, whose graph is a parabola. Such polynomials are fundamental to the study of sections, which are characterized by equating the expression for f to zero. Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces, in linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space. F = a 2 + k is called the vertex form, the coefficient a is the same value in all three forms. To convert the standard form to factored form, one only the quadratic formula to determine the two roots r1 and r2. To convert the standard form to form, one needs a process called completing the square. To convert the factored form to form, one needs to multiply
20.
Cubic function
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In algebra, a cubic function is a function of the form f = a x 3 + b x 2 + c x + d, where a is nonzero. Setting f =0 produces an equation of the form. The solutions of this equation are called roots of the polynomial f, If all of the coefficients a, b, c, and d of the cubic equation are real numbers then there will be at least one real root. All of the roots of the equation can be found algebraically. The roots can also be found trigonometrically, alternatively, numerical approximations of the roots can be found using root-finding algorithms like Newtons method. The coefficients do not need to be complex numbers, much of what is covered below is valid for coefficients of any field with characteristic 0 or greater than 3. The solutions of the cubic equation do not necessarily belong to the field as the coefficients. For example, some cubic equations with rational coefficients have roots that are complex numbers. Cubic equations were known to the ancient Babylonians, Greeks, Chinese, Indians, Babylonian cuneiform tablets have been found with tables for calculating cubes and cube roots. The Babylonians could have used the tables to solve cubic equations, the problem of doubling the cube involves the simplest and oldest studied cubic equation, and one for which the ancient Egyptians did not believe a solution existed. Methods for solving cubic equations appear in The Nine Chapters on the Mathematical Art, in the 3rd century, the Greek mathematician Diophantus found integer or rational solutions for some bivariate cubic equations. In the 11th century, the Persian poet-mathematician, Omar Khayyám, in an early paper, he discovered that a cubic equation can have more than one solution and stated that it cannot be solved using compass and straightedge constructions. He also found a geometric solution, in the 12th century, the Indian mathematician Bhaskara II attempted the solution of cubic equations without general success. However, he gave one example of an equation, x3 + 12x = 6x2 +35. He used what would later be known as the Ruffini-Horner method to approximate the root of a cubic equation. He also developed the concepts of a function and the maxima and minima of curves in order to solve cubic equations which may not have positive solutions. He understood the importance of the discriminant of the equation to find algebraic solutions to certain types of cubic equations. Leonardo de Pisa, also known as Fibonacci, was able to approximate the positive solution to the cubic equation x3 + 2x2 + 10x =20
21.
PostScript
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PostScript is a page description language in the electronic publishing and desktop publishing business. It is a typed, concatenative programming language and was created at Adobe Systems by John Warnock, Charles Geschke, Doug Brotz, Ed Taft. The concepts of the PostScript language were seeded in 1976 when John Warnock was working at Evans & Sutherland, at that time John Warnock was developing an interpreter for a large three-dimensional graphics database of New York harbor. Warnock conceived the Design System language to process the graphics, concurrently, researchers at Xerox PARC had developed the first laser printer and had recognized the need for a standard means of defining page images. In 1975-76 Bob Sproull and William Newman developed the Press format, but Press, a data format rather than a language, lacked flexibility, and PARC mounted the Interpress effort to create a successor. In 1978 Evans & Sutherland asked Warnock to move from the San Francisco Bay Area to their headquarters in Utah. He then joined Xerox PARC to work with Martin Newell and they rewrote Design System to create J & M which was used for VLSI design and the investigation of type and graphics printing. This work later evolved and expanded into the Interpress language, Warnock left with Chuck Geschke and founded Adobe Systems in December 1982. They, together with Doug Brotz, Ed Taft and Bill Paxton created a language, similar to Interpress, called PostScript. At about this time they were visited by Steve Jobs, who urged them to adapt PostScript to be used as the language for driving laser printers. In March 1985, the Apple LaserWriter was the first printer to ship with PostScript, the combination of technical merits and widespread availability made PostScript a language of choice for graphical output for printing applications. For a time an interpreter for the PostScript language was a component of laser printers. However, the cost of implementation was high, computers output raw PS code that would be interpreted by the printer into an image at the printers natural resolution. This required high performance microprocessors and ample memory, the LaserWriter used a 12 MHz Motorola 68000, making it faster than any of the Macintosh computers to which it attached. When the laser printer engines themselves cost over a thousand dollars the added cost of PS was marginal, the first version of the PostScript language was released to the market in 1984. The term Level 1 was added when Level 2 was introduced, PostScript 3 came at the end of 1997, and along with many new dictionary-based versions of older operators, introduced better color handling, and new filters. Prior to the introduction of PostScript, printers were designed to print character output given the text—typically in ASCII—as input and this changed to some degree with the increasing popularity of dot matrix printers. The characters on these systems were drawn as a series of dots, dot matrix printers also introduced the ability to print raster graphics
22.
CorelDRAW
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CorelDraw is a vector graphics editor developed and marketed by Corel Corporation. It is also the name of Corels Graphics Suite, which bundles CorelDraw with bitmap-image editor Corel Photo-Paint as well as other graphics-related programs, the latest version is designated X8, and was released in March 2016. CorelDraw is designed to edit two-dimensional images such as logos and posters, in 1987, Corel hired software engineers Michel Bouillon and Pat Beirne to develop a vector-based illustration program to bundle with their desktop publishing systems. That program, CorelDraw, was released in 1989. CorelDraw 1. x and 2. x ran under Windows 2. x and 3.0, CorelDraw 3.0 came into its own with Microsofts release of Windows 3.1. The inclusion of TrueType in Windows 3, CorelDraw was originally developed for Microsoft Windows 3 and currently runs on Windows XP, Windows Vista, Windows 7, Windows 8 and Windows 10. The latest version, X8, was released on 15 March 2016, versions for Mac OS and Mac OS X were at one time available, but due to poor sales these were discontinued. The last port for Linux was version 9 and the last version for OS X was version 11, also, up until version 5, CorelDraw was developed for Windows 3. 1x, CTOS and OS/2. With version 6, CorelDraw introduced the automation of tasks using a Corel proprietary scripting language, with version 10, support for VBA was introduced for scripting by what Corel calls now macros. Corel recommends to no longer use the COREL Script language but only VBA, in its first versions, the CDR file format was a completely proprietary file format primarily used for vector graphic drawings, recognizable by the first two bytes of the file being WL. The actual data chunk of the RIFF remains a Corel proprietary format, F was the last valid hex digit, and the fver now indicates that the letter before does no longer stand for a hex digit. There is no publicly available CDR file format specification, other CorelDraw file formats include CorelDraw Compressed, CorelDraw Template and Corel Presentation Exchange. In December 2006 the sK1 open source project started to reverse-engineer the CDR format. The results and the first working snapshot of the CDR importer were presented at the Libre Graphics Meeting 2007 conference taking place in May 2007 in Montreal, later on the team parsed the structure of other Corel formats with the help of the open source CDR Explorer. As of 2008, the sK1 project claims to have the best import support for CorelDraw file formats among open source software programs, the sK1 project developed also the UniConvertor, a command line open source tool which supports conversion from CorelDraw ver. 7-X4 formats to other formats. UniConvertor is also used in Inkscape and Scribus open source projects as an tool for CorelDraw files importing. In 2007, Microsoft blocked CDR file format in Microsoft Office 2003 with the release of Service Pack 3 for Office 2003, Microsoft later apologized for inaccurately blaming the CDR file format and other formats for security problems in Microsoft Office and released some tools for solving this problem. In 2012 the joint LibreOffice/re-lab team implemented libcdr, a library for reading CDR files from v1 to the currently latest X7 version, the library has extensive support for shapes and their properties, including support for color management and spot colors, and has a basic support for text
23.
Inkscape
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Inkscape is a free and open-source vector graphics editor, it can be used to create or edit vector graphics such as illustrations, diagrams, line arts, charts, logos and complex paintings. Inkscapes primary vector graphics format is Scalable Vector Graphics, however many other formats can be imported and exported, Inkscape can render primitive vector shapes and text. These objects may be filled with colors, patterns, radial or linear color gradients and their borders may be stroked. Embedding and optional tracing of raster graphics is supported, enabling the editor to create vector graphics from photos. Created shapes can be manipulated with transformations, such as moving, rotating, scaling and skewing. Inkscape began in 2003 as a fork of the Sodipodi project. Sodipodi, developed since 1999, was based on Raph Leviens Gill. The Inkscape FAQ interprets the word Inkscape as a compound of ink, four former Sodipodi developers led the fork, they identified differences over project objectives, openness to third-party contributions, and technical disagreements as their reasons for forking. Notably, Inkscapes implementation of the SVG standard, although incomplete, has shown gradual improvement, since 2005 Inkscape has participated in the Google Summer of Code program. Up until the end of November 2007, Inkscapes bug tracking system was hosted on SourceForge, the basic objects in Inkscape are, Rectangles & Squares tool, creates rectangles and squares, corners of squares and rectangles can be rounded. 3D Boxes tool, creates 3D boxes that have adjustable XYZ perspectives, 3D boxes are in fact groups of paths and after ungrouping can be further modified. Circles/Ellipses/Arcs tool, circles and ellipses can be transformed into arcs, stars & Polygons tool, Multi-pointed stars with two radius control handles can be used to emulate spirographs. Polygons with one control handle can be used to create items based on the number of hexagons, pentagons. Spirals tool, creates spirals that have a number of turns, divergence, inner radius Pencil tool. Pen tool, creates a Bézier node-by-node curve and or line segments in the same path, calligraphy tool, creates freehand calligraphic or brush-like strokes, optionally the tool can use pressure and tilt readings from a graphics tablet. Text tool, creates texts that can use any of the Operating Systems outline, Text conversion to paths, Normal, Bold, Italic, Condensed and Heavy, Alignments, Superscript, Subscript, Vertical and Horizontal text are implemented. All text objects can be transformed via Line Spacing, Letter Spacing, Word Spacing, Horizontal Kerning, Vertical Shift, Text can be put along a path, flowed into a shape or spell checked. Bullet lists, numbered lists, indentations, and underlined text are not available as of version 0.91
24.
Smoothness
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In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain, differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives, consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer, the function f is said to be of class Ck if the derivatives f′, f′′. The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth, Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω, to put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous, thus, a C1 function is exactly a function whose derivative exists and is of class C0. In particular, Ck is contained in Ck−1 for every k, C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers. The function f = { x if x ≥0,0 if x <0 is continuous, because cos oscillates as x →0, f ’ is not continuous at zero. Therefore, this function is differentiable but not of class C1, the functions f = | x | k +1 where k is even, are continuous and k times differentiable at all x. But at x =0 they are not times differentiable, so they are of class Ck, the exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined, the function f is an example of a smooth function with compact support. Let n and m be some positive integers, if f is a function from an open subset of Rn with values in Rm, then f has component functions f1. Each of these may or may not have partial derivatives, the classes C∞ and Cω are defined as before. These criteria of differentiability can be applied to the functions of a differential structure. The resulting space is called a Ck manifold, if one wishes to start with a coordinate-independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is a map which approximates it at that point
25.
Adobe Flash
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Adobe Flash is a multimedia software platform used for production of animations, rich Internet applications, desktop applications, mobile applications and mobile games. Flash displays text, vector graphics and raster graphics to provide animations, video games and it allows streaming of audio and video, and can capture mouse, keyboard, microphone and camera input. Artists may produce Flash graphics and animations using Adobe Animate, Software developers may produce applications and video games using Adobe Flash Builder, FlashDevelop, Flash Catalyst, or any text editor when used with the Apache Flex SDK. End-users can view Flash content via Flash Player, AIR or third-party players such as Scaleform, Adobe Flash Player enables end-users to view Flash content using web browsers. Adobe Flash Lite enabled viewing Flash content on older smartphones, but has discontinued and superseded by Adobe AIR. The ActionScript programming language allows the development of interactive animations, video games, web applications, desktop applications, programmers can implement Flash software using an IDE such as Adobe Animate, Adobe Flash Builder, Adobe Director, FlashDevelop and Powerflasher FDT. Adobe AIR enables full-featured desktop and mobile applications to be developed with Flash, and published for Windows, macOS, Android, iOS, Xbox One, PlayStation 4, Nintendo Switch, and Wii U. Content-providers frequently used to use Flash to display streaming video, advertising and interactive content on web pages. However, after the 2000s, the usage of Flash on Web sites declined, in the early 2000s, Flash was widely installed on desktop computers, and was commonly used to display interactive web pages, online games, and to playback video and audio content. In 2005, YouTube was founded by former PayPal employees, between 2000 and 2010, numerous businesses used Flash-based websites to launch new products, or to create interactive company portals. Notable users include Nike, Hewlett-Packard, Nokia, General Electric, World Wildlife Fund, HBO, Cartoon Network, after Adobe introduced hardware-accelerated 3D for Flash, Flash websites saw a growth of 3D content for product demonstrations and virtual tours. In 2007, YouTube offered videos in HTML5 format to support the iPhone and iPad, after a controversy with Apple, Adobe stopped developing Flash Player for Mobile, focussing its efforts on Adobe AIR applications and HTML5 animation. In 2015, Google introduced Google Swiffy to convert Flash animation to HTML5, in 2015, YouTube switched to HTML5 technology on all devices, however it will preserve the Flash-based video player for older web browsers. After Flash 5 introduced ActionScript in 2000, developers combined the visual and programming capabilities of Flash to produce interactive experiences, such Web-based applications eventually came to be known as Rich Internet Applications. In 2004, Macromedia Flex was released, and specifically targeted the application development market, Flex introduced new user interface components, advanced data visualization components, data remoting, and a modern IDE. Flex competed with Asynchronous JavaScript and XML and Microsoft Silverlight during its tenure, Flex was upgraded to support integration with remote data sources, using AMF, BlazeDS, Adobe LiveCycle, Amazon Elastic Compute Cloud, and others. As of 2015, Flex applications can be published for desktop platforms using Adobe AIR, between 2006 and 2016, the Speedtest. net web service conducted over 9.0 billion speed tests using an RIA built with Adobe Flash. In 2016, the service shifted to HTML5 due to the availability of Adobe Flash Player on PCs
26.
Synfig
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Synfig Studio is a free and open source 2D vector graphics and timeline-based computer animation program created by Robert Quattlebaum with additional contributions by Adrian Bentley. Synfig began as the custom animation platform for Voria Studios, and in 2005 was released as free/open source software, the goal of the developers is to create a program that is capable of producing feature-film quality animation with fewer people and resources. The program offers an alternative to manual tweening so that the animator doesnt have to draw each, the software is capable of simulating soft-shading using curved gradients within an area so that the animator doesnt have to draw shading into every single frame. There is also a variety of other real-time effects that can be applied to layers or groups of layers like radial blurs. Other features include the ability to control and animate the width of lines at their control points. Synfig also works with High Dynamic Range Imaging, Synfig stores its animations in its own XML file format, often compressed with gzip. These files use the filename extension. sif. sifz or. sfg, the files store vector graphics data, embed or reference external bitmap images, and also a revision history of the project. Synfig can render to video formats such as AVI, Theora and MPEG, as well as animated graphics formats such as MNG and it can also render to a sequence of numbered image files, using formats such as PNG, BMP, PPM and OpenEXR. From version 0.62.00, Synfig has basic support for SVG import, from version 0.91 Inkscape can save as. sif file format. They have been making regular updates since then on the Morevna project site, free Software Magazine contained an article on the project. It was originally called SINFG, an acronym for SINFG Is Not A Fractal Generator. List of 2D animation software Official website
27.
Cascading Style Sheets
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Cascading Style Sheets is a style sheet language used for describing the presentation of a document written in a markup language. It can also display the web page differently depending on the size or viewing device. Readers can also specify a different style sheet, such as a CSS file stored on their own computer, to override the one the author specified. Changes to the design of a document can be applied quickly and easily, by editing a few lines in the CSS file they use. The CSS specification describes a priority scheme to determine which style rules apply if more than one rule matches against a particular element, in this so-called cascade, priorities are calculated and assigned to rules, so that the results are predictable. The CSS specifications are maintained by the World Wide Web Consortium, Internet media type text/css is registered for use with CSS by RFC2318. The W3C operates a free CSS validation service for CSS documents, CSS has a simple syntax and uses a number of English keywords to specify the names of various style properties. A style sheet consists of a list of rules, each rule or rule-set consists of one or more selectors, and a declaration block. In CSS, selectors declare which part of the markup a style applies to by matching tags, Selectors may apply to, all elements of a specific type, e. g. Classes and IDs are case-sensitive, start with letters, and can include alphanumeric characters and underscores. A class may apply to any number of instances of any elements, an ID may only be applied to a single element. Pseudo-classes are used in CSS selectors to permit formatting based on information that is not contained in the document tree. One example of a widely used pseudo-class is, hover, which identifies content only when the points to the visible element. It is appended to a selector as in a, hover or #elementid, a pseudo-class classifies document elements, such as, link or, visited, whereas a pseudo-element makes a selection that may consist of partial elements, such as, first-line or, first-letter. Selectors may be combined in ways to achieve great specificity and flexibility. Multiple selectors may be joined in a spaced list to specify elements by location, element type, id, class, the order of the selectors is important. For example, div. myClass applies to all elements of class myClass that are inside div elements, the following table provides a summary of selector syntax indicating usage and the version of CSS that introduced it. A declaration block consists of a list of declarations in braces, each declaration itself consists of a property, a colon, and a value. If there are multiple declarations in a block, a semi-colon must be inserted to separate each declaration, properties are specified in the CSS standard
28.
JavaScript
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JavaScript is a high-level, dynamic, untyped, and interpreted programming language. It has been standardized in the ECMAScript language specification, JavaScript is prototype-based with first-class functions, making it a multi-paradigm language, supporting object-oriented, imperative, and functional programming styles. JavaScript was influenced by programming languages such as Self and Scheme, JavaScript is also used in environments that are not Web-based, such as PDF documents, site-specific browsers, and desktop widgets. Newer and faster JavaScript virtual machines and platforms built upon them have increased the popularity of JavaScript for server-side Web applications. On the client side, developers have traditionally implemented JavaScript as an interpreted language, programmers also use JavaScript in video-game development, in crafting desktop and mobile applications, and in server-side network programming with run-time environments such as Node. js. In 1994, a company called Mosaic Communications was founded in Mountain View, California, however, it intentionally shared no code with NCSA Mosaic. The internal codename for the browser was Mozilla, which stood for Mosaic killer. The first version of the Web browser, Mosaic Netscape 0.9, was released in late 1994, within four months it had already taken three-quarters of the browser market and became the main browser for Internet in the 1990s. To avoid trademark problems with the NCSA, the browser was subsequently renamed Netscape Navigator in the same year. Netscape Communications realized that the Web needed to more dynamic. In 1995, the company recruited Brendan Eich with the goal of embedding the Scheme programming language into its Netscape Navigator, to defend the idea of JavaScript against competing proposals, the company needed a prototype. Eich wrote one in 10 days, in May 1995, there is a common misconception that JavaScript was influenced by an earlier Web page scripting language developed by Nombas named C--. Brendan Eich, however, had never heard of C-- before he created LiveScript, Nombas did pitch their embedded Web page scripting to Netscape, though Web page scripting was not a new concept, as shown by the ViolaWWW Web browser. Nombas later switched to offering JavaScript instead of C-- in their ScriptEase product and was part of the TC39 group that standardized ECMAScript, in December 1995, soon after releasing JavaScript for browsers, Netscape introduced an implementation of the language for server-side scripting with Netscape Enterprise Server. Since the mid-2000s, additional server-side JavaScript implementations have been introduced, Microsoft script technologies including VBScript and JScript were released in 1996. JScript, an implementation of Netscapes JavaScript, was part of Internet Explorer 3. JScript was also available for server-side scripting in Internet Information Server, JavaScript began to acquire a reputation for being one of the roadblocks to a cross-platform and standards-driven Web. Some developers took on the task of trying to make their sites work in both major browsers, but many could not afford the time
29.
TrueType
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TrueType is an outline font standard developed by Apple and Microsoft in the late 1980s as a competitor to Adobes Type 1 fonts used in PostScript. It has become the most common format for fonts on the classic Mac OS, macOS, with widely varying rendering technologies in use today, pixel-level control is no longer certain in a TrueType font. TrueType was known during its development stage, first by the codename Bass, the system was developed and eventually released as TrueType with the launch of Mac System 7 in May 1991. The initial TrueType outline fonts, four-weight families of Times Roman, Helvetica, Courier, Apple also replaced some of their bitmap fonts used by the graphical user-interface of previous Macintosh System versions with scalable TrueType outline-fonts. For compatibility with older systems, Apple shipped these fonts, a TrueType Extension, for compatibility with the Laserwriter II, Apple developed fonts like ITC Bookman and ITC Chancery in TrueType format. All of these fonts could now scale to all sizes on screen and printer, the early TrueType systems — being still part of Apples QuickDraw graphics subsystem — did not render Type 1 fonts on-screen as they do today. At the time, many users had already invested money in Adobes still proprietary Type 1 fonts. As part of Apples tactic of opening the font format versus Adobes desire to keep it closed to all but Adobe licensees, Apple licensed TrueType to Microsoft. Meanwhile, in exchange for TrueType, Apple got a license for TrueImage and this was never actually included in any Apple products when a later deal was struck between Apple and Adobe, where Adobe promised to put a TrueType interpreter in their PostScript printer boards. Part of Adobes response to learning that TrueType was being developed was to create the Adobe Type Manager software to scale Type 1 fonts for anti-aliased output on-screen. Although ATM initially cost money, rather than coming free with the operating system, Apple extended TrueType with the launch of TrueType GX in 1994, with additional tables in the sfnt which formed part of QuickDraw GX. This offered powerful extensions in two main areas, first was font axes, for example allowing fonts to be smoothly adjusted from light to bold or from narrow to extended — competition for Adobes multiple master technology. While maintaining the backing store of characters necessary for spell-checkers and text searching, however, the lack of user-friendly tools for making TrueType GX fonts meant there were no more than a handful of GX fonts. Much of the technology in TrueType GX, including morphing and substitution, few font-developers outside Apple attempt to make AAT fonts, instead, OpenType has become the dominant sfnt format despite its lack of support for axes or multiple masters. To ensure its adoption, Apple licensed TrueType to Microsoft for free. By 1991 Microsoft added TrueType into the Windows 3.1 operating system and this included the fonts that are standard with Windows to this day, Times New Roman, Arial and Courier New. Microsoft has heavily marketed ClearType, and sub-pixel rendering techniques for text are now used on all platforms. Microsoft also developed a smart font technology, named TrueType Open in 1994, while some fonts provided with the new operating systems are now in the OpenType format, most free or inexpensive third-party fonts use plain TrueType
30.
Asymptote (vector graphics language)
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Asymptote is a descriptive vector graphics language — developed by Andy Hammerlindl, John C. Bowman, and Tom Prince — which provides a natural coordinate-based framework for technical drawing, Asymptote runs on all major platforms. It is free software, available under the terms of the GNU Lesser General Public License, Asymptote typesets labels and equations with LaTeX, producing high-quality PostScript, PDF, SVG, or 3D PRC output. It is inspired by Metapost, but has a C++-like syntax and it provides a language for typesetting mathematical figures, just as TeX/LaTeX provides a language for typesetting equations. It is mathematically oriented, and uses the method and deferred drawing to solve overall size constraint issues between fixed-sized objects and objects that should scale with figure size. Asymptote fully generalizes MetaPost path construction algorithms to three dimensions, and compiles commands into virtual machine code for speed without sacrificing portability, high-level graphics commands are implemented in the Asymptote language itself, allowing them to be easily tailored to specific applications. It also appears to be the first software package to lift TeX into three dimensions and this allows Asymptote to be used as a 3D vector file format. The following source code allows you to draw a graph of the Heaviside function by means of the Asymptote language
31.
Metafont
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Metafont is a description language used to define vector fonts. It is also the name of the interpreter that executes Metafont code, Metafont was devised by Donald Knuth as a counterpart to his TeX typesetting system. One of the characteristics of Metafont is that all of the shapes of the glyphs are defined with geometrical equations, in particular, one can define a given point to be the intersection of a line segment and a Bézier cubic. Unlike more common font formats, a Metafont font is primarily made up of strokes with finite-width pens. Thus, rather than describing the outline of the glyph directly, some simpler Metafont fonts, such as the calligraphic mathematics fonts in the Computer Modern family, use a single pen stroke with a relatively large pen to define each visual stroke of the glyphs. Thus, by changing the value of one of these parameters at one location in the Metafont file, in addition, the Computer Modern typewriter and sans-serif fonts are defined using essentially the same Metafont file as the Roman font, but with different global parameters. Curves in Metafont are defined as cubic splines rather than quadratic, Metafont can render any kind of graphical output, not just glyphs. However, MetaPost and Asymptote are preferred for mathematical illustrations, Metafont is most commonly invoked without a direct request from the user. DVI files can contain references to typefaces, rather than the sets of raster or vector glyphs that other formats like PostScript allow. Consequently, the glyphs in the typefaces need to be accessed whenever a request is made to view, most TeX distributions are configured so that any fonts not currently available at the required resolution are generated by calls to Metafont. The fonts are stored for later reuse. Metafont can also be run interactively, and has commands for displaying on the screen the images it produces, Knuth has said that he uses Metafont as a kind of desk calculator for solving complicated equations, though he now uses MetaPost for mathematical illustrations. There are several tools for converting Metafont programs to PostScript Type 1 fonts, most make use of MetaPosts ability to convert a subset of Metafonts language into EPS outlines, which can subsequently be converted to PostScript Type 1 fonts. Generating vector outlines of pen strokes directly in Metafont is nontrivial, the Polish JNS team developed METATYPE1 for creating PostScript Type 1 fonts. Its big disadvantage is that no pens may be used which reduces the power of Metafont language. Initially, tool has been used by the Polish GUST typeface foundry to produce a modernized and extended variant of Computer Modern, mf2pt1, which is also based on Metapost, was developed by Scott Pakin with some contributions from Werner Lemberg. It also has several limitations detailed in its manual, the most significant one being the restriction that each glyph must be drawn entirely from closed paths, since FontForge supports scripting, this post-processing step can also be automated. This approach was used to produce Type 1 outlines for the MnSymbol mathematical font developed by Achim Blumensath, metaFog is a proprietary converter that can analytically convert pen strokes, but it requires manual post processing to eliminate degenerate cases, and it is not publicly available
32.
OpenType
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OpenType is a format for scalable computer fonts. It was built on its predecessor TrueType, retaining TrueTypes basic structure, OpenType is a registered trademark of Microsoft Corporation. The specification germinated at Microsoft, with Adobe Systems also contributing by the time of the announcement in 1996. OpenTypes origins date to Microsofts attempt to license Apples advanced typography technology GX Typography in the early 1990s and those negotiations failed, motivating Microsoft to forge ahead with its own technology, dubbed TrueType Open in 1994. Adobe joined Microsoft in those efforts in 1996, adding support for the glyph outline technology used in its Type 1 fonts and these efforts were intended by Microsoft and Adobe to supersede both Apples TrueType and Adobes Type 1 font formats. The name OpenType was chosen for the technologies, and the technology was announced later that year. Adobe and Microsoft continued to develop and refine OpenType over the next decade, adoption of the new standard reached formal approval in March 2007 as ISO Standard ISO/IEC 14496-22 called Open Font Format. It is also referred to as Open Font Format Specification. The initial standard was technically equivalent to OpenType 1.4 specification, the second edition of the Open Font Format was published in 2009 and was declared technically equivalent to the OpenType font format specification. Since then, the Open Font Format and the OpenType specification have continued to be maintained in sync, OFF is a free, publicly available standard. By 2001 hundreds of OpenType fonts were on the market, Adobe finished converting their entire font library to OpenType toward the end of 2002. As of early 2005, around 10,000 OpenType fonts had become available, by 2006, every major font foundry and many minor ones were developing fonts in OpenType format. In 2014, Adobe announced the creation of OpenType Collections, an OTC file bundles together multiple OpenType font files. This allows for efficient storage. For example, the Noto fonts CJK OTC is ~10MB smaller than the sum of the four separate OTFs of which it is comprised, additionally, OTFs can contain a maximum of 65,535 glyphs, the use of an OTC overcomes this limitation. OpenType uses the general sfnt structure of a TrueType font, but it adds several smartfont options that enhance the fonts typographic and language support capabilities. The glyph outline data in an OpenType font may be in one of two formats, either TrueType format outlines in a table, or Compact Font Format outlines in a CFF table. CFF outline data is based on the PostScript language Type 2 font format, however, the OpenType specification does not support the use of PostScript outlines in a TrueType Collection font file
33.
Microsoft Excel
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Microsoft Excel is a spreadsheet developed by Microsoft for Windows, macOS, Android and iOS. It features calculation, graphing tools, pivot tables, and a programming language called Visual Basic for Applications. It has been a widely applied spreadsheet for these platforms, especially since version 5 in 1993. Excel forms part of Microsoft Office, Microsoft Excel has the basic features of all spreadsheets, using a grid of cells arranged in numbered rows and letter-named columns to organize data manipulations like arithmetic operations. It has a battery of supplied functions to answer statistical, engineering, in addition, it can display data as line graphs, histograms and charts, and with a very limited three-dimensional graphical display. It allows sectioning of data to view its dependencies on various factors for different perspectives, Excel was not designed to be used as a database. Microsoft allows for a number of optional command-line switches to control the manner in which Excel starts, the Windows version of Excel supports programming through Microsofts Visual Basic for Applications, which is a dialect of Visual Basic. Programming with VBA allows spreadsheet manipulation that is awkward or impossible with standard spreadsheet techniques, programmers may write code directly using the Visual Basic Editor, which includes a window for writing code, debugging code, and code module organization environment. A common and easy way to generate VBA code is by using the Macro Recorder, the Macro Recorder records actions of the user and generates VBA code in the form of a macro. These actions can then be repeated automatically by running the macro, the macros can also be linked to different trigger types like keyboard shortcuts, a command button or a graphic. The actions in the macro can be executed from these types or from the generic toolbar options. The VBA code of the macro can also be edited in the VBE, advanced users can employ user prompts to create an interactive program, or react to events such as sheets being loaded or changed. Macro Recorded code may not be compatible between Excel versions, some code that is used in Excel 2010 can not be used in Excel 2003. Making a Macro that changes the colors and making changes to other aspects of cells may not be backward compatible. User-created VBA subroutines execute these actions and operate like macros generated using the macro recorder, from its first version Excel supported end user programming of macros and user defined functions.0. Beginning with version 5.0 Excel recorded macros in VBA by default, after version 5.0 that option was discontinued. All versions of Excel, including Excel 2010 are capable of running an XLM macro, Excel supports charts, graphs, or histograms generated from specified groups of cells. The generated graphic component can either be embedded within the current sheet and these displays are dynamically updated if the content of cells change
34.
Non-uniform rational B-spline
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Non-uniform rational basis spline is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces. It offers great flexibility and precision for handling both analytic and modeled shapes, NURBS are commonly used in computer-aided design, manufacturing, and engineering and are part of numerous industry wide standards, such as IGES, STEP, ACIS, and PHIGS. NURBS tools are found in various 3D modeling and animation software packages. They can be handled by the computer programs and yet allow for easy human interaction. NURBS surfaces are functions of two parameters mapping to a surface in three-dimensional space, the shape of the surface is determined by control points. NURBS surfaces can represent, in a form, simple geometrical shapes. T-splines and subdivision surfaces are more suitable for complex shapes because they reduce the number of control points twofold in comparison with the NURBS surfaces. In general, editing NURBS curves and surfaces is highly intuitive, control points are always either connected directly to the curve/surface, or act as if they were connected by a rubber band. Before computers, designs were drawn by hand on paper with various drafting tools, rulers were used for straight lines, compasses for circles, and protractors for angles. But many shapes, such as the curve of a ships bow. Although such curves could be drawn freehand at the drafting board, such large drawings were done with the help of flexible strips of wood, called splines. The shape could be tweaked by moving the ducks, in 1946, mathematicians started studying the spline shape, and derived the piecewise polynomial formula known as the spline curve or spline function. I. J. Schoenberg gave the spline function its name after its resemblance to the mechanical spline used by draftsmen. As computers were introduced into the process, the physical properties of such splines were investigated so that they could be modelled with mathematical precision. Pioneering work was done in France by Renault engineer Pierre Bézier, at first NURBS were only used in the proprietary CAD packages of car companies. Later they became part of computer graphics packages. Real-time, interactive rendering of NURBS curves and surfaces was first made available on Silicon Graphics workstations in 1989. In 1993, the first interactive NURBS modeller for PCs, called NöRBS, was developed by CAS Berlin, today most professional computer graphics applications available for desktop offer NURBS technology, which is most often realized by integrating a NURBS engine from a specialized company
35.
FreeType
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FreeType is a popular software development library used to render text onto bitmaps, and provides support for other font-related operations. The FreeType font rasterization engine is free and open source software with the source code dual-licensed under a BSD-like license, Ability to access the embedding and subsetting restriction information of fonts was added in 2.3.8. LCD filter became normalized and color-balanced in 2.6.2, TrueType TrueType bytecode interpreter was turned off in 2.0 Beta 8, and then it was turned off again in 2.0.2. The TrueType bytecode interpreter has been fixed to produce exactly the same output as FreeType 1. x in 2.1.0, the unpatented hinter became default in 2.3.0, which detects the need of the hinter by font name. The ability to detect patented bytecode instructions and toggling of unpatented hinter were added in 2.3.5, TrueType bytecode interpreter is enabled by default in 2.4.0. TrueType subpixel hinting support is implemented in 2.4.11, the subpixel hinting mode in version 40 of the bytecode engine was added in 2.6.4, and was disabled in 2.6.5, then enabled by default in 2.7. The GETVARIATION bytecode operator was implemented in 2.7, PostScript PostScript hinter was removed in 2.0.5. A new PostScript hinter was introduced in 2.0.6 to support native hints in PostScript Type 1, PostScript CID, hinting engine was updated in 2.1.3. New CFF parsing and hinting engine by Dave Arnold was introduced in 2.4.12, old CFF engine was disabled by default. CFF stem darkening can be configured via darkening-parameters property, CFF Stem darkening behaviour became configurable at build time in 2.5.4. Auto-hinter New auto-hinting module was added in 2.0 Beta 8, hinting engine was updated in 2.1.3. In 2.1.8, autohinting is disabled for glyphs that are distorted or mirrored. Auto-hinter was replaced by auto-fitter in FreeType 2.1.10, a new algorithm for CJK fonts was introduced in FreeType 2.2, based on Akito Hirais patch. It only works for fonts with a Unicode character map, improved support for serifs, corner cases, and spacing adjustments were introduced in 2.3.2. When computing blue zones, single-point contours are ignored in 2.3.3, in 2.3.5, default hinter for non-Latin scripts was switched to CJK hinting module. In 2.4.5, if automatic hinting is not explicitly disabled, in 2.5.1, Auto-hinter rendering for TrueType fonts was improved, especially in the range 20-40ppem. Support of HarfBuzz library 0.9.19 or newer was added in 2.5.3, auto-hinting was added to Devanagari and Telugu scripts in 2.5.4. Auto-hinting was added to Arabic and Thai scripts in 2.6, auto-hinting was added to Lao script, with Arabic script support being enhanced in 2.6.1
36.
Font rasterization
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Font rasterization is the process of converting text from a vector description to a raster or bitmap description. This often involves some anti-aliasing on screen text to make it smoother and easier to read and it may also involve hinting—information embedded in the font data that optimizes rendering details for particular character sizes. The simplest form of rasterization is simple line-drawing with no anti-aliasing of any sort, in Microsofts terminology, this is called bi-level rendering because no intermediate shades are used to draw the glyphs. This form of rendering is also called aliased or jagged and this is the fastest rendering method in the sense that it requires the least computational effort. However, it has the disadvantage that rendered glyphs may lose definition, as prototypical example, all versions of Microsoft Windows prior to Windows 95 only provided this type of built-in rasterizer. A more complicated approach is to use standard anti-aliasing techniques from computer graphics and this can be thought of as determining, for each pixel at the edges of the character, how much of that pixel the character occupies, and drawing that pixel with that degree of opacity. For example, when drawing a black letter on a white background, over-simple application of this procedure can produce blurry glyphs. For example, if the letter includes a line that should be one pixel wide. This blurriness trades clarity for accuracy, most computer displays have pixels made up of multiple subpixels. This is generally known as subpixel rendering, one proprietary implementation of subpixel rendering is Microsofts ClearType. In modern operating systems, rasterization is normally provided by a shared library common to many applications, such a shared library may be built into the operating system or the desktop environment, or may be added later. In principle, each application may use a different font rasterization library, Microsoft Windows has supported subpixel rendering since Windows XP. This has changed with Direct2D/DirectWrite shipping on Windows 7 and Windows Vista platform update, the result is that the on-screen display looks extremely similar to printed output, but can occasionally be difficult to read at smaller point sizes. Contrary to other rasterizers, Quartz ignores any Postscript or TrueType hints in the font, a simpler type of font antialiasing was introduced in Mac OS8.5, in 1998. Apples technique can be seen on Windows in older versions of Safari for Windows, in more recent versions, however, Apple has switched to using system settings by default. RISC OS includes font anti-aliasing, first introduced before January 1989 and it uses its own font rendering system, which favours accurate shapes over readability, with features such as scaffolding and hinting, sub-pixel positioning and background blending. PDF documents are usually rendered with Adobe CoolType, most other systems use the FreeType library, which falls somewhere between Microsofts and Apples implementations, it supports hinting and anti-aliasing, and optionally performs subpixel rendering. The Free fonts included with most Linux distributions look better with FreeTypes auto-hinting mode, d-Type Font Engine is an independent, proprietary and portable font rasterization library
37.
Affine space
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A Euclidean space is an affine space over the reals, equipped with a metric, the Euclidean distance. Therefore, in Euclidean geometry, a property is a property that may be proved in affine spaces. In an affine space, there is no distinguished point that serves as an origin, hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead displacement vectors, also called translation vectors or simply translations, thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, Any vector space may be considered as an affine space, and this amounts to forgetting the special role played by the zero vector. In this case, the elements of the space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the vector is called the origin. Adding a fixed vector to the elements of a subspace of a vector space produces an affine subspace. One commonly says that this affine subspace has been obtained by translating the linear subspace by the translation vector, in finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that space are the solutions of the corresponding homogeneous linear system. Linear subspaces, in contrast, always contain the origin of the vector space, the dimension of an affine space is defined as the dimension of the vector space of its translations. An affine space of one is an affine line. An affine space of dimension 2 is an affine plane, an affine subspace of dimension n –1 in an affine space or a vector space of dimension n is an affine hyperplane. The following characterization may be easier to understand than the formal definition. Imagine that Alice knows that a point is the actual origin. Two vectors, a and b, are to be added, similarly, Alice and Bob may evaluate any linear combination of a and b, or of any finite set of vectors, and will generally get different answers. However, if the sum of the coefficients in a combination is 1, then Alice. If Alice travels to λa + b then Bob can similarly travel to p + λ + = λa + b, under this condition, for all coefficients λ + =1, Alice and Bob describe the same point with the same linear combination, despite using different origins
38.
Line (geometry)
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The notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects, the straight line is that which is equally extended between its points. In modern mathematics, given the multitude of geometries, the concept of a line is tied to the way the geometry is described. When a geometry is described by a set of axioms, the notion of a line is left undefined. The properties of lines are determined by the axioms which refer to them. One advantage to this approach is the flexibility it gives to users of the geometry, thus in differential geometry a line may be interpreted as a geodesic, while in some projective geometries a line is a 2-dimensional vector space. This flexibility also extends beyond mathematics and, for example, permits physicists to think of the path of a light ray as being a line, to avoid this vicious circle certain concepts must be taken as primitive concepts, terms which are given no definition. In geometry, it is frequently the case that the concept of line is taken as a primitive, in those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy, in a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that a description or mental image of a notion is provided to give a foundation to build the notion on which would formally be based on the axioms. Descriptions of this type may be referred to, by some authors and these are not true definitions and could not be used in formal proofs of statements. The definition of line in Euclids Elements falls into this category, when geometry was first formalised by Euclid in the Elements, he defined a general line to be breadthless length with a straight line being a line which lies evenly with the points on itself. These definitions serve little purpose since they use terms which are not, themselves, in fact, Euclid did not use these definitions in this work and probably included them just to make it clear to the reader what was being discussed. In an axiomatic formulation of Euclidean geometry, such as that of Hilbert, for example, for any two distinct points, there is a unique line containing them, and any two distinct lines intersect in at most one point. In two dimensions, i. e. the Euclidean plane, two lines which do not intersect are called parallel, in higher dimensions, two lines that do not intersect are parallel if they are contained in a plane, or skew if they are not. Any collection of many lines partitions the plane into convex polygons. Lines in a Cartesian plane or, more generally, in affine coordinates, in two dimensions, the equation for non-vertical lines is often given in the slope-intercept form, y = m x + b where, m is the slope or gradient of the line. B is the y-intercept of the line, X is the independent variable of the function y = f