Raeda Naamnieh 1

Outline Subdivision of Bezier Curves Restricted proof for Bezier Subdivision Convergence of Refinement Strategies 2

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Definitions Definition 5.7 Definition 5.7 For, the functions for where n is any nonnegative integer, are called the generalized Bernstein blending functions. 4

Definitions Definition Definition We call the Bezier curve with control points on the interval. 5

For defined as above then where 6

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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 Lemma

Restricted Proof for Bezier Subdivision Proof: Proof: 11

Restricted Proof for Bezier Subdivision Proof: Proof: 12

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: induction on n, and for arbitrary c, a<c<b. If n=1 13

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: Now, assume the theorem holds for all 14

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: Now using the results from 15

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: - The second part of the proof is almost identical, hence left as exercise 16

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: We now show the theorem for the other part of the interval. Once again, set and 17

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: The argument for the right end of the interval proceeds the same way as above for the left side. 18

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: The argument for the right end of the interval proceeds the same way as above for the left side. 19

Restricted Proof for Bezier Subdivision Proof for Bezier Subdivision: Proof for Bezier Subdivision: The conclusion follows since 20

Outline Subdivision of Bezier Curves Restricted proof for Bezier Subdivision Convergence of Refinement Strategies 21

Convergence of Refinement Strategies 22

Convergence of Refinement Strategies 23

Convergence of Refinement Strategies 24

Convergence of Refinement Strategies 25

Convergence of Refinement Strategies 26

Convergence of Refinement Strategies 27

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. 28

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. 29

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. 30

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. 31

Convergence of Refinement Strategies o The subdivided Bezier curve at level is over the interval: and has vertices: for o We shall write has distinct points which define it. 32

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. 33

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 34

Convergence of Refinement Strategies Proof: By induction on the superscript, for, 35

Convergence of Refinement Strategies Proof: Now, suppose that the conclusion has been shown for superscripts up to.Then, 36

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. 37

Convergence of Refinement Strategies Proof: Induction on, Let First consider and 38

Convergence of Refinement Strategies Proof: Let where 39

Convergence of Refinement Strategies Proof: Now, suppose 40

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. 41

Convergence of Refinement Strategies Proof: We shall prove the results for Let us fix And call By the subdivision Theorem

Convergence of Refinement Strategies Proof: Since this is proved for all the conclusion of the lemma holds for all 43

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. 44

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 45

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. 46

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. 47

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 48

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. 49

Summary Subdivision of Bezier CurvesRestricted proof for Bezier SubdivisionConvergence of Refinement Strategies 50

Appendix Geometric Modeling with Splines Chapter 16 Elaine Cohen Richard F. Riesenfeld Gershon Elber 51

Q&A 52