NURBS, short for nonuniform rational B-spline, is a computer graphics technique for drawing curves. A NURBS curve is defined by a set of weighted control points, the curve's order and a knot vector. NURBS are generalizations of both B-splines and Bézier curves, with the primary difference being the weighting of the control points which makes them rational (non-rational B-splines are a special case of rational B-splines, in practice most NURBS curves are non-rational).
NURBS curves are easily generalized into NURBS patches.
One of the first common uses of NURBS was in 3D modeling packages, notably Alias's line starting in the early 1990s. NURBS are useful for a number of reasons: they
offer one common mathematical form for both standard analytical shapes (e.g. conics) and free form shapes;
provide the flexibility to design a large variety of shapes;
can be evaluated reasonably fast by numerically stable and accurate algorithms;
are generalizations of non-rational B-splines and non-rational and rational Bézier curves and surfaces.
They can accurately represent standard geometric objects like lines or conic sections as well as free form geometry.
The knot vector is a sequence of parameter values that determine the continuity along the NURBS curve. The number of knots is always equal to the number of control points plus the order of the curve.
The values of the knot vector must be in ascending order: the following vector is valid [0 0 1 2 3] while this one is not [0 0 2 1 3]. Also note that the only significant factor is the ratio of the values to each other: the knot vectors [0 0 1 2 3], [0 0 2 4 6] and [1 1 2 3 4] produce the same curve. There can be no more duplicate values than the degree: knots multiplicity <= degree. For degree 1 NURBS each knot is paired with a control point.
The order of the curve is greater or equal to 2, corresponding to a linear curve (order = 2), a quadratic curve (order = 3) and a cubic curve (order = 4). The curve is represented mathematically by a polynomial of same order, a cubic curve is represented by a degree 3 polynomial which order is 4. Besides, the number of control points must be equal to or greater than the order of the curve.
we need more on the "rational" aspect of NURBS, including a formula, and what distinguishes them from ordinary non-uniform B-splines
For example, NURBSgeometry makes it possible to treat the hood of an automobile or the wing of an airplane as a single surface and create more realistic shapes of human bodies.
With NURBS it is easier to create virtual worlds in VRML with smooth surfaces and reduced download size of VRML files because of the compact NURBS description at once.
This description corresponds to NURBS Extension for VRML97 Discussion & Node proposal 12 March, 1999 by Blaxxun interactive except the fields distance, quality, uTessellation, vTessellation, and texCoord.
NURBS, Non-Uniform Rational B-Splines, are mathematical representations of 3-D geometry that can accurately describe any shape from a simple 2-D line, circle, arc, or curve to the most complex 3-D organic free-form surface or solid.
Because of their flexibility and accuracy, NURBS models can be used in any process from illustration and animation to manufacturing.
NURBS lines and polylines are usually degree 1, NURBS circles are degree 2, and most free-form curves are degree 3 or 5.
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