On-Line Geometric Modeling Notes
In 1972, Carl DeBoor and M.G. Cox independently discovered the relationship between the analytic and geometric definitions of B-splines. Starting with the definition of the normalized B-spline blending functions, these two researchers were able to develop the geometric definition of the B-spline. It is this calculation that is discussed in this paper.
To get a postscript version of these notes look here.
Definition of the B-Spline Curve
A B-spline curve , is defined by
We note that if, in equation (3), either of the N terms on the right hand side of the equation are zero, or the subscripts are out of the range of the summation limits, then the associated fraction is not evaluated and the term becomes zero. This is to avoid a zero-over-zero evaluation problem.
The order k is independent of the number of control points (n+1). In the B-Spline curve, unlike the Bézier Curve, we have the flexibility of using many control points, and restricting the degree of the polymonial segments.
The DeBoor-Cox Calculation
In the DeBoor-Cox calculation, we substitute the definition of
given in equation (3), into the
right-hand side of
and simplify. This will give us the definition of in terms of , which is of lower degree. The general idea is to continue this process until the sum is written with functions, which we can evaluate easily.
So here we go. If [), then by substituting equation (3) into the equation (1), we have
Now since the support of a B-spline blending function is
the interval , we have that
is non-zero only if
[), which is outside the interval
defined). Thus, . Also is
non-zero only if [), which is
outside the interval . Thus,
and we have
If we denote
then the above result may be written
Now we have written the summation terms of equation (1) in terms of blending functions of lower degree. Of course, we have transferred some of the complexity to the s, but we retain a similar form with control points s weighted by blending functions, and we can repeat this calculation again.
Once again then, repeating the calculation and manipulating the sums,
(Note the appearance of the 2s on the right-hand side of the equation.)
If we continue with this process again, we will manipulate the
sum so that the blending functions have order k-3. Then again with
give us k-4, and eventually we will obtain blending functions of
We are lead to the following result:
If we define
Then, if t is in the interval , we have
This can be shown by continuing the DeBoor-Cox
calculation k-1 times. When complete, we arrive at the formula
where is given in equation (4). (Note the algebraic simplification that the 's provide.) If , then then the only nonzero term of the sum is the lth term, which is one, and the sum must equal .
This enables us to define the geometrical definition of the B-spline curve.
Geometric Definition of the B-Spline Curve
Given a set of Control Points
and a set of knots
The B-Spline curve of order k is defined to be
It is useful to view the geometric construction as
the following pyramid
Any in this pyramid is calculated as a convex combination of the two functions immediately to it's left.
The DeBoor-Cox calculation is a fundamental result in the geometric modeling field. It was used to exhibit the relation between the analytic definition of the B-Spline curve and the geometric definition of the curve. The geometric definition of the curve, because of its computational stability has become the primary technique by which points on these curves are calculated.
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