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A B-spline function is a combination of flexible bands that is controlled by a number of points that are called control points, creating smooth curves. These functions are used to create and manage complex shapes and surfaces using a number of points. B-spline function and Bézier functions are applied extensively in shape optimization methods. [5]
In the mathematical subfield of numerical analysis, de Boor's algorithm [1] is a polynomial-time and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of de Casteljau's algorithm for Bézier curves. The algorithm was devised by German-American mathematician Carl R. de Boor. Simplified ...
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 ...
It was devised by Edwin Catmull and Jim Clark in 1978 as a generalization of bi-cubic uniform B-spline surfaces to arbitrary topology. [1] In 2005/06, Edwin Catmull, together with Tony DeRose and Jos Stam, received an Academy Award for Technical Achievement for their invention and application of subdivision surfaces. DeRose wrote about ...
The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker Citroën ...
Non-uniform rational basis spline (NURBS) is a mathematical model using basis splines (B-splines) that is commonly used in computer graphics for representing curves and surfaces. It offers great flexibility and precision for handling both analytic (defined by common mathematical formulae ) and modeled shapes .
A box spline is a multivariate function defined for a set of vectors, , usually gathered in a matrix := […]. When the number of vectors is the same as the dimension of the domain (i.e., N = d {\displaystyle N=d} ) then the box spline is simply the (normalized) indicator function of the parallelepiped formed by the vectors in Ξ {\displaystyle ...
B C Cubic spline Common implementations 0 Any: Cardinal splines: 0 0.5 Catmull-Rom spline: Bicubic filter in GIMP: 0 0.75 Unnamed: Bicubic filter in Adobe Photoshop [5] 1/3 1/3 Mitchell–Netravali Mitchell filter in ImageMagick [4] 1 0 B-spline: Bicubic filter in Paint.net