© Dr Moudar Zgoul 2010-2011

Non-uniform rational basis spline (NURBS)

An Introduction to

Non-uniform rational basis spline (NURBS)

It’s is a mathematical model commonly used in computer graphics for generating and representing curves and surfaces which offers great flexibility and precision for handling both analytic and freeform shapes.

Development of NURBS began in the 1950s by engineers who were in need of a mathematically precise representation of freeform surfaces like those used for ship hulls, aerospace exterior surfaces, and car bodies, which could be exactly reproduced whenever technically needed.

Freeform surface, or freeform surfacing, is used in CAD and other computer graphics software to describe the skin of a 3D geometric element.

Freeform surfaces do not have rigid radial dimensions, unlike regular surfaces such as planes, cylinders and conic surfaces.

They are used to describe forms such as turbine blades, car bodies and boat hulls.

The forms of freeform surfaces (and curves) are not stored or defined in CAD software in terms of polynomial equations, but by their poles, degree, and number ofpatches (segments with spline curves).

The degree of a surface determines its mathematical properties, and can be seen as representing the shape by a polynomial with variables to the power of the degree value.

For example, a surface with a degree of 1 would be a flat cross section surface.

A surface with degree 2 would be curved in one direction, while a degree 3 surface could (but does not necessarily) change once from concave to convex curvature.

Some CAD systems use the term order instead of degree.

The pioneers of this development were Engineers at Renault, and Citroën.

At first NURBS were only used in the proprietary CAD packages of car companies. Later they became part of standard computer graphics packages.

Real-time, interactive rendering of NURBS curves and surfaces was first made available on Silicon Graphics workstations in 1989. In 1993, the first interactive NURBS modeller for PCs, called NöRBS, was developed by a small startup company cooperating with the Technical University of Berlin.

Today most professional computer graphics applications available for desktop use offer NURBS technology, which is most often realized by integrating a NURBS engine from a specialized company.

NURBS are commonly used in computer-aided design (CAD), manufacturing (CAM), and engineering (CAE) and are part of numerous industry wide used standards, such as IGES, STEP, ACIS, and PHIGS

To fully exploit the flexibility of NURBS, a thorough working knowledge of the underpinning mathematics is necessary.

Frequently, NURBS are considered a very mathematical subject beyondthe means of many working professionals and students.

Curves and surfaces description in general

Curves and surfaces are mathematically represented either explicitly, implicitly or parametrically.

Explicit representations of the form y = f(x), althoughuseful in many applications, are axis dependent, cannot adequately represent multiple-valued functions, and cannot be used where a constraint involves an infinite derivative.

Hence, they are little used in computer graphics or computer aided design.

Curves and surfaces description in general

Implicit representations of the form:f(x, y) = 0 and f(x, y, z) = 0

for curves and surfaces, respectively, are capable of representing multiple-valued functions but are still axis dependent.

However, they have a variety of uses in computer graphics and computer aided design.

Parametric Curves and surfaces

Parametric curve representations of the formx- f(t); y- g(t); z- h(t)

where t is the parameter, are extremely flexible.

They are axis independent, easily represent multiple-valued functions and infinite derivatives, and have additional degrees of freedom compared to either explicit or implicit formulations.

Parametric Curves and surfaces

Consider the explicit cubic equationy - ax 3 + bx 2 + cx + d

Here, four degrees of freedom exist, one for each of the four constant coefficients a, b, c, d.

Rewriting this equation in parametric form, we have

Parametric Curves and surfaces

Here, eight degrees of freedom exist, one for each of the eight constant coefficients: α, β, γ, δ and α, β, γ, δ.

Although not necessary, the parameter range is frequently normalized to 0 ≤ t ≤ 1.

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Parametric Curves and surfaces

The derivative of y with respect to x is given by

where, for example, from the equation above we have

Parametric Curves and surfaces

The derivative is infinite when the denominator is zero.

Note that simply testing if the denominator is zero shows that the derivative is infinite.

Conversely, setting the denominator to zero imposes the constraint of an infinite derivative.

One example is the requirement that the derivative beinfinite for a circular arc tangent to a vertical edge.

Extension to Three Dimensions

The parametric form easily extends to three dimensions (or more) by specifying z = z(t),

where again t is the parameter. An example of a simple three-dimensional parametric space curve is the circular helix shown .

The circular helix is represented by the parametric equations:

x(t) - r cos t; y(t) - r sin t; z(t) - bt

Extension to Three Dimensions

x(t) - r cos t; y(t) - r sin t; z(t) - bt

or any point on the helix is given by the vector-valued function, i.e., the function has x, y, z components. Thus, we write

P(t) = [ x(t) y(t) z(t) ] = [ r cos t rsint bt ]

for r and b ≠ 0 and -∞ ≤ t ≤ ∞.

This curve lies on the surface of a right circular cylinder of radius Irl.

Extension to Three Dimensions

x(t) - r cos t; y(t) - r sin t; z(t) - bt

The effect of the equation z = bt is to move the points of the curve infinitely in the z direction.

After each 2π interval in the parameter t, the variables x and y return to their initial values; but z increases or decreases by 2 π lbl, depending upon the sign of b. This change in z is called the pitch of the helix.

Parametric Line

The simplest 'curve' is a straight line. The general parametric equation for a straight line segment is

P(t) = P1 + (P2 - P1)t 0 ≤ t ≤ 1

where P1 and P2 are vector-valued functions typically called position vectors and t is the parameter.

A position vector has components in some coordinate system.

Parametric Line

The values of the components are specified with respect to the origin of the coordinate system.

The position vector in row matrix form as

P [x y z] or P= [x y z] or even just [x y z].

The origin of a typical Cartesian coordinate system is

P[0 0 0] or P = [0 0 0] or [0 0 0].

Parametric Line

Each component of the position vector also has a parametric representation:

x(t) = x1 + (x2 - x1)t 0 ≤ t ≤ 1y(t) = y1 + (y2 - y1)t 0 ≤ t ≤ 1z(t) = z1 + (z2 - z1)t 0 ≤ t ≤ 1

Example

Determine the parametric representation for the line segment between the position vectorsP1[x1=1 y1=1 z1=1] and P2[x2=20.1 y2=5.6 z2=10.8].

Calculate the point on the line halfway between P1 and P2.

Example

P1[1 1 1]P2[20.1 5.6 10.8]

Performing the subtraction of the components of P1 and P2 yields:

P(t)= [1 1 1] + [19.1 4.6 9.8]t

Because this is a straight line and the parameterization is linear the midpoint of the line occurs for a parameter value t= 1/2.

Example

P(½) = [1 1 1] + ½[19.1 4.6 9.8] = [10.55 3.3 5.9]

Recalling that each component of the position vector also has a parametric representation, we havex(t) = 1 + (20.1 - 1)t = 1 + 19.1ty(t) = 1+ (5.6 - 1)t = 1 +4.6tz(t) = 1 + (10.8 - 1) t = 1 + 9.8t

and for t = 1/2 we havex(t) = 1 + 1/2(19.1) = 10.55y(t) = 1 + 1/2(4.6) = 3.3z(t) = 1 + 1/2(9.8) = 5.9

Reparameterization

Reparameterize the line segment defined earlier to the range 0-2.

let the new parameter be u. The conditions or constraints on the new parameter aret = 0 u = 0 ; t = 1 u=2Assuming a linear form of the reparameterization function, i.e. u=a+bt

and imposing the constraints yields u=2 t ≤ t ≤ 1Thus, the new equation of the line segment isP(u) + 1/2(P2 - P1)u 0 ≤ u ≤ 2

Parametric Surfaces

x=x(u,w) y=y(u,w) z=z(u,w)