The antonym of rational is non-rational. Non-rational B-splines are a special case of rational B-splines, just as uniform B-splines are a special case of non-uniform B-splines. Thus, non-uniform rational B-splines encompass almost every other possible 3D shape definition. Non-uniform rational B-spline is a bit of a mouthful and so it is generally abbreviated to NURBS.
We have already learnt all about the the B-spline bit of NURBS and about the non-uniform bit. So now all we need to know is the meaning of the rational bit and we will fully(?) understand NURBS.
Rational B-splines are defined simply by applying the B-spline equation
(Equation 87) to homogeneous coordinates, rather
than normal 3D coordinates. We discussed homogeneous coordinates in
the IB course. You will remember that these are 4D
coordinates where the transformation from 4D to 3D is:
A NURBS curve is thus defined as:
We now want to see what a NURBS curve looks like in normal 3D
coordinates, so we need to apply Equation 94 to
Equation 97. In order to better explain what is
going on, we first write Equation 97 in terms of its
individual components. Equation 97 is equivalent to:
So now, we need to define an additional parameter, hi, for each control point, . The default is to set . This results in the denominator of Equation 105 becoming one, and the NURBS equation (Equation 105) therefore reducing to the non-rational B-spline equation (Equation 87).
Increasing hi pulls the curve closer to point . Decreasing hi pushes the curve farther from point . Setting hi=0 means that has no effect on the curve at all.
A non-rational B-spline or a Bezier curve cannot exactly represent a circle. An interesting exercise is to place a cubic Bezier curve's end points at (0,1) and (1,0), with the other control points at and . Now see how close this ``quarter circle'' comes to the real quarter circle defined by x2 + y2 = 1, i.e. what is the value of for which the Bezier curve most closely matches the quarter circle.
NURBS can be used to represent circles, and all of the other conics. NURBS surfaces can be used to represent quadric surfaces. As an example, let us consider one way in which NURBS can be used to describe a true circle. Rogers and Adams cover this on pages 371-375.
Construct eight control points in a square. Let , , , and be the vertices of the square. Let , , , and be the midpoints of the respective sides, so that the vertices are numbered sequentially as you proceed around the square. Finally, you need a ninth point to join up the curve, so let .
Use a quadratic B-spline basis function with the knot vector
[0,0,0,1,1,2,2,3,3,4,4,4]. This means that the curve will pass through , , , and , and allows us to essentially treat each quarter of the circle independently.
We finally need to specify the homogeneous co-ordinates. As a circle is symmetrical is should be obvious that that and . As we would like the curve to pass through the even numbered points we know that . All we therefore need to determine is , the value of the odd numbered homogeneous co-ordinates.
then the NURBS curve will bulge out more than a
circle. If ,
it will bow in. This gives us limits on the
value of .
To find the exact value we take one quarter of the
NURBS curve definition:
A NURBS surface is defined as the same two-dimensional extension to NURBS curves described in Equation 93, though obviously carried out in homogeneous co-ordinates. You can define sweeps using NURBS curves by using one NURBS curve as the sweep path, and another NURBS curve as the cross-section. You take the tensor product of the two curves. This will be described more fully in the lectures and is covered in Rogers & Adams, pages 445-456, 465-477.