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Bézier Curve
A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces. The Bézier triangle is a special case of the latter. In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are not bound by the limi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adobe Flash
Adobe Flash (formerly Macromedia Flash and FutureSplash) is a multimedia Computing platform, software platform used for production of Flash animation, animations, rich web applications, application software, desktop applications, mobile apps, mobile games, and embedded web browser video players. Flash displays text, vector graphics, and raster graphics to provide animations, video games, and applications. It allows streaming of Flash Video, audio and video, and can capture mouse, keyboard, microphone, and camera input. Digital art, Artists may produce Flash graphics and animations using Adobe Animate (formerly known as Adobe Flash Professional). Programmer, Software developers may produce applications and video games using Adobe Flash Builder, FlashDevelop, Flash Catalyst, or any text editor combined with the Apache Flex SDK. End users view Flash content via Adobe Flash Player, Flash Player (for web browsers), Adobe AIR (for desktop or mobile apps), or third-party players such as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Citroën
Citroën () is a French automobile brand. The "Automobiles Citroën" manufacturing company was founded in March 1919 by André Citroën. Citroën is owned by Stellantis since 2021 and previously was part of the PSA Group after Peugeot acquired 89.95% share in 1976. Citroën's head office is located in the Stellantis Poissy Plant in Saint-Ouen-sur-Seine since 2021 (previously in Rueil-Malmaison) and its offices studies and research in Vélizy-Villacoublay, Poissy (CEMR), Carrières-sous-Poissy and Sochaux-Montbéliard. In 1934, the firm established its reputation for innovative technology with the Citroën Traction Avant, Traction Avant. This was the world's first car to be mass-produced with front-wheel drive, four-wheel independent suspension, as well as unibody construction, omitting a separate chassis, and instead using the body of the car itself as its main load-bearing structure. In 1954, they produced the world's first hydropneumatic self-levelling suspension system then, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
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, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Casteljau's Algorithm
In the mathematical field of numerical analysis, De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau. De Casteljau's 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. Definition A Bézier curve B (of degree n, with control points \beta_0, \ldots, \beta_n) can be written in Bernstein form as follows :B(t) = \sum_^\beta_b_(t), where b is a Bernstein basis polynomial :b_(t) = (1-t)^t^i. The curve at point t_0 can be evaluated with the recurrence relation :\beta_i^ := \beta_i,\ \ i=0,\ldots,n :\beta_i^ := \beta_i^ (1-t_0) + \beta_^ t_0,\ \ i = 0,\ldots,n-j,\ \ j= 1,\ldots,n Then, the evaluation of B at point t_0 can be evaluated in \binom operations. The result B(t_0) is given by :B(t_0) = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul De Casteljau
Paul de Casteljau (19 November 1930 – 24 March 2022) was a French physicist and mathematician. In 1959, while working at Citroën, he developed an algorithm for evaluating calculations on a certain family of curves, which would later be formalized and popularized by engineer Pierre Bézier, leading to the curves widely known as Bézier curves. He studied at École Normale Supérieure, and worked at Citroën from 1958 until his retirement in 1992. When he arrived there, "Specialists admitted that all electrical, electronic and mechanical problems had more or less been solved. All—except for one single formality which made up for 5%, but certainly not for 20% of the problem; in other words, how to express component parts by equations." He continued publishing in retirement. De Casteljau curves De Casteljau's algorithm is widely used, with some modifications, as it is the most robust and numerically stable method for evaluating polynomials. Other methods, such as Horner's me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate is . An example with three indeterminates is . Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry. Etymology The word ''polynomial'' join ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein Polynomial
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein. A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval , 1 became important in the form of Bézier curves. Definition The ''n''+1 Bernstein basis polynomials of degree ''n'' are defined as : b_(x) = \binom x^ \left( 1 - x \right)^, \quad \nu = 0, \ldots, n, where \tbinom is a binomial coefficient. So, for example, b_(x) = \tbinomx^2(1-x)^3 = 10x^2(1-x)^3. The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are: : \begin b_(x) & = 1, \\ b_(x) & = 1 - x, & b_(x) & = x \\ b_(x) & = (1 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graphical User Interface
The GUI ( "UI" by itself is still usually pronounced . or ), graphical user interface, is a form of user interface that allows users to interact with electronic devices through graphical icons and audio indicator such as primary notation, instead of text-based UIs, typed command labels or text navigation. GUIs were introduced in reaction to the perceived steep learning curve of CLIs ( 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. The term ''GUI'' tends not to be applied to other lower-display resolution types of interfaces, such as video games (where HUD (''head-up display'') is preferred), or not including flat screens like volumetric displays because ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
User Interface
In the industrial design field of human–computer interaction, a user interface (UI) is the space where interactions between humans and machines occur. The goal of this interaction is to allow effective operation and control of the machine from the human end, while the machine simultaneously feeds back information that aids the operators' decision-making process. Examples of this broad concept of user interfaces include the interactive aspects of computer operating systems, hand tools, heavy machinery operator controls and process 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 that makes it easy, efficient, and enjoyable (user-friendly) to operate a machine in the way which produces the desired result (i.e. maximum usability). This generally means that the operator needs to provide minimal input ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autodesk 3ds Max
Autodesk 3ds Max, formerly 3D Studio and 3D Studio Max, is a professional 3D computer graphics program for making 3D animations, models, games and images. It is developed and produced by Autodesk Media and Entertainment. It has modeling capabilities and a flexible plugin architecture and must be used on the Microsoft Windows platform. It is frequently used by video game developers, many TV commercial studios, and architectural visualization studios. It is also used for movie effects and movie pre-visualization. 3ds Max features shaders (such as ambient occlusion and subsurface scattering), dynamic simulation, particle systems, radiosity, normal map creation and rendering, global illumination, a customizable user interface, and its own scripting language. History The original 3D Studio product was created for the DOS platform by the Yost Group, and published by Autodesk. The release of 3D Studio made Autodesk's previous 3D rendering package AutoShade obsolete. After 3D Stu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Autodesk Maya
Autodesk Maya, commonly shortened to just Maya ( ), is a 3D computer graphics application that runs on Windows, macOS and Linux, originally developed by Alias and currently owned and developed by Autodesk. It is used to create assets for interactive 3D applications (including video games), animated films, TV series, and visual effects. History Maya was originally an animation product based on code from The Advanced Visualizer by Wavefront Technologies, Thomson Digital Image (TDI) Explore, PowerAnimator by Alias, and ''Alias Sketch!''. The IRIX-based projects were combined and animation features were added; the project codename was Maya. Walt Disney Feature Animation collaborated closely with Maya's development during its production of ''Dinosaur''. Disney requested that the user interface of the application be customizable so that a personalized workflow could be created. This was a particular influence in the open architecture of Maya, and partly responsible for it becoming popul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
