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Bézier Surface
Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points; rather, it is "stretched" toward them as though each were an attractive force. They are visually intuitive, and for many applications, mathematically convenient. History Bézier surfaces were first described in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies. Bézier surfaces can be of any degree, but bicubic Bézier surfaces generally provide enough degrees of freedom for most applications. Equation A given Bézier surface of degree (''n'', ''m'') is defined by a set of (''n'' + 1)(''m'' + 1) control points k''i'',''j'' where ''i'' = 0, ..., ''n'' and ''j'' = 0 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spline (mathematics)
In mathematics, a spline is a special function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees. In the computer science subfields of computer-aided design and computer graphics, the term ''spline'' more frequently refers to a piecewise polynomial ( parametric) curve. Splines are popular curves in these subfields because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design. The term spline comes from the flexible spline devices used by shipbuilders and draftsmen to draw smooth shapes. Introduction The term "spline" is used to refer to a wide class of functions that are used in applications requiring data interpolation and/or smoothing. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Transformation
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) '' endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term '' linear function'' has the same meaning as ''linea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rendering Pipeline
In computer graphics, a computer graphics pipeline, rendering pipeline or simply graphics pipeline, is a conceptual model that describes what steps a graphics system needs to perform to render a 3D scene to a 2D screen. Once a 3D model has been created, for instance in a video game or any other 3D computer animation, the graphics pipeline is the process of turning that 3D model into what the computer displays. Because the steps required for this operation depend on the software and hardware used and the desired display characteristics, there is no universal graphics pipeline suitable for all cases. However, graphics application programming interfaces (APIs) such as Direct3D and OpenGL were created to unify similar steps and to control the graphics pipeline of a given hardware accelerator. These APIs abstract the underlying hardware and keep the programmer away from writing code to manipulate the graphics hardware accelerators (AMD/Intel/NVIDIA etc. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ray Tracing (graphics)
In 3D computer graphics, ray tracing is a technique for modeling light transport for use in a wide variety of rendering algorithms for generating digital images. On a spectrum of computational cost and visual fidelity, ray tracing-based rendering techniques, such as ray casting, recursive ray tracing, distribution ray tracing, photon mapping and path tracing, are generally slower and higher fidelity than scanline rendering methods. Thus, ray tracing was first deployed in applications where taking a relatively long time to render could be tolerated, such as in still computer-generated images, and film and television visual effects (VFX), but was less suited to real-time applications such as video games, where speed is critical in rendering each frame. Since 2018, however, hardware acceleration for real-time ray tracing has become standard on new commercial graphics cards, and graphics APIs have followed suit, allowing developers to use hybrid ray tracing and rasteriz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cylinder (geometry)
A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infinite curvilinear surface in various modern branches of geometry and topology. The shift in the basic meaning—solid versus surface (as in ball and sphere)—has created some ambiguity with terminology. The two concepts may be distinguished by referring to solid cylinders and cylindrical surfaces. In the literature the unadorned term cylinder could refer to either of these or to an even more specialized object, the ''right circular cylinder''. Types The definitions and results in this section are taken from the 1913 text ''Plane and Solid Geometry'' by George Wentworth and David Eugene Smith . A ' is a surface consisting of all the points on all the lines which are parallel to a given line and which pass through a fixed plane curve in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres roll smoothly in any direction, so most balls used in sports and toys are spherical, as are ball bearings. Basic terminology As mentioned earlier is th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Up until the 19th century, mathematicians largely relied on intuitive notions of continuity, and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicubic Patches
In mathematics, bicubic interpolation is an extension of cubic interpolation (not to be confused with cubic spline interpolation, a method of applying cubic interpolation to a data set) for interpolating data points on a two-dimensional regular grid. The interpolated surface (meaning the kernel shape, not the image) is smoother than corresponding surfaces obtained by bilinear interpolation or nearest-neighbor interpolation. Bicubic interpolation can be accomplished using either Lagrange polynomials, cubic splines, or cubic convolution algorithm. In image processing, bicubic interpolation is often chosen over bilinear or nearest-neighbor interpolation in image resampling, when speed is not an issue. In contrast to bilinear interpolation, which only takes 4 pixels (2×2) into account, bicubic interpolation considers 16 pixels (4×4). Images resampled with bicubic interpolation can have different interpolation artifacts, depending on the b and c values chosen. Computation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézier Triangle
A Bézier triangle is a special type of Bézier surface that is created by (linear, quadratic, cubic or higher degree) interpolation of control points. ''n''th-order Bézier triangle A general ''n''th-order Bézier triangle has (''n'' +1)(''n'' + 2)/2 control points ''α''''i''''β''''j''''γ''''k'' where ''i'', ''j'', ''k'' are non-negative integers such that ''i'' + ''j'' + ''k'' = ''n''. The surface is then defined as : (\alpha s + \beta t + \gamma u)^n = \sum_ s^i t^j u^k \alpha^i \beta^j \gamma^k = \sum_ \frac s^i t^j u^k \alpha^i \beta^j \gamma^k for all non-negative real numbers ''s'' + ''t'' + ''u'' = 1. With linear order (n=1), the resulting Bézier triangle is actually a regular flat triangle, with the triangle vertices equaling the three control points. A quadratic (n=2) Bézier triangle features 6 control points which are all located on the edges. The cubic (n=3) Bézier triang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B-spline
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data. In computer-aided design and computer graphics, spline functions are constructed as linear combinations of B-splines with a set of control points. Introduction The term "B-spline" was coined by Isaac Jacob Schoenberg and is short for basis spline. A spline function of order n is a piecewise polynomial function of degree n - 1 in a variable x. The places where the pieces meet are known as knots. The key property of spline functions is that they and their derivatives may be continuous, depending on the multiplicities of the k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
B-spline Surface
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data. In computer-aided design and computer graphics, spline functions are constructed as linear combinations of B-splines with a set of control points. Introduction The term "B-spline" was coined by Isaac Jacob Schoenberg and is short for basis spline. A spline function of order n is a piecewise polynomial function of degree n - 1 in a variable x. The places where the pieces meet are known as knots. The key property of spline functions is that they and their derivatives may be continuous, depending on the multiplicities of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate System
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
