3B) B-splines.

B-splines are covered in some detail in <A HREF="../SMAG/node4.html"><B>SMAG</B> section 3</A> and in <B>R&amp;A</B> Section 5-9.

##NOTESONLY##

Parts of this Section of <B>R&amp;A</B> are included in the
handout. Beware that none of the worked examples are in the
handout. These may come in useful, and you will need to get hold of a
real copy of <B>R&amp;A</B> if you wish to work your way through them.

##-NOTESONLY##

<H2>Why B-splines?</H2>

B-splines have many nice properties when compared to other families of
curves which could be used. They:
<UL>
<LI>minimise the order of the polynomial pieces (order k)
<LI>maximise the continuity between pieces (continuity C(k-2))
<LI>minimise the number of control points controlling a piece (k points)
<LI>have positive basis functions
<LI>have basis functions which partition unity, implying that each piece lies
    inside its control points' convex hull
<LI>are invarient with respect to affine transforms
</UL>

<P>
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Exercises
<OL>

<LI>How many control points are required for a quartic Bezier and how
many for a quartic B-spline?

<LI>Why are cubics the default for B-spline use?

<LI>Explain the difference between Uniform, Open Uniform, and
Non-Uniform knot vectors. What are the advantages of each type?

</OL>
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</TD>
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