Bezier surface

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Bezier surface

(graphics)
A surface defined by mathematical formulae, used in computer graphics. A surface P(u, v), where u and v vary orthogonally from 0 to 1 from one edge of the surface to the other, is defined by a set of (n+1)*(m+1) "control points" (X(i, j), Y(i, j), Z(i, j)) for i = 0 to n, j = 0 to m.

P(u, v) = Sum i=0..n Sum j=0..m [

B(i, n, u) = C(n, i) * u^i * (1-u)^(n-i)

C(n, i) = n!/i!/(n-i)!

Bezier surfaces are an extension of the idea of Bezier curves, and share many of their properties.
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Neutrosophic Bezier surfaces are generated by the control points from one of
Every set of TC, IC and FC determines a Bezier surface. Thus, we obtain three Bezier surfaces.
In this paper, we consider Bezier surface modeling of neutrosophic data problems and applications in real life.
As an essential part of this quantitative analysis, formulated Bezier surfaces were used as resulting surfaces, which indicate the goodness of the tactical solutions and performance of an attack.
Because the surfaces are produced from measured data into the form of Bezier surfaces, it is possible to add mathematical comparison and pattern recognition in the CMEP method.
In this paper we focus particularlyon the case of polynomial Bezier surfaces, a kind of free-form splines very popular in fields such as CAD/CAM and computer graphics.
In this context, the present paper describes a new method to solve this challenging parameterization problem for freeform polynomial Bezier surfaces. Our method applies a powerful nature-inspired metaheuristic algorithm, called firefly algorithm, introduced recently by Professor Yang (Cambridge University) to solve difficult optimization problems.
FreeDimension is based on patented N-sided surfacing (NSS) technology, which frees users from quadrilateral surfacing technologies like NURBS and Bezier surfaces. According to the company, the NSS technology permits FreeDimension to present a curve-based interface to the user that closely mimics a pencil-and-paper method of design, giving designers more freedom in generating 3D shapes.
FreeDimension is based on the patent-pending N-Sided Surfacing (NSS) technology, which frees users from quadrilateral surfacing technologies like NURBS and Bezier surfaces. The NSS technology enables FreeDimension to present a curve-based interface to the user that more closely mimics the pencil and paper method of design, creating more freedom and innovation for designers in product development, industrial design, 3D gaming, entertainment and more.
Bernard said Dassault had held off on implementing NURBS in Catia because Bezier surfaces have better continuity for complex curves.
In this paper we introduce a class of surface patch representations, called S-patches, that unify and generalize triangular and tensor product Bezier surfaces by allowing patches to be defined over any convex polygonal domain; hence, S-patches may have any number of boundary curves.
The 3-D capability will allow the analysis to handle the contact of deformable bodies idealized as curved shells and brick elements, and of rigid bodies modeled as flat patches, ruled surfaces, surfaces of revolution, or Bezier surfaces. All of the 2-D capabilities available with the current program will be retained by the 3-D release: the prescription of nonlinear velocities to rigid surfaces, the use of Coulomb or shear friction, and the performance of coupled or uncoupled thermal-structural analysis.