, is defined by
On-Line Geometric Modeling Notes
Overview
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
are the control points,
are the ``normalized B-spline blending functions''.
They are described by the order k and by a non-decreasing sequence of
real numbers
functions are described as follows
.
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 
(where 
is
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,
we obtain
where
(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
order 1.
We are lead to the following result:
If we define
where
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
where
and
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.
Summary
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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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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