Editing Non-uniform rational B-spline (section)

=== Control points ===
[[File:NURBS 3-D surface.gif|250px|thumb|Three-dimensional NURBS surfaces can have complex, organic shapes. Control points influence the directions the surface takes. A separate square below the control cage delineates the X and Y extents of the surface.]]
The control points determine the shape of the curve.<ref>{{cite book |last=Gershenfeld |first=Neil |author-link=Neil Gershenfeld |year=1999 |page=141 |title=The Nature of Mathematical Modeling |publisher=[[Cambridge University Press]] |isbn=0-521-57095-6}}</ref> Typically, each point of the curve is computed by taking a [[weighted]] sum of a number of control points. The weight of each point varies according to the governing parameter. For a curve of degree d, the weight of any control point is only nonzero in d+1 intervals of the parameter space. Within those intervals, the weight changes according to a polynomial function (''basis functions'') of degree d. At the boundaries of the intervals, the basis functions go smoothly to zero, the smoothness being determined by the degree of the polynomial.

As an example, the basis function of degree one is a triangle function. It rises from zero to one, then falls to zero again. While it rises, the basis function of the previous control point falls. In that way, the curve interpolates between the two points, and the resulting curve is a polygon, which is [[continuous function|continuous]], but not [[Differentiable function|differentiable]] at the interval boundaries, or knots. Higher degree polynomials have correspondingly more continuous derivatives. Note that within the interval the polynomial nature of the basis functions and the linearity of the construction make the curve perfectly smooth, so it is only at the knots that discontinuity can arise.

In many applications the fact that a single control point only influences those intervals where it is active is a highly desirable property, known as '''local support'''. In modeling, it allows the changing of one part of a surface while keeping other parts unchanged.

Adding more control points allows better approximation to a given curve, although only a certain class of curves can be represented exactly with a finite number of control points. NURBS curves also feature a scalar '''weight''' for each control point. This allows for more control over the shape of the curve without unduly raising the number of control points.  In particular, it adds conic sections like circles and ellipses to the set of curves that can be represented exactly. The term ''rational'' in NURBS refers to these weights.

The control points can have any [[dimensionality]]. One-dimensional points just define a [[scalar (mathematics)|scalar]] function of the parameter. These are typically used in image processing programs to tune the brightness and color curves. Three-dimensional control points are used abundantly in 3D modeling, where they are used in the everyday meaning of the word 'point', a location in 3D space.
Multi-dimensional points might be used to control sets of time-driven values, e.g. the different positional and rotational settings of a robot arm. NURBS surfaces are just an application of this. Each control 'point' is actually a full vector of 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.