Raeda Naamnieh 1. Outline Subdivision of Bezier Curves Restricted proof for Bezier Subdivision Convergence of Refinement Strategies 2.

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Subdivision of Bezier curves

Raeda Naamnieh

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Outline

Subdivision of Bezier Curves

Restricted proof for Bezier Subdivision

Convergence of Refinement Strategies

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MOTIVATION

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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 16.11We call

the Bezier curve with control points on the interval .

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• For defined as above then

where

THE BEZIER CURVE SUBDIVISION THEOREM

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THE BEZIER CURVE SUBDIVISION THEOREM

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THE BEZIER CURVE SUBDIVISION THEOREM

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Outline

Subdivision of Bezier Curves

Restricted proof for Bezier Subdivision

Convergence of Refinement Strategies

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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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SUBDIVISION AT THE MIDPOINT

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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

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Convergence of Refinement Strategies

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Convergence of Refinement Strategies

o Bezier polygon defined on .o 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 is a pisewise linear function defined by the ordered vertices of the control polygons of the Bezier curves whose composite is the original curve.

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Convergence of Refinement Strategies

o is a pisewise linear function defined by the ordered vertices of the control polygons of the Bezier curves whose composite is the original curve.

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Convergence of Refinement Strategies

o is a pisewise linear function defined by the ordered vertices of the control polygons of the Bezier curves whose composite is the original curve.

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Convergence of Refinement Strategies

o The subdivided Bezier curve at level is over the interval:

and has vertices: for oWe shall write

has distinct points which define it.

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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 and

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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 the conclusion of the lemma holds for all

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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 16Elaine CohenRichard F. RiesenfeldGershon Elber

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Q&A