Concepts inReal-time GPU rendering of piecewise algebraic surfaces
Real-time computer graphics
Real-time computer graphics is the subfield of computer graphics focused on producing and analyzing images in real time. The term is most often used in reference to interactive 3D computer graphics, typically using a GPU, with video games the most noticeable users. The term can also refer to anything from rendering an application's GUI to real-time image processing and image analysis.
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Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two and so of dimension four as a smooth manifold. The theory of algebraic surfaces is much more complicated than that of algebraic curves (including the compact Riemann surfaces, which are genuine surfaces of dimension two).
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Piecewise
In mathematics, a piecewise-defined function (also called a piecewise function) is a function whose definition changes depending on the value of the independent variable. Mathematically, a real-valued function f of a real variable x is a relationship whose definition is given differently on disjoint subsets of its domain (known as subdomains).
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Graphics processing unit
A graphics processing unit or GPU (also occasionally called visual processing unit or VPU) is a specialized electronic circuit designed to rapidly manipulate and alter memory in such a way so as to accelerate the building of images in a frame buffer intended for output to a display. GPUs are used in embedded systems, mobile phones, personal computers, workstations, and game consoles.
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Root-finding algorithm
A root-finding algorithm is a numerical method, or algorithm, for finding a value x such that f(x) = 0, for a given function f. Such an x is called a root of the function f. This article is concerned with finding scalar, real or complex roots, approximated as floating point numbers. Finding integer roots or exact algebraic roots are separate problems, whose algorithms have little in common with those discussed here.
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Nth root
In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals x where n is the degree of the root. A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred to using ordinal numbers, as in fourth root, twentieth root, etc. For example: 2 is a square root of 4, since 2 = 4. ¿2 is also a square root of 4, since (¿2) = 4. A real number or complex number has n roots of degree n.
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Shader
In the field of computer graphics, a shader is a computer program that is used primarily to calculate rendering effects on graphics hardware with a high degree of flexibility. Shaders are used to program the graphics processing unit (GPU) programmable rendering pipeline, which has mostly superseded the fixed-function pipeline that allowed only common geometry transformation and pixel-shading functions; with shaders, customized effects can be used.
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Bernstein polynomial
In the mathematical field of numerical analysis, a Bernstein polynomial, named after Sergei Natanovich Bernstein, is a polynomial in the Bernstein form, that is a linear combination of Bernstein basis polynomials. 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 Stone¿Weierstrass approximation theorem.
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