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Raeda Naamnieh
Outline
Subdivision of Bezier Curves
Restricted proof for Bezier Subdivision
Convergence of Refinement Strategies
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Definitions
• Definition 5.7
For
, the functions
for
where n is any nonnegative
integer, are called the generalized Bernstein
blending functions.
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Definitions
• Definition 16.11
We call
the Bezier curve with control points
on the interval
.
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• For
where
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defined as above then
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Outline
Subdivision of Bezier Curves
Restricted proof for Bezier Subdivision
Convergence of Refinement Strategies
9
Restricted Proof for Bezier
Subdivision
• Lemma 16.22
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Restricted Proof for Bezier
Subdivision
• Proof:
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Restricted Proof for Bezier
Subdivision
• Proof:
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Restricted Proof for Bezier
Subdivision
• Proof for Bezier Subdivision:
induction on n, and for arbitrary c, a<c<b.
If n=1
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Restricted Proof for Bezier
Subdivision
• Proof for Bezier Subdivision:
Now, assume the theorem holds for all
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Restricted Proof for Bezier
Subdivision
• Proof for Bezier Subdivision:
Now using the results from
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Restricted Proof for Bezier
Subdivision
• Proof for Bezier Subdivision:
-The second part of the proof is almost identical, hence
left as exercise
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Outline
Subdivision of Bezier Curves
Restricted proof for Bezier Subdivision
Convergence of Refinement Strategies
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Convergence of Refinement Strategies
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Convergence of Refinement Strategies
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Convergence of Refinement Strategies
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Convergence of Refinement Strategies
o
o
Bezier polygon defined on
.
the piecewise linear function given by the
original polygon.
o
the piecewise linear function formed with
vertices defined by concatenating together the
control polygons for the two subdivided curves
and
at the midpoint.
o It has 2n+1 distinct points.
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Convergence of Refinement Strategies
o
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is a pisewise linear function defined by the
ordered vertices of the control polygons of the
Bezier curves whose composite is the original
curve.
Convergence of Refinement Strategies
o
30
is a pisewise linear function defined by the
ordered vertices of the control polygons of the
Bezier curves whose composite is the original
curve.
Convergence of Refinement Strategies
o
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is a pisewise linear function defined by the
ordered vertices of the control polygons of the
Bezier curves whose composite is the original
curve.
Convergence of Refinement Strategies
o The subdivided Bezier curve at level
interval:
is over the
and has vertices:
for
o We shall write
has
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distinct points which define it.
Convergence of Refinement Strategies
Theorem 16.17:
That is, the polyline consisting of the union of all the
sub polygons converges to the Bezier curve.
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Convergence of Refinement Strategies
Lemma 16.18:
If
is a Bezier curve, define
. If
Are defined by the rule in Theorem 16.12, then
for
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Convergence of Refinement Strategies
Proof:
By induction on the superscript, for
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,
Convergence of Refinement Strategies
Proof:
Now, suppose that the conclusion has been shown
for superscripts up to
.Then,
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Convergence of Refinement Strategies
Lemma 16.19:
Any two consecutive vertices of are no farther
apart than
,where is independent of .
That is, if and are two consecutive vertices of
Then
.
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Convergence of Refinement Strategies
Proof:
Induction on ,
Let
First consider
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and
Convergence of Refinement Strategies
Proof:
Let
where
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Convergence of Refinement Strategies
Proof:
Now, suppose
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Convergence of Refinement Strategies
Proof:
Assume
for
. Now we show it is
true for
.
The vertices in are defined by subdividing the
Bezier polygons in
.
We see that
are formed by
subdividing the Bezier curve with control polygon
where
respectively.
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Convergence of Refinement Strategies
Proof:
We shall prove the results for
Let us fix
And call
By the subdivision Theorem 16.12
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Convergence of Refinement Strategies
Proof:
Since this is proved for all
lemma holds for all
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the conclusion of the
Convergence of Refinement Strategies
Proof for convergence theorem:
The subdivision theorem showed that over each
subinterval
,
the Bezier curve resulting from the appropriate sub
collection of
is identical to the original
We denote this by
.
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Convergence of Refinement Strategies
Proof for convergence theorem:
Any arbitrary value in the original interval is then
contained in an infinite sequence of intervals,
for which
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Convergence of Refinement Strategies
Proof for convergence theorem:
Hence, the curve value,
lies within the
convex hull of the
vertices of
which
correspond to the Bezier polygon over
, for each .
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Convergence of Refinement Strategies
Proof for convergence theorem:
Since
the spacial extent of the convex
hull of each Bezier polygon over
, all and , gets smaller and converges to zero.
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Convergence of Refinement Strategies
Proof for convergence theorem:
Consider the subsequence of polygons corresponding
to the intervals containing .
is contained in all of them, for all
Further, if any other curve point were contained in all
of them, say
, then would be in
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Convergence of Refinement Strategies
Proof for convergence theorem:
Since is the only point in that intersection,
is the only point in the intersection of
the convex hull of the Bezier polygons of these
selected subintervals.
The polygonal approximation converges.
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Summary
• Subdivision of Bezier Curves
• Restricted proof for Bezier Subdivision
• Convergence of Refinement Strategies
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Appendix
Geometric Modeling with
Splines
Chapter 16
Elaine Cohen
Richard F. Riesenfeld
Gershon Elber
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Q&A
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