Checkpoints

The control points determine the shape of the curve. Generally, each point on the curve is calculated by taking the weighted sum of a series of control points. The weight of each point varies according to the parameter that governs it. To obtain a curve of degree d, the weight of any control point is nonzero only in D +1 intervals of the parameter space. Within these intervals, the weight changes according to a polynomial function (basis functions) of degree d. At the limits of the intervals, the basis functions slowly approach zero, this speed determining the degree of the polynomial.

As an example, the basis function of degree one is a triangular function. Increments from zero to one, then decrements back to zero. As it increases, the basis function of the previous checkpoint falls. In this way, the interpolation between the two points is a curve, and this resulting curve is a polygon, which is continuous, but not differentiable at the limits of the interval, or the nodes. Polynomials of higher degree, consequently, have more continuous derivatives. It should be noted that within the interval the polynomial nature of the basis functions and the linearity of the construction makes the curve perfectly smooth, so it is only at the nodes that discontinuity can arise.

The fact that a single control point only influences those intervals in which it is active is a very desirable property, it is known as local support. In modeling, this allows part of a surface to change, while the other parts stay the same.

Adding more control points allows for a better approximation of a given curve, although only certain classes of curves can be represented exactly with a finite number of control points. NURBS curves also have a scalar weight for each control point.

This allows greater control over the shape of the curve without unduly increasing the number of control points. In particular, conic sections such as circles and ellipses are added to the set of curves that can be represented exactly. The term rational in NURBS refers to these weights.

Control points can have any dimension. Points in a dimension only define a scalar function "Scalar (mathematics)") of the parameter. These are typically used in image processing programs to adjust brightness and color curves. Three-dimensional control points are widely used in 3D modeling, where they are used daily as a reference to the word "point", a place in 3D space. Multidimensional points can be used to control sets of values ​​based on time, for example, the different position and rotation settings of a robot arm. NURBS surfaces are just one application of this. Each control "point" is actually a vector filled with control points, defining a curve.

These curves share their degree and the number of control points, and span one dimension of the parameter space. By interpolating these control vectors over the other dimension of the parameter space, a continuous set of curves is obtained, defining the surface.

Node Vector

The node vector is a sequence of parameter values ​​that determine where and how the control points affect the NURBS curve. The number of nodes is always equal to the number of control points plus the degree of the curve plus one. The node vector divides the parametric space into the intervals mentioned above, usually known as node segments. Each time the parameter value introduces a new node segment, a new checkpoint is activated, while an old checkpoint is discarded. It follows that the values ​​in the node vector must be in ascending order, so (0, 0, 1, 2, 3, 3) is valid, while (0, 0, 2, 1, 3, 3) is not.

Consecutive nodes can have the same value. This then defines a node segment of zero length, which implies that two checkpoints are activated at the same time (and of course two old checkpoints will be deactivated). This has an impact on the continuity of the resulting curve or its higher derivatives, for example allowing the creation of corners on a smoothed NURBS curve. A number of matching nodes is sometimes called a node with a certain multiplicity. Nodes with multiplicity two or three are known as double or triple nodes. The multiplicity of a node is limited to the degree of the curve, since a greater multiplicity would divide the curve into disjointed parts and leave the control points unused. For first-degree NURBS, each node is associated with a checkpoint.

The node vector usually starts with a node that has a multiplicity equal to the order. This makes sense, as it activates the checkpoints that influence the first node segment. Likewise, the vector of nodes usually ends with a node of that multiplicity. Curves with such node vectors start and end at a control point.

The individual node values ​​are not significant by themselves, only the proportion of difference between the node values ​​matters. Therefore, the node vectors (0, 0, 1, 2, 3, 3) and (0, 0, 2, 4, 6, 6) produce the same curve. The values ​​of the node positions influence the mapping from the parameter space to the curve space. Rendering a NURBS curve is generally done in steps with fixed distance across the parameter range. By changing the lengths of the node segments, more sample points can be used in regions where the curvature is maximum. Another use is in situations where the parameter value has some physical meaning, for example, if the parameter is time and the curve describes the movement of a robot arm. The node segment lengths are then translated into speed and acceleration, which are essential to prevent damage to the robotic arm or its environment. This flexibility in mapping is what the phrase non-uniform in NURBS refers to.

Needed only for internal calculations, nodes are generally not useful to software modeling users. Therefore, many modeling applications do not make nodes editable or even visible. It is usually possible to establish reasonable node vectors by observing the variation in the control points. Newer versions of software for NURBS (e.g. Autodesk Maya and Rhinoceros 3D) allow interactive editing of node positions, but this is significantly less intuitive than editing control points.

Degree

The order of a NURBS curve defines the number of nearby control points that influence any point on the curve. The curve is represented mathematically by a polynomial of degree one minus the order of the curve. Therefore, second-order curves (which are represented by linear polynomials) are called linear curves, third-order curves are called quadratic curves, and fourth-order curves are called cubic curves. The number of control points must be greater than or equal to the order of the curve.

In practice, cubic curves are the most used. Fifth- and sixth-order curves are sometimes useful, especially for obtaining higher-order continuous derivatives, but higher-order curves are practically never used because they lead to internal numerical problems and tend to require disproportionately large calculation times.

Contenido

Una curva NURBS se define por su grado, un conjunto de puntos de control ponderados, y un vector de nodos. Las curvas y superficies NURBS son generalizaciones de curvas B-splines y curvas de Bézier, así como de superficies, siendo su diferencia principal la ponderación de los puntos de control que hacen a las curvas NURBS racionales (las curvas B-splines racionales no uniformes son un caso especial de las curvas B-splines racionales). Mientras que las curvas de Bézier se desarrollan en una sola dirección paramétrica, normalmente llamada s o u, las superficies NURBS evolucionan en dos direcciones paramétricas, llamada s y t o u y v.

Mediante la evaluación de una curva de Bézier o una curva NURBS en diversos valores del parámetro, la curva se puede representar en un espacio Cartesiano de dos o tres dimensiones. Asimismo, mediante la evaluación de una superficie NURBS en diversos valores de los dos parámetros, la superficie se puede representar en el espacio cartesiano.

Las curvas y superficies NURBS son útiles por varias razones:.

En las siguientes secciones, las curvas NURBS se analizan en una dimensión. Debe tenerse en cuenta que todo esto se puede generalizar a dos o incluso más dimensiones.

Checkpoints

The control points determine the shape of the curve. Generally, each point on the curve is calculated by taking the weighted sum of a series of control points. The weight of each point varies according to the parameter that governs it. To obtain a curve of degree d, the weight of any control point is nonzero only in D +1 intervals of the parameter space. Within these intervals, the weight changes according to a polynomial function (basis functions) of degree d. At the limits of the intervals, the basis functions slowly approach zero, this speed determining the degree of the polynomial.

As an example, the basis function of degree one is a triangular function. Increments from zero to one, then decrements back to zero. As it increases, the basis function of the previous checkpoint falls. In this way, the interpolation between the two points is a curve, and this resulting curve is a polygon, which is continuous, but not differentiable at the limits of the interval, or the nodes. Polynomials of higher degree, consequently, have more continuous derivatives. It should be noted that within the interval the polynomial nature of the basis functions and the linearity of the construction makes the curve perfectly smooth, so it is only at the nodes that discontinuity can arise.

The fact that a single control point only influences those intervals in which it is active is a very desirable property, it is known as local support. In modeling, this allows part of a surface to change, while the other parts stay the same.

Adding more control points allows for a better approximation of a given curve, although only certain classes of curves can be represented exactly with a finite number of control points. NURBS curves also have a scalar weight for each control point.

This allows greater control over the shape of the curve without unduly increasing the number of control points. In particular, conic sections such as circles and ellipses are added to the set of curves that can be represented exactly. The term rational in NURBS refers to these weights.

Control points can have any dimension. Points in a dimension only define a scalar function "Scalar (mathematics)") of the parameter. These are typically used in image processing programs to adjust brightness and color curves. Three-dimensional control points are widely used in 3D modeling, where they are used daily as a reference to the word "point", a place in 3D space. Multidimensional points can be used to control sets of values ​​based on time, for example, the different position and rotation settings of a robot arm. NURBS surfaces are just one application of this. Each control "point" is actually a vector filled with control points, defining a curve.

These curves share their degree and the number of control points, and span one dimension of the parameter space. By interpolating these control vectors over the other dimension of the parameter space, a continuous set of curves is obtained, defining the surface.

Node Vector

The node vector is a sequence of parameter values ​​that determine where and how the control points affect the NURBS curve. The number of nodes is always equal to the number of control points plus the degree of the curve plus one. The node vector divides the parametric space into the intervals mentioned above, usually known as node segments. Each time the parameter value introduces a new node segment, a new checkpoint is activated, while an old checkpoint is discarded. It follows that the values ​​in the node vector must be in ascending order, so (0, 0, 1, 2, 3, 3) is valid, while (0, 0, 2, 1, 3, 3) is not.

Consecutive nodes can have the same value. This then defines a node segment of zero length, which implies that two checkpoints are activated at the same time (and of course two old checkpoints will be deactivated). This has an impact on the continuity of the resulting curve or its higher derivatives, for example allowing the creation of corners on a smoothed NURBS curve. A number of matching nodes is sometimes called a node with a certain multiplicity. Nodes with multiplicity two or three are known as double or triple nodes. The multiplicity of a node is limited to the degree of the curve, since a greater multiplicity would divide the curve into disjointed parts and leave the control points unused. For first-degree NURBS, each node is associated with a checkpoint.

The node vector usually starts with a node that has a multiplicity equal to the order. This makes sense, as it activates the checkpoints that influence the first node segment. Likewise, the vector of nodes usually ends with a node of that multiplicity. Curves with such node vectors start and end at a control point.

The individual node values ​​are not significant by themselves, only the proportion of difference between the node values ​​matters. Therefore, the node vectors (0, 0, 1, 2, 3, 3) and (0, 0, 2, 4, 6, 6) produce the same curve. The values ​​of the node positions influence the mapping from the parameter space to the curve space. Rendering a NURBS curve is generally done in steps with fixed distance across the parameter range. By changing the lengths of the node segments, more sample points can be used in regions where the curvature is maximum. Another use is in situations where the parameter value has some physical meaning, for example, if the parameter is time and the curve describes the movement of a robot arm. The node segment lengths are then translated into speed and acceleration, which are essential to prevent damage to the robotic arm or its environment. This flexibility in mapping is what the phrase non-uniform in NURBS refers to.

Needed only for internal calculations, nodes are generally not useful to software modeling users. Therefore, many modeling applications do not make nodes editable or even visible. It is usually possible to establish reasonable node vectors by observing the variation in the control points. Newer versions of software for NURBS (e.g. Autodesk Maya and Rhinoceros 3D) allow interactive editing of node positions, but this is significantly less intuitive than editing control points.

Degree

The order of a NURBS curve defines the number of nearby control points that influence any point on the curve. The curve is represented mathematically by a polynomial of degree one minus the order of the curve. Therefore, second-order curves (which are represented by linear polynomials) are called linear curves, third-order curves are called quadratic curves, and fourth-order curves are called cubic curves. The number of control points must be greater than or equal to the order of the curve.

In practice, cubic curves are the most used. Fifth- and sixth-order curves are sometimes useful, especially for obtaining higher-order continuous derivatives, but higher-order curves are practically never used because they lead to internal numerical problems and tend to require disproportionately large calculation times.