What does NURBS mean and why should I care?
More details
http://en.wikipedia.org/wiki/NURBS
Non-Uniform Rational B-Spline (NURBS)
What are NURBS?
The word NURBS is an acronym for non-uniform rational B-spline. Non uniform
rational B-splines can represent 3D geometry.
Why use NURBS to represent 3D geometry?
NURBS geometry has five important qualities that make it an ideal choice for
computer aided modeling.
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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 has a precise and well-known definition. Most major universities teach the
mathematics and computer science of NURBS geometry. 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 able to work with NURBS geometry.
NURBS accurately represents 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
efficiently and accurately.
What is NURBS geometry?
There are lots of ways to answer this question. If you are comfortable reading
mathematical formulae, you can get more detailed information at the Books and
papers on NURBS section at the openNURBS web site.
Rhino uses NURBS to represent curves and surfaces. NURBS curves and surfaces
behave in similar ways and share a lot of terminology. Since curves are easiest to
describe, we’ll cover them in detail. Rhino has surface tools that are analogous to
the curve tools mentioned below.
A NURBS curve is defined by four things: degree, control points, knots, and an
evaluation rule.
The degree is a positive whole number.
This number is usually 1, 2, 3 or 5. Rhino lines and polylines are degree 1. Rhino
circles are degree 2. And most Rhino free-form curves are degree 3 or 5. Rhino will
let you work with NURBS that have degrees from 1 to 32. 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. It
is not possible to reduce a NURBS curve’s degree without changing its shape. Rhino
provides tools that can change degrees to any value in the range from 1 to 32.
The control points are a list of at least (degree+1) points.
One of easiest ways to change the geometry of a NURBS curve is to move its control
points. Rhino provides several ways to move control points. To perform large freeform adjustments you simply use the mouse to drag the control point. Rhino
provides other tools tailored for small precise adjustments.
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 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.
Rhino provides tools for examining and changing control point weights.
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 3D direction.
This list of knot numbers must satisfy several technical conditions. The standard way
to satisfy the technical conditions 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:
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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 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 7 has multiplicity two, and the knot value 9 has multiplicity three. A
knot value is a full multiplicity knot if it is duplicated many times. In the example,
the knot values 0, 2, and 9 have full multiplicity. A knot value that appears only
once is 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 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 NU in NURBS stands for non uniform and indicates 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 smoother or kinkier. Rhino has tools for removing and adding knots.
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. Rhino provides an
advanced knot removing interface that automatically performs appropriate knot
removal when you delete a control point.
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.
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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, if needed, it automatically adds and removes these two
superfluous knots.
The evaluation rule uses a mathematical formula that takes a number and assigns
a point.
The formula involves the degree, control points, and knots. In the formula there are
some things called B-spline basis functions. The BS in NURBS stands for B-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. The
degree, knots, and control points determine how the black box works.
Rhino has evaluation tools. You can select a NURBS curve, type in the value of the
parameter, and produce the corresponding point.
Conceptually, the knots determine the B-spline basis functions. The values of the Bspline basis functions at the parameter determine how the control points and
weights are averaged together to produce a point. Detailed discussions of the
evaluation rule and B-spline basis functions are available in many textbooks and web
pages.
rhino/nurbs.txt · Last modified: 2020/08/14 (external edit)