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Non-uniform rational B-spline 81
The knot span lengths then translate into velocity and acceleration, which are essential to get right to prevent damage
to the robot arm or its environment. This flexibility in the mapping is what the phrase non uniform in NURBS refers
to.
Necessary only for internal calculations, knots are usually not helpful to the users of modeling software. Therefore,
many modeling applications do not make the knots editable or even visible. It's usually possible to establish
reasonable knot vectors by looking at the variation in the control points. More recent versions of NURBS software
(e.g., Autodesk Maya and Rhinoceros 3D) allow for interactive editing of knot positions, but this is significantly less
intuitive than the editing of control points.
Order
The order of a NURBS curve defines the number of nearby control points that influence any given point on the
curve. The curve is represented mathematically by a polynomial of degree one less than the order of the curve.
Hence, 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 ones most commonly used. Fifth- and sixth-order curves are sometimes useful,
especially for obtaining continuous higher order derivatives, but curves of higher orders are practically never used
because they lead to internal numerical problems and tend to require disproportionately large calculation times.
Construction of the basis functions [4]
The basis functions used in NURBS curves are usually denoted as , in which corresponds to the -th
control point, and corresponds with the degree of the basis function. The parameter dependence is frequently left
out, so we can write . The definition of these basis functions is recursive in . The degree-0 functions
are piecewise constant functions. They are one on the corresponding knot span and zero everywhere else.
Effectively, is a linear interpolation of and . The latter two functions are non-zero for
knot spans, overlapping for knot spans. The function is computed as
From bottom to top: Linear basis functions
(blue) and (green), their weight
functions and and the resulting quadratic
basis function. The knots are 0, 1, 2 and 2.5
rises linearly from zero to one on the interval where is non-zero, while falls from one to zero on
the interval where is non-zero. As mentioned before, is a triangular function, nonzero over two knot
spans rising from zero to one on the first, and falling to zero on the second knot span. Higher order basis functions