has compact support. This
means that the function is zero outside of some interval. In these
notes, We
find this interval explicitly in terms of the knot sequence.
On-Line Geometric Modeling Notes
Overview
A B-spline blending function has compact support. This
means that the function is zero outside of some interval. In these
notes, We
find this interval explicitly in terms of the knot sequence.
To get a postscript version of these notes look here.
The Support of the Function
Given an order k, and a knot sequence
,
the normalized B-spline blending function 
is positive
if and only if 
.
We can show that this is true by considering the
following pyramid structure.
The definition of the normalized blending function
as a
weighted sum of 
and 
.
Thus for any of the N functions in the pyramid,
it is a weighted sum of the two items immediately to its right.
If we follow the pyramid to its right edge, we see that the only
blending functions 
that contribute to 
are
those with 
, and these function are collectively
nonzero when 
.
Summary
A B-spline blending function has compact support. The support of this function depends on the knot sequence and always covers an interval of containing several knots - containing k+1 knots if the curve is or order k.
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This document maintained by
Ken Joy
All contents copyright (c) 1996, 1997 |
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