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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 geometry has five important qualities that make it an ideal choice for
computer-aided modeling.
- There are several industry standard ways to exchange NURBS geometry. This
means that customers can and should expect to be able to move their valuable
geometric models between various modeling, rendering, animation, and
engineering analysis programs. They can store geometric information in a way
that will be usable 20 years from now.
- NURBS have a precise and well-known definition. The mathematics and
computer science of NURBS geometry is taught in most major universities.
This means that specialty software vendors, engineering teams, industrial
design firms, and animation houses that need to create custom software
applications, can find trained programmers who are able to work with NURBS
geometry.
- NURBS can accurately represent both standard geometric objects like lines,
circles, ellipses, spheres, and tori, and free-form geometry like car bodies
and human bodies.
- The amount of information required for a NURBS representation of a piece
of geometry is much smaller than the amount of information required by
common faceted approximations.
- The NURBS evaluation rule, discussed below, can be implemented on a
computer in a way that is both efficient and accurate.
What is NURBS Geometry?
NURBS curves and surfaces behave in similar ways and share terminology.
Since curves are easiest to describe, we will cover them in detail. A NURBS curve is defined by four things: degree, control points, knots, and
an evaluation rule.
Degree
The degree is a positive whole number. This number is usually 1, 2, 3 or 5, but can be any positive whole number.
NURBS lines and polylines are usually degree 1, NURBS circles are
degree 2, and most free-form curves are degree 3 or 5. Sometimes the terms
linear, quadratic, cubic, and quintic are used. Linear means degree 1,
quadratic means degree 2, cubic means degree 3, and quintic means degree 5.
You may see references to the order of a NURBS curve. The order of a NURBS
curve is positive whole number equal to (degree+1). Consequently, the degree
is equal to order-1. It is possible to increase the degree of a NURBS curve and not change its
shape. Generally, it is not possible to reduce a NURBS curve’s degree
without changing its shape.
Control Points
The control points are a list of at least degree+1 points. One of easiest ways to change the shape of a NURBS curve is to move its
control points.
The control points have an associated number called a weight . With a few
exceptions, weights are positive numbers. When a curve’s control points all
have the same weight (usually 1), the curve is called non-rational,
otherwise the curve is called rational. The R in NURBS stands for rational
and indicates that a NURBS curve has the possibility of being rational. In
practice, most NURBS curves are non-rational. A few NURBS curves, circles
and ellipses being notable examples, are always rational.
Knots The knots are a list of degree+N-1 numbers, where N is the number of control
points. Sometimes this list of numbers is called the knot vector. In this
term, the word vector does not mean 3‑D direction.
This list of knot numbers must satisfy several technical conditions. The
standard way to ensure that the technical conditions are satisfied is to
require the numbers to stay the same or get larger as you go down the list
and to limit the number of duplicate values to no more than the degree. For
example, for a degree 3 NURBS curve with 11 control points, the list of
numbers 0,0,0,1,2,2,2,3,7,7,9,9,9 is a satisfactory list of knots. The list
0,0,0,1,2,2,2,2,7,7,9,9,9 is unacceptable because there are four 2s and four
is larger than the degree. The number of times a knot value is duplicated is called the knot’s
multiplicity. In the preceding example of a satisfactory list of knots, the
knot value 0 has multiplicity three, the knot value 1 has multiplicity one,
the knot value 2 has multiplicity three, the knot value 3 has multiplicity
one, the knot value 7 has multiplicity
two, and the knot value 9 has multiplicity three. A knot value is said to be
a full-multiplicity knot if it is duplicated degree many times. In the
example, the knot values 0, 2, and 9 have full multiplicity. A knot value
that appears only once is called a simple knot. In the example, the knot
values 1 and 3 are simple knots. If a list of knots starts with a full multiplicity knot, is followed by
simple knots, terminates with a full multiplicity knot, and the values are
equally spaced, then the knots are called uniform. For example, if a degree
3 NURBS curve with 7 control points has knots 0,0,0,1,2,3,4,4,4, then the
curve has uniform knots. The knots 0,0,0,1,2,5,6,6,6 are not uniform. Knots
that are not uniform are called non‑uniform. The N and U in NURBS stand for
non‑uniform and indicate that the knots in a NURBS curve are permitted to be
non-uniform. Duplicate knot values in the middle of the knot list make a NURBS curve less
smooth. At the extreme, a full multiplicity knot in the middle of the knot
list means there is a place on the NURBS curve that can be bent into a sharp
kink. For this reason, some designers like to add and remove knots and then
adjust control points to make curves have smoother or kinkier shapes. Since the number of knots is equal
to (N+degree‑1), where N is the number of control points, adding knots also
adds control points and removing knots removes control points. Knots can be
added without changing the shape of a NURBS curve. In general, removing
knots will change the shape of a curve.
Knots and Control Points
A common misconception is that each knot is paired with a control point.
This is true only for degree 1 NURBS (polylines). For higher degree NURBS,
there are groups of 2 x degree knots that correspond to groups of degree+1
control points. For example, suppose we have a degree 3 NURBS with 7 control
points and knots 0,0,0,1,2,5,8,8,8. The first four control points are
grouped with the first six knots. The second through fifth control points
are grouped with the knots 0,0,1,2,5,8. The third through sixth control
points are grouped with the knots 0,1,2,5,8,8. The last four control points
are grouped with the last six knots. Some modelers that use older algorithms for NURBS evaluation require two
extra knot values for a total of degree+N+1 knots. When Rhino is exporting
and importing NURBS geometry, it automatically adds and removes these two
superfluous knots as the situation requires.
Evaluation Rule
A curve evaluation rule is a mathematical formula that takes a number and
assigns a point. The NURBS evaluation rule is a formula that involves the degree, control
points, and knots. In the formula there are some things called B-spline
basis functions. The B and S in NURBS stand for “basis spline.” The number
the evaluation rule starts with is called a parameter. You can think of the
evaluation rule as a black box that eats a parameter and produces a point
location. The degree, knots, and control points determine how the black box
works.
More reading
If you are comfortable reading mathematical formulae, here are some
white papers with more technical details:
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