Control points Non-uniform rational B-spline

three-dimensional nurbs surfaces can have complex, organic shapes. control points influence directions surface takes. outermost square below delineates x/y extents of surface.

the control points determine shape of curve. typically, each point of curve computed taking weighted sum of number of control points. weight of each point varies according governing parameter. curve of degree d, weight of control point nonzero in d+1 intervals of parameter space. within intervals, weight changes according polynomial function (basis functions) of degree d. @ boundaries of intervals, basis functions go smoothly zero, smoothness being determined degree of polynomial.

as example, basis function of degree 1 triangle function. rises 0 one, falls 0 again. while rises, basis function of previous control point falls. in way, curve interpolates between 2 points, , resulting curve polygon, continuous, not differentiable @ interval boundaries, or knots. higher degree polynomials have correspondingly more continuous derivatives. note within interval polynomial nature of basis functions , linearity of construction make curve smooth, @ knots discontinuity can arise.

in many applications fact single control point influences intervals active highly desirable property, known local support. in modeling, allows changing of 1 part of surface while keeping other parts unchanged.

adding more control points allows better approximation given curve, although class of curves can represented finite number of control points. nurbs curves feature scalar weight each control point. allows more control on shape of curve without unduly raising number of control points. in particular, adds conic sections circles , ellipses set of curves can represented exactly. term rational in nurbs refers these weights.

the control points can have dimensionality. one-dimensional points define scalar function of parameter. these typically used in image processing programs tune brightness , color curves. three-dimensional control points used abundantly in 3d modeling, used in everyday meaning of word point , location in 3d space.

multi-dimensional points might used control sets of time-driven values, e.g. different positional , rotational settings of robot arm. nurbs surfaces application of this. each control point full vector of control points, defining curve. these curves share degree , number of control points, , span 1 dimension of parameter space. interpolating these control vectors on other dimension of parameter space, continuous set of curves obtained, defining surface.