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Subdivision Surface
In the field of 3D computer graphics, a subdivision surface (commonly shortened to SubD surface or Subsurf) is a curved Computer representation of surfaces, surface represented by the specification of a coarser polygon mesh and produced by a recursive algorithmic method. The curved surface, the underlying ''inner mesh'', can be calculated from the coarse mesh, known as the ''control cage'' or ''outer mesh'', as the functional Limit (mathematics), limit of an iterative process of subdividing each polygonal Face (geometry), face into smaller faces that better approximate the final underlying curved surface. Less commonly, a simple algorithm is used to add geometry to a mesh by subdividing the faces into smaller ones without changing the overall shape or volume. The opposite is reducing polygons or un-subdividing. Overview A subdivision surface algorithm is recursive in nature. The process starts with a base level polygonal mesh. A refinement scheme is then applied to this mesh. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Computer Graphics
3D computer graphics, sometimes called Computer-generated imagery, CGI, 3D-CGI or three-dimensional Computer-generated imagery, computer graphics, are graphics that use a three-dimensional representation of geometric data (often Cartesian coordinate system#Cartesian coordinates in three dimensions, Cartesian) that is stored in the computer for the purposes of performing calculations and rendering digital images, usually 2D images but sometimes 3D images. The resulting images may be stored for viewing later (possibly as an Computer animation, animation) or displayed in Real-time computer graphics, real time. 3D computer graphics, contrary to what the name suggests, are most often displayed on two-dimensional displays. Unlike 3D film and similar techniques, the result is two-dimensional, without visual depth perception, depth. More often, 3D graphics are being displayed on 3D displays, like in virtual reality systems. 3D graphics stand in contrast to 2D computer graphics which t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bi-cubic
In mathematics, bicubic interpolation is an extension of 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 Suppose the function values f and the derivat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Subdivision Surfaces
Butterflies are winged insects from the lepidopteran superfamily Papilionoidea, characterized by large, often brightly coloured wings that often fold together when at rest, and a conspicuous, fluttering flight. The oldest butterfly fossils have been dated to the Paleocene, about 56 million years ago, though molecular evidence suggests that they likely originated in the Cretaceous. Butterflies have a four-stage life cycle, and like other holometabolous insects they undergo complete metamorphosis. Winged adults lay eggs on the food plant on which their larvae, known as caterpillars, will feed. The caterpillars grow, sometimes very rapidly, and when fully developed, pupate in a chrysalis. When metamorphosis is complete, the pupal skin splits, the adult insect climbs out, expands its wings to dry, and flies off. Some butterflies, especially in the tropics, have several generations in a year, while others have a single generation, and a few in cold locations may take several ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterfly Scheme
Butterflies are winged insects from the lepidopteran Superfamily (taxonomy), superfamily Papilionoidea, characterized by large, often brightly coloured wings that often fold together when at rest, and a conspicuous, fluttering flight. The oldest butterfly fossils have been dated to the Paleocene, about 56 million years ago, though molecular evidence suggests that they likely originated in the Cretaceous. Butterflies have a four-stage Biological life cycle, life cycle, and like other Holometabola, holometabolous insects they undergo Holometabolism, complete metamorphosis. Winged adults lay eggs on the food plant on which their larvae, known as caterpillars, will feed. The caterpillars grow, sometimes very rapidly, and when fully developed, pupate in a chrysalis. When metamorphosis is complete, the pupal skin splits, the adult insect climbs out, expands its wings to dry, and flies off. Some butterflies, especially in the tropics, have several generations in a year, while othe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
√3 Subdivision Scheme
The square root of 3 is the positive real number that, when multiplied by itself, gives the number 3 (number), 3. It is denoted mathematically as \sqrt or 3^. It is more precisely called the principal square root of 3 to distinguish it from the negative number with the same property. The square root of 3 is an irrational number. It is also known as Theodorus' constant, after Theodorus of Cyrene, who proved its irrationality. In 2013, its numerical value in decimal notation was computed to ten billion digits. Its decimal expansion, written here to 65 decimal places, is given by : : The fraction \frac (...) can be used as a good approximation. Despite having a denominator of only 56, it differs from the correct value by less than \frac (approximately 9.2\times 10^, with a relative error of 5\times 10^). The rounded value of is correct to within 0.01% of the actual value. The fraction \frac (...) is accurate to 1\times 10^. Archimedes reported a range for its value: (\frac)^>3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box Spline
In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines are considered as a multivariate generalization of basis splines (B-splines) and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space. Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general polytopes. Definition A box spline is a multivariate function (\mathbb^d \to \mathbb) defined for a set of vectors, \xi \in \mathbb^d, usually gathered in a matrix \mathbf := \left xi_1 \dots \xi_N\right When the number of vectors is the same as the dimension of the domain (i.e., N = d ) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in \mathbf: : M_(\mathbf) := \frac\chi_(\mathbf) = \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Box-spline
In the mathematical fields of numerical analysis and approximation theory, box splines are piecewise polynomial functions of several variables. Box splines are considered as a multivariate generalization of basis splines (B-splines) and are generally used for multivariate approximation/interpolation. Geometrically, a box spline is the shadow (X-ray) of a hypercube projected down to a lower-dimensional space. Box splines and simplex splines are well studied special cases of polyhedral splines which are defined as shadows of general polytopes. Definition A box spline is a multivariate function (\mathbb^d \to \mathbb) defined for a set of vectors, \xi \in \mathbb^d, usually gathered in a matrix \mathbf := \left xi_1 \dots \xi_N\right When the number of vectors is the same as the dimension of the domain (i.e., N = d ) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in \mathbf: : M_(\mathbf) := \frac\chi_(\mathbf) = \begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Subdivision Surface
In computer graphics, the Loop method for subdivision surfaces is an approximating subdivision scheme developed by Charles Loop in 1987 for triangular meshes. Prior methods, namely Catmull-Clark and Doo-Sabin (1978), focused on quad meshes. Loop subdivision surfaces are defined recursively, dividing each triangle into four smaller ones. The method is based on a quartic box spline. It generates C2 continuous limit surfaces everywhere except at extraordinary vertices, where they are C1 continuous. Applications Geologists have applied Loop subdivision surfaces to model erosion on mountain faces, specifically in the Appalachians. See also * Geodesic polyhedron A geodesic polyhedron is a convex polyhedron made from triangles. They usually have icosahedral symmetry, such that they have 6 triangles at a vertex, except 12 vertices which have 5 triangles. They are the dual of corresponding Goldberg po ... * Catmull-Clark subdivision surface * Doo-Sabin subdivision sur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Biquadratic
In algebra, a quartic function is a function of the form :f(x)=ax^4+bx^3+cx^2+dx+e, where ''a'' is nonzero, which is defined by a polynomial of degree four, called a quartic polynomial. A ''quartic equation'', or equation of the fourth degree, is an equation that equates a quartic polynomial to zero, of the form :ax^4+bx^3+cx^2+dx+e=0 , where . The derivative of a quartic function is a cubic function. Sometimes the term biquadratic is used instead of ''quartic'', but, usually, biquadratic function refers to a quadratic function of a square (or, equivalently, to the function defined by a quartic polynomial without terms of odd degree), having the form :f(x)=ax^4+cx^2+e. Since a quartic function is defined by a polynomial of even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If ''a'' is positive, then the function increases to positive infinity at both ends; and thus the function has a global minimum. Likewise, if ''a'' is ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
