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Smoothstep
Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, video game engines, and machine learning. The function depends on three parameters, the input ''x'', the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge. The function receives a real number ''x'' as an argument and returns 0 if ''x'' is less than or equal to the left edge, 1 if x is greater than or equal to the right edge, and smoothly interpolates, using a Hermite polynomial, between 0 and 1 otherwise. The gradient of the smoothstep function is zero at both edges. This is convenient for creating a sequence of transitions using smoothstep to interpolate each segment as an alternative to using more sophisticated or expensive interpolation techniques. In HLSL and GLSL, smoothstep implements the \operatorname_1(x), the cubic Hermite interpolation after clamping: : \operatorname(x) = S_1(x) = \begin 0 & x \le 0 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smoothstep And Smootherstep
Smoothstep is a family of sigmoid-like interpolation and clamping functions commonly used in computer graphics, video game engines, and machine learning. The function depends on three parameters, the input ''x'', the "left edge" and the "right edge", with the left edge being assumed smaller than the right edge. The function receives a real number ''x'' as an argument and returns 0 if ''x'' is less than or equal to the left edge, 1 if x is greater than or equal to the right edge, and smoothly interpolates, using a Hermite polynomial, between 0 and 1 otherwise. The gradient of the smoothstep function is zero at both edges. This is convenient for creating a sequence of transitions using smoothstep to interpolate each segment as an alternative to using more sophisticated or expensive interpolation techniques. In HLSL and GLSL, smoothstep implements the \operatorname_1(x), the cubic Hermite interpolation after clamping: : \operatorname(x) = S_1(x) = \begin 0 & x \le 0 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigmoid Function
A sigmoid function is a mathematical function having a characteristic "S"-shaped curve or sigmoid curve. A common example of a sigmoid function is the logistic function shown in the first figure and defined by the formula: :S(x) = \frac = \frac=1-S(-x). Other standard sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as an alias for the logistic function. Special cases of the sigmoid function include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. A wide variety of sigmoid functions including the logistic and hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
RenderMan Shading Language
Renderman Shading Language (abbreviated RSL) is a component of the RenderMan Interface Specification, and is used to define shaders. The language syntax is C-like. A shader written in RSL can be used without changes on any RenderMan-compliant renderer, such as Pixar's PhotoRealistic RenderMan, DNA Research's 3Delight, Sitexgraphics' Air or an open source solution such as Pixie or Aqsis. RenderMan Shading Language defines standalone functions and five types of shaders: surface, light, volume, imager and displacement shaders. An example of a surface shader that defines a metal surface is: surface metal (float Ka = 1; float Ks = 1; float roughness = 0.1;) Shaders express their work by reading and writing special variables such as Cs (surface color), N (normal at given point), and Ci (final surface color). The arguments to the shaders are global parameters that are attached to objects of the model (so one metal shader can be used for different metals and so on). Shaders have no ret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maclaurin Series
Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin (1878–1915), Australian general * Ian MacLaurin, Baron MacLaurin of Knebworth * Richard Cockburn Maclaurin (1870–1920), US physicist and educator See also * Taylor series in mathematics, a special case of which is the ''Maclaurin series'' * Maclaurin (crater), a crater on the Moon * McLaurin (other) * MacLaren (surname) * McLaren (other) McLaren is a Formula One racing team, part of the McLaren Group. McLaren or MacLaren may also refer to: * McLaren (surname) * MacLaren (surname) * Clan MacLaren, a Scottish clan Places * McLaren Flat, South Australia * McLaren Park, New Zeal ... {{surname, Maclaurin Clan MacLaren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Even And Odd Functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses. They are important in many areas of mathematical analysis, especially the theory of power series and Fourier series. They are named for the parity of the powers of the power functions which satisfy each condition: the function f(x) = x^n is an even function if ''n'' is an even integer, and it is an odd function if ''n'' is an odd integer. Definition and examples Evenness and oddness are generally considered for real functions, that is real-valued functions of a real variable. However, the concepts may be more generally defined for functions whose domain and codomain both have a notion of additive inverse. This includes abelian groups, all rings, all fields, and all vector spaces. Thus, for example, a real function could be odd or even (or neither), as could a complex-valued function of a vector variable, and so on. The given e ... [...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] |
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives can be generalized to functions of several real variables. In this generalization, the derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ken Perlin
Kenneth H. Perlin is a professor in the Department of Computer Science at New York University, founding director of the Media Research Lab at NYU, director of the Future Reality Lab at NYU, and the Director of the Games for Learning Institute. He holds a BA. degree in Theoretical Mathematics from Harvard University (7/1979), a MS degree in Computer Science from the Courant Institute of Mathematical Sciences, New York University (6/1984), and a PhD degree in Computer Science from the same institution (2/1986). His research interests include graphics, animation, multimedia, and science education. He developed or was involved with the development of techniques such as Perlin noise, real-time interactive character animation, and computer-user interfaces. He is best known for the development of Perlin noise and Simplex noise, both of which are algorithms for realistic-looking Gradient noise. He is a collaborator of the World Building Institute. Awards In 1996, K. Perlin received a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clipping (signal Processing)
Clipping is a form of distortion that limits a signal once it exceeds a threshold. Clipping may occur when a signal is recorded by a sensor that has constraints on the range of data it can measure, it can occur when a signal is digitized, or it can occur any other time an analog or digital signal is transformed, particularly in the presence of gain or overshoot and undershoot. Clipping may be described as hard, in cases where the signal is strictly limited at the threshold, producing a flat cutoff; or it may be described as soft, in cases where the clipped signal continues to follow the original at a reduced gain. Hard clipping results in many high-frequency harmonics; soft clipping results in fewer higher-order harmonics and intermodulation distortion components. Audio In the frequency domain, clipping produces strong harmonics in the high-frequency range (as the clipped waveform comes closer to a squarewave). The extra high-frequency weighting of the signal could make ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GLSL
OpenGL Shading Language (GLSL) is a high-level shading language with a syntax based on the C programming language. It was created by the OpenGL ARB (OpenGL Architecture Review Board) to give developers more direct control of the graphics pipeline without having to use ARB assembly language or hardware-specific languages. Background With advances in graphics cards, new features have been added to allow for increased flexibility in the rendering pipeline at the vertex and fragment level. Programmability at this level is achieved with the use of fragment and vertex shaders. Originally, this functionality was achieved by writing shaders in ARB assembly language – a complex and unintuitive task. The OpenGL ARB created the OpenGL Shading Language to provide a more intuitive method for programming the graphics processing unit while maintaining the open standards advantage that has driven OpenGL throughout its history. Originally introduced as an extension to OpenGL 1.4, GLSL wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermite Interpolation
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than that takes the same value at given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than such that the polynomial and its first derivatives have the same values at given points as a given function and its first derivatives. Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both are derived from the calculation of divided differences. However, there are other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see . Statement of the pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
HLSL
The High-Level Shader Language or High-Level Shading Language (HLSL) is a proprietary shading language developed by Microsoft for the Direct3D 9 API to augment the shader assembly language, and went on to become the required shading language for the unified shader model of Direct3D 10 and higher. HLSL is analogous to the GLSL shading language used with the OpenGL standard. It is very similar to the Nvidia Cg shading language, as it was developed alongside it. Early versions of the two languages were considered identical, only marketed differently. HLSL shaders can enable profound speed and detail increases as well as many special effects in both 2D and 3D computer graphics. HLSL programs come in six forms: pixel shaders (fragment in GLSL), vertex shaders, geometry shaders, compute shaders, tessellation shaders (Hull and Domain shaders), and ray tracing shaders (Ray Generation Shaders, Intersection Shaders, Any Hit/Closest Hit/Miss Shaders). A vertex shader is executed for each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
