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B Spline vs Bezier curves

In CAGD applications, a curve may have a so complicated shape that it cannot be represented by a single Bézier cubic curve since the shape of a cubic curve is not rich enough. Increasing the degree of a Bézier curve adds flexibility to the curve for shape design. However, this will significantly increase processing effort for curve evaluation and manipulation. Furthermore, a Bézier curve of high degree may cause numerical noise in computation. For these reasons, we often split the curve such that each subdivided segment can be represented by a lower degree Bézier curve. This technique is known as piecewise representation. A curve that is made of several Bézier curves is called a composite Bézier curve or a Bézier spline curve. In some area (e.g., computer data exchange), a composite Bézier cubic curve is known as the PolyBézier. If a composite Bézier curve of degree n has m Bézier curves, then the composite Bézier curve has in total m×n+1 control vertices.

A curve with complex shape may be represented by a composite Bézier curve formed by joining a number of Bézier curves with some constraints at the joints. The default constraint is that the curves are jointed smoothly. This in turn requires the continuity of the first-order derivative at the joint, which is known as the first-order parametric continuity. We may relax the constraint to require only the continuity of the tangent directions at the joint, which is known as the first-order geometric continuity. Increasing the order of continuity usually improves the smoothness of a composite Bézier curve. Although a composite Bézier curve may be used to describe a complex shape in CAGD applications, there are primarily two disadvantages associated the use of the composite Bézier curve:

  1. It is considerably involved to join Bézier curves with some order of derivatives continuity.
  2. For the reason that will become clear later, a composite Bézier curve requires more control vertices than a B-spline curve.

These disadvantages can be eliminated by working with spline curves. Originally, a spline curve was a draughtsman's aid. It was a thin elastic wooden or metal strip that was used to draw curves through certain fixed points (called nodes). The resulting curve minimizes the internal strain energy in the splines and hence is considered to be smooth. The mathematical equivalent is the cubic polynomial spline. However, conventional polynomial splines are not popular in CAD systems since they are not intuitive for iterative shape design. B-splines (sometimes, interpreted as basis splines) were investigated by a number of researchers in the 1940s. But B-splines did not gain popularity in industry until de Boor and Cox published their work in the early 1970s. Their recurrence formula to derive B-splines is still the most useful tool for computer implementation.

It is beyond the scope of this section to discuss different ways of deriving B-splines and their generic properties. Instead, we shall take you directly to the definition of a B-spline curve and then explain to you the mathematics of B-splines. Given M control vertices (or de Boor points) di (i = 0,1,,M-1), a B-spline curve of order k (or degree n = k-1) is defined as

r(u) = M-1

i = 0 
di Ni,k(u),
where, Ni,k(u) are polynomials of degree n and known as B-splines of order k. Analogous to Bézier curves, Ni,k(u) are also called the B-spline basis functions. In most practical applications, the B-spline basis functions are derived from the following knot vector (or knot sequence):
u0 = u1 = = un, uk uj uM-1, uM = uM+1 = = uM+n
where, ui are called knots. The reason to select the first and last k knots to be equal is that the control vertices d0 and dM-1 are the points on the curve. Furthermore, the tangent direction of the curve at d0 is from d0 to d1 and the tangent direction at dM-1 is from dM-2 to dM-1. Due to these properties, the shape of a B-spline curve resembles the shape of its control polygon formed by the control vertices di, although such resemblance is not as intuitive as that of a Bézier curve. In the literature, the M-k knots uk, uk+1, , uM-1 are sometimes called the interior knots. If the interior knots are all distinct, then the B-spline curve has M-k non-vanishing spans (i.e., the arc length of each span is not zero).

As we said previously, there are several methods to derive the B-spline basis functions Ni,k(u) in terms of the knot vector. We present only the recursive formula derived by de Boor and Cox as follows:

Ni,k(u) = u-ui
ui+k-1-ui
Ni,k-1(t)+ ui-u
ui+k-ui+1
Ni+1,k-1(u),
with
Ni,1 =

1, u [ui,ui+1)
0, elsewhere
These basis functions have the following properties:

 


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Pascal Vuylsteker / eScience / Computer Science / ANU
Last modified: 20/4/2004
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