六點(diǎn)Binary逼近細(xì)分法

朱　洪

(安徽三聯(lián)學(xué)院基礎(chǔ)部，合肥230601)

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六點(diǎn)Binary逼近細(xì)分法

朱洪

(安徽三聯(lián)學(xué)院基礎(chǔ)部，合肥230601)

[摘要]提出了一種新的細(xì)分算法——六點(diǎn)Binary逼近細(xì)分法.利用生成多項(xiàng)式等方法對(duì)細(xì)分法的一致收斂性和Ck連續(xù)性進(jìn)行了分析，通過(guò)對(duì)細(xì)分法中張力參數(shù)μ的不同取值，極限曲線可達(dá)到C0～C7連續(xù).特別是當(dāng)μ=11/1024時(shí)，極限曲線可達(dá)到C9連續(xù).數(shù)值算例表明，該方法是合理有效的.

[關(guān)鍵詞]Binary細(xì)分法； 逼近； 生成多項(xiàng)式； Ck連續(xù)性

1引言

細(xì)分法是根據(jù)初始數(shù)據(jù)或初始控制多邊形由計(jì)算機(jī)直接生成曲線曲面或其他幾何形體的一類方法，其處理方法簡(jiǎn)單、易于實(shí)現(xiàn)，因此在幾何造型中得到廣泛的應(yīng)用.Dyn等[1-2]從理論上對(duì)Binary細(xì)分格式及其極限曲線的存在性和光滑性進(jìn)行研究，利用三次Lagrange插值提出了四點(diǎn)Binary逼近細(xì)分法，可以構(gòu)造出C2連續(xù)曲線.Hassan等[3-4]提出了三點(diǎn)Ternary逼近細(xì)分法和四點(diǎn)Ternary插值細(xì)分法，并給出了三重格式的充分條件，其生成的極限曲線均達(dá)到C2連續(xù).Siddiqi等[5]利用三次均勻B樣條基函數(shù)構(gòu)造了一種三點(diǎn)Ternary逼近細(xì)分法，并生成了C2連續(xù)的極限曲線.Siddiqi等[6-7]提出了五點(diǎn)Binary逼近細(xì)分法和五點(diǎn)Ternary逼近細(xì)分法，其生成的極限曲線分別達(dá)到C3和C4連續(xù).Siddiqi等[8]提出了改進(jìn)的三點(diǎn)Binary細(xì)分算法.Ko等[9]將文獻(xiàn)[2]推廣到三重格式的情形，提出了四點(diǎn)Ternary逼近細(xì)分法.Daniel等[10]將靜態(tài)格式推廣到動(dòng)態(tài)格式，提出了動(dòng)態(tài)的三點(diǎn)Binary逼近細(xì)分法，可以產(chǎn)生C1連續(xù)的極限曲線.Bao-jun LI等[11]定義了一類帶有松弛參數(shù)列的動(dòng)態(tài)細(xì)分格式，并使得這種方法可以重構(gòu)指數(shù)多項(xiàng)式空間.檀結(jié)慶等[12]提出了基于插值細(xì)分的逼近細(xì)分法，在Hassan四點(diǎn)Ternary插值細(xì)分法中引入一個(gè)偏移參數(shù)，推導(dǎo)出一種逼近細(xì)分法，從而使Ternary插值細(xì)分和逼近細(xì)分統(tǒng)一到一個(gè)細(xì)分格式中.鄭紅蟬等[13]介紹了雙參數(shù)四點(diǎn)細(xì)分法及其性質(zhì).本文提出了六點(diǎn)Binary逼近細(xì)分法，并利用生成多項(xiàng)式方法討論了極限曲線的收斂性和Ck連續(xù)性，使造型曲線具有更大的光滑度，以滿足實(shí)際應(yīng)用的需要.

2預(yù)備知識(shí)

(1)

其中a={ai|i∈}為該細(xì)分法的掩模，將細(xì)分法記為S，則S的生成多項(xiàng)式為

定理1[1]若Binary細(xì)分法S一致收斂，則其掩模a={ai}滿足

(2)

定理2[1]設(shè)Binary細(xì)分法S的掩模a={ai}滿足式(2)，則存在一個(gè)Binary細(xì)分法S1，滿足

dPk=S1dPk-1,

(3)

3六點(diǎn)Binary逼近細(xì)分法的收斂性和Ck連續(xù)性分析

3.1　六點(diǎn)Binary逼近細(xì)分法

(4)

其中

μ為張力參數(shù)，且有

λ1+λ2+λ3+λ4+λ5+λ6=1.

3.2　收斂性和Ck連續(xù)性分析

證由細(xì)分法(4)生成的多項(xiàng)式為

根據(jù)定理2知

根據(jù)定理3知，六點(diǎn)Binary逼近細(xì)分法是一致收斂的.

又由定理2知

根據(jù)定理3知，六點(diǎn)Binary逼近細(xì)分法是C1連續(xù)的.

證根據(jù)定理2知，

又S4的生成多項(xiàng)式為

根據(jù)定理3知，六點(diǎn)Binary逼近細(xì)分法是C3連續(xù)的.

證根據(jù)定理2知

又S6的生成多項(xiàng)式為

根據(jù)定理3知，六點(diǎn)Binary逼近細(xì)分法是C5連續(xù)的.

證根據(jù)定理2知，

又S8的生成多項(xiàng)式為

4結(jié)論和數(shù)值算例

圖1　六點(diǎn)Binary逼近細(xì)分法算例

[參考文獻(xiàn)]圖2本文算法與其它幾種算法的比較

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A Six-Point Binary Approximating Subdivision Scheme

ZHUHong

(Department of Basic Courses, Anhui Sanlian University, hefei 230009, China)

Abstract:A binary six-point approximating subdivision scheme is presented. Using the generating polynomial method,the uniform convergence and Ck-continuity of subdivision scheme are analyzed. The subdivision scheme can be used to generate a family of Ck(k=1,2,…7) limit curves in certain range of tension parameter μ and C9limit curves forμ=11/1024. The numerical examples show that the proposed method is proper and effective.

Key words:binary subdivision scheme; approximation; generating polynomial; Ck-continuity

[中圖分類號(hào)]TP391

[文獻(xiàn)標(biāo)識(shí)碼]C

[文章編號(hào)]1672-1454(2015)05-0108-06

[基金項(xiàng)目]安徽三聯(lián)學(xué)院校級(jí)自然科學(xué)基金(2014Z002)

[收稿日期]2015-07-26