The NURBS (Non-Uniform Rational B-Splines) are mathematical entities that define exactly one curve or surface from several points of control, not necessarily belonging to the path, and a weight associated with them .
The higher the complexity of the curve, the greater the number of control points to be specified, but in any case, the number of points earned will be less than that required using the traditional approach straight.
Nurbs Curves are an important tool for working in 3D. Can be a simple way to model a surface or an advanced control for animation, defined by control points that control the shape of the curve. It also contains other types of item as editable, which indicates the beginning or end of a segment, and lines connecting the control points on a curve.
curve geometry is defined by degree, control points, knots, and slide rule. Surface Nurbs
have the principles of NURBS curves and apply the same, however a difference between curves and surfaces is that the curves sometena one direction and the surface is bidirectional, these bidirectional has a source called normal, that determines the front or back of it.
basic B-spline functions
Let U = {u0, u1,. . . , Ul} be a nondecreasing sequence of real numbers, ie
ui ≤ ui +1, i = 0,. . . , L-1. The ui are called nodes and U vector of nodes. The i-th B-spline basic function of degree p (order p + 1), denoted by Ni, p (u), is defined recursively:
Note that the functions Ni, 0 (u), i = 0,. . . , M are functions bound, identically zero except in the half-open interval [ui, ui +1) (which can have zero length, then both nodes can be the same). In contrast, for p> 0 the function Ni, p (u) is a linear combination of two basic functions of degree (p-1). Of course, the computation of the basic functions required to specify the vector of nodes U and the degree p.
The derivative of a basic function is given by successive derivatives
times the expression (2) we obtain Nk i, p (u), the k-th derivative of Ni, p (u), as
2.2
The NURBS Surfaces more widely used in surface design processes in industry are NURBS surfaces. This is explained by its great advantages, which include capabilities for interactive design and its ability to accurately represent closed shapes such as conics and quadrics. Moreover, NURBS surfaces include B-spline surfaces as special cases. De
Indeed, many applications of CAD / CAM, virtual reality, visualization and animation using models based on NURBS surfaces and these surfaces are included in many of the most popular formats in the industry, such as IGES. Also many newer graphics standards such as PHIGS + and OpenGL NURBS surfaces include among its primitive graphics. NURBS9
A surface S (u, v) of degree (p, q) is a bivariate rational function of the form:
where {wij} i, j represent the scalar values \u200b\u200bof the weights associated with control points {Pij } i = 0 ,..., n, j = 0 ,..., m and vectors of nodes U and V defined.
2.3 Derivatives of NURBS surfaces
The derivatives a NURBS surface can be calculated from the derivatives of A (u, v) and w (u, v) (the numerator and the denominator of the expression (4) respectively) as those derived
which can be easily calculated From expression (3). The expression (5) also indicates that the derivatives of a NURBS surface are obtained recursively and its computation is possible by applying a distributed scheme.
The higher the complexity of the curve, the greater the number of control points to be specified, but in any case, the number of points earned will be less than that required using the traditional approach straight.
Nurbs Curves are an important tool for working in 3D. Can be a simple way to model a surface or an advanced control for animation, defined by control points that control the shape of the curve. It also contains other types of item as editable, which indicates the beginning or end of a segment, and lines connecting the control points on a curve.
curve geometry is defined by degree, control points, knots, and slide rule. Surface Nurbs
have the principles of NURBS curves and apply the same, however a difference between curves and surfaces is that the curves sometena one direction and the surface is bidirectional, these bidirectional has a source called normal, that determines the front or back of it.
basic B-spline functions
Let U = {u0, u1,. . . , Ul} be a nondecreasing sequence of real numbers, ie
ui ≤ ui +1, i = 0,. . . , L-1. The ui are called nodes and U vector of nodes. The i-th B-spline basic function of degree p (order p + 1), denoted by Ni, p (u), is defined recursively:
Note that the functions Ni, 0 (u), i = 0,. . . , M are functions bound, identically zero except in the half-open interval [ui, ui +1) (which can have zero length, then both nodes can be the same). In contrast, for p> 0 the function Ni, p (u) is a linear combination of two basic functions of degree (p-1). Of course, the computation of the basic functions required to specify the vector of nodes U and the degree p. The derivative of a basic function is given by successive derivatives
times the expression (2) we obtain Nk i, p (u), the k-th derivative of Ni, p (u), as 2.2 The NURBS Surfaces more widely used in surface design processes in industry are NURBS surfaces. This is explained by its great advantages, which include capabilities for interactive design and its ability to accurately represent closed shapes such as conics and quadrics. Moreover, NURBS surfaces include B-spline surfaces as special cases. De
Indeed, many applications of CAD / CAM, virtual reality, visualization and animation using models based on NURBS surfaces and these surfaces are included in many of the most popular formats in the industry, such as IGES. Also many newer graphics standards such as PHIGS + and OpenGL NURBS surfaces include among its primitive graphics. NURBS9
A surface S (u, v) of degree (p, q) is a bivariate rational function of the form:
where {wij} i, j represent the scalar values \u200b\u200bof the weights associated with control points {Pij } i = 0 ,..., n, j = 0 ,..., m and vectors of nodes U and V defined. 2.3 Derivatives of NURBS surfaces
The derivatives a NURBS surface can be calculated from the derivatives of A (u, v) and w (u, v) (the numerator and the denominator of the expression (4) respectively) as those derived
which can be easily calculated From expression (3). The expression (5) also indicates that the derivatives of a NURBS surface are obtained recursively and its computation is possible by applying a distributed scheme. Computer Graphics Group 2 Mr. Santiago Igor Valiente Blog: Jazmin Gonzalez Cruz Aide 
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