矩形域上的參數(shù)曲面自由變形算法

張　莉　葛先玉　檀結(jié)慶,2

1(合肥工業(yè)大學(xué)數(shù)學(xué)學(xué)院　合肥　230009)2(合肥工業(yè)大學(xué)計(jì)算機(jī)與信息學(xué)院　合肥　230009)(lizhang@hfut.edu.cn)

矩形域上的參數(shù)曲面自由變形算法

張莉1葛先玉1檀結(jié)慶1,2

1(合肥工業(yè)大學(xué)數(shù)學(xué)學(xué)院合肥230009)2(合肥工業(yè)大學(xué)計(jì)算機(jī)與信息學(xué)院合肥230009)(lizhang@hfut.edu.cn)

摘要針對(duì)參數(shù)曲面的自由變形,構(gòu)造了矩形域上一類帶平臺(tái)的、分片多項(xiàng)式形式的伸縮函數(shù).新的伸縮函數(shù)具備了對(duì)稱性、單點(diǎn)峰值性、線性峰值性和區(qū)域峰值性等良好的性質(zhì);且具備了簡(jiǎn)單的多項(xiàng)式形式,易于構(gòu)造和操控;由此構(gòu)造的伸縮因子中的各個(gè)參數(shù)具備明顯的幾何意義,特別適用于交互設(shè)計(jì).將基于伸縮因子構(gòu)造的變形矩陣作用于待變形的曲面,可以獲得諸如凹凸、剪切、伸縮和改變變形中心等各類變形效果,借助于疊加變形可進(jìn)一步實(shí)現(xiàn)豐富多樣的復(fù)雜變形效果.該文付出了大量的數(shù)值實(shí)例,展示了各類變形效果以及伸縮函數(shù)中各個(gè)不同參數(shù)對(duì)形狀的調(diào)控.

關(guān)鍵詞參數(shù)曲面;伸縮因子;自由變形;矩形區(qū)域;變形效果

在計(jì)算機(jī)圖形學(xué)領(lǐng)域中,物體的變形一直是一個(gè)熱點(diǎn)問題.從眾多的相關(guān)研究來看,我們可以將物體變形大致分為基于物理的變形[1]和基于幾何的變形.基于物理的變形方法是將物體看作物理實(shí)體而非單純的幾何體,根據(jù)客觀規(guī)律對(duì)物體進(jìn)行變形,得到較為逼真的效果.而基于幾何的變形方法則是將物體抽象為空間中的幾何體,運(yùn)用CADCAM中的數(shù)字幾何處理技術(shù)對(duì)物體形狀進(jìn)行修改、編輯.其數(shù)學(xué)本質(zhì)就是對(duì)其逐點(diǎn)映射以生成滿足特定條件的新的幾何形狀,與基于物理的變形方法相比較,該類方法計(jì)算量小、簡(jiǎn)單易行、但逼真度欠佳.

有鑒于此,本文針對(duì)參數(shù)曲面的自由變形,通過構(gòu)造雙變量分片多項(xiàng)式型的伸縮函數(shù),給出了矩形域上一種交互良好,參數(shù)幾何意義明確,且能在點(diǎn)、線、矩形區(qū)域上達(dá)到峰值的參數(shù)曲面變形算法.這種算法無需依賴新的輔助工具,表達(dá)式中的每個(gè)參數(shù)都具備各自明顯的幾何屬性,可以在交互設(shè)計(jì)中靈活調(diào)用.

1伸縮因子的定義和性質(zhì)

1.1伸縮函數(shù)

首先給出二元伸縮函數(shù)的定義.

定義1. 設(shè)n為正整數(shù),r1,r2,s1,s2均為非負(fù)實(shí)

數(shù),其中0≤r1

D1={(x,y)||x-x0|≤r1,|y-y0|≤s1};

D2={(x,y)|r1<|x-x0|≤r2,|y-y0|≤s1};

D3={(x,y)||x-x0|≤r1,s1<|y-y0|≤s2};

D4={(x,y)|r1<|x-x0|≤r2,s1<|y-y0|≤s2};

記D=D1∪D2∪D3∪D4,定義2上伸縮函數(shù)如下:

(1)

為表述簡(jiǎn)便起見,這里的r=|x-x0|,s=|y-y0|.

由定義1可見,二元伸縮函數(shù)g(x,y)的支撐區(qū)域D共有9個(gè)部分,其中D1為峰值區(qū)域,區(qū)域D2,D3,D4環(huán)繞在峰值區(qū)域D1的四周,具體分布如圖1所示:

Fig. 1　Support area and peak region.圖1　支撐區(qū)域和峰值區(qū)域圖

二元伸縮函數(shù)g(x,y)具有5個(gè)性質(zhì):

Fig. 2 　Peak properties of extension function g(x,y).圖2　伸縮函數(shù)g(x,y)的峰值性

1) 區(qū)域峰值.(x,y)∈D1時(shí),g(x,y)取最大值1；(x,y)?D時(shí),g(x,y)取最小值0,取r1>0,s1>0,g(x,y)在D1上達(dá)到峰值,如圖2(a)所示.

2) 線型峰值.任取r1和s1中一個(gè)為0但不同時(shí)為0,這時(shí)峰值區(qū)域D1退化為線段,g(x,y)在線段上達(dá)到峰值,如圖2(b)所示.

3) 單點(diǎn)峰值.取r1=0,s1=0,這時(shí)峰值區(qū)域退化為單點(diǎn),g(x,y)具有單峰值,如圖2(c)所示.

容易證明性質(zhì)1～3是成立的,性質(zhì)4，5的證明見附錄A.

1.2伸縮因子

定義2. 設(shè)h為實(shí)數(shù),定義帶參數(shù)n,h的二元伸縮因子:

(2)

其中，n為光滑參數(shù),h為伸縮參數(shù),D為支撐區(qū)域,最值區(qū)域?yàn)镈1.

在式(2)的定義基礎(chǔ)上,可以得到伸縮因子具有5個(gè)與伸縮函數(shù)類似的性質(zhì):

1) 區(qū)域最值.取r1和s1均大于0,當(dāng)h≥0時(shí),E(x,y)有最大值1+h,h<0時(shí),E(x,y)有最小值1+h.

2) 線型最值.任取r1和s1中一個(gè)為0但不同時(shí)為0,E(x,y)在線段上達(dá)到最值.

3) 單點(diǎn)最值.r1=0,s1=0,E(x,y)在單點(diǎn)取最值.

4) 支撐區(qū)域D的邊界?D處的偏導(dǎo)值：

5) 內(nèi)部交界處的偏導(dǎo)值：

2空間曲面的變形與控制

2.1變形的數(shù)學(xué)模型

給定一張空間中的Cu(u≥1)類曲面P(x,y)=f(x,y)(亦可采用參數(shù)曲面形式),由定義2,伸縮因子為

Eij(x,y)=1+hij×g(x,y),i,j=1,2,3,

這組伸縮因子具有相同的支撐區(qū)域及光滑指數(shù),其中光滑指數(shù)n≤u+1.我們稱矩陣

則稱F=(e1,e2,e3)MN(e1,e2,e3)T為變形矩陣,特殊地,取e1=(1,0,0)T,e2=(0,1,0)T,e3=(0,0,1)T時(shí),F=M.

定義3. 在曲面定義域上選取中心點(diǎn)O′,變形后曲面Pd(x,y)與變形前曲面P(x,y)滿足關(guān)系:

Pd(x,y)=F×(P(x,y)-O′)+O′,

(3)

P(x,y)-O′=(l1,l2,l3)(c1,c2,c3)T,

則式(3)可以表示為

(4)

由此可以看出定義3中的變形有明確的幾何意義:在仿射坐標(biāo)系[O′,l1,l2,l3]下,對(duì)P(x,y)-O′的坐標(biāo)(c1,c2,c3)T作仿射變換,變換矩陣就是變形矩陣F.

定理1. 待變形曲面為Bézier或NURBS曲面時(shí),變形后曲面為分片Bézier或分片NURBS曲面,即形式上仍為Bézier或NURBS曲面.

證明. 作變換:

(5)

則定義2中伸縮因子可表示為

(6)

若待變形的曲面為K×S次Bézier曲面:

它的控制網(wǎng)格為bi,j(i=0,1,…,K;j=0,1,…,S),在區(qū)域D4={(x,y)|r1<|x-x0|≤r2,s1<|y-y0|≤s2},此時(shí),0≤u<1,0≤v<1,作形如式(5)的變換可得:

(7)

(1+hun(2-u)nvn(2-v)n)×

(8)

注意到式(8)中的簡(jiǎn)單變形,最后2步借助于冪基和Bernstein基的相互轉(zhuǎn)換[20],由于伸縮因子為分片多項(xiàng)式,故而變形后的曲面為分片Bézier曲面.復(fù)雜的變形可類似處理.關(guān)于NURBS曲面的討論類似.

2.2疊加變形

通常一次變形不能得到理想的曲面,我們需要對(duì)同一個(gè)曲面進(jìn)行多次變形,這里假定為k次,令P0,d(x,y)=P(x,y). 由定義3可知,相鄰2次變形Pi,d(x,y),Pi-1,d(x,y),i=1,2,…,k,滿足遞推關(guān)系:

(9)

Fi=(ei1,ei2,ei3)MiNi(ei1,ei2,ei3)T,

其中ei1,ei2,ei3和Mi分別為第i變形的主方向及伸縮矩陣,這里:

特別地,當(dāng)變形中心均取同一點(diǎn)O′時(shí),經(jīng)過k次變形后曲面Pd(x,y)與初始曲面P(x,y)滿足關(guān)系:

(10)

2.3變形與控制

在變形過程中,通過改變控制變形中心、峰值區(qū)域、光滑度、主方向等各個(gè)參數(shù),可以對(duì)待變形的初始曲面形狀進(jìn)行靈活的控制,以得到理想的曲面.

1) 要控制曲面的變形中心,只需改變O′.

2) 要控制曲面的峰值區(qū)域,可通過改變r(jià)1,s1.另外,r2和s2控制曲面變形的支撐區(qū)域.

3) 要控制變形曲面在支撐區(qū)域邊界的光滑度,可通過改變n的大小,n越大,變形后曲面與待變曲面越靠近.

4) 要控制曲面的變形主方向,只需改變e1,e2,e3.

5) 改變hi i的符號(hào),可以控制變形曲面沿主方向ei的正向或負(fù)向變形,改變|hi i|的大小,控制曲面沿ei變形的幅度,其中i=1,2,3.改變h12,h13的符號(hào),可以控制曲面沿e1的正向或負(fù)向剪切,改變|h12|, |h13|的大小,可以得到不同的剪切效果,值越大,剪切效果越明顯.類似地,可以通過h21,h23控制曲面沿e2的剪切效果,通過h31,h32控制曲面沿e3的剪切效果.在實(shí)際應(yīng)用中,通常同時(shí)改變多個(gè)參數(shù),以達(dá)到理想的變形效果.

3數(shù)值實(shí)例

例1. 圖3、圖4以拋物面z=(2-x2-y2)8+2為例,取光滑指數(shù)n=4,依次展示了改變控制參數(shù)引起的實(shí)際效果.為了方便起見,在本例中取主方向?yàn)樽鴺?biāo)軸方向.圖3(a)為原始拋物面.取變形中心1=(0,0,0),參數(shù)r1=1.5,s1=1.5,r2=3.5,s2=3.5來確定變形的峰值區(qū)域及支撐區(qū)域,取參數(shù)hi j(i,j=1,2,3)中h33=2.5,其余為0,獲得圖3(b)所示變形效果,原始拋物面在支撐區(qū)域內(nèi)沿主方向e3(本例中即為z軸)正向伸展變形.上述參數(shù)不變,僅改變變形中心為2=(1,-1,0),所得伸展變形效果見圖3(c).取變形中心O′=(0,0,0),通過對(duì)參數(shù)r1,s1的調(diào)控可以讓初始拋物面達(dá)到不同類型峰值區(qū)域的效果,如圖4(a)(b)(c)所示,依次取得區(qū)域峰值、線型峰值、單點(diǎn)峰值.給定相同的支撐區(qū)域和峰值區(qū)域,取h11=2,得到曲面沿x軸凸出的效果圖(如圖4(d)所示),分別取h12,h13=1,得到如圖4(e)和圖4(f)兩種沿x軸方向不同的剪切效果.圖4中各個(gè)子圖具體參數(shù)賦值情況如表1所示.

Fig. 3　Effects of changing deformation center O′.圖3　改變變形中心O′的效果

Iconr1s1r2s2hij(i,j=1,2,3)Fig.4(a)1.51.533h33=-1,otherhij=0Fig.4(b)1.5042h33=2.5,otherhij=0Fig.4(c)0044h33=2.5,otherhij=0Fig.4(d)1122h33=2,h11=2,otherhij=0Fig.4(e)1122h33=2,h12=1,otherhij=0Fig.4(f)1122h33=2,h13=1,otherhij=0

Fig. 4　Deformation effects after changing different parameters at the deformation center O′=(0,0,0)(see Table 1).圖4　O′=(0,0,0)處不同參數(shù)(如表1所示)變化得到的曲面變形效果

例2. 以平面z=1(如圖5(a)所示)為例,展示了復(fù)雜變形效果.圖5(b)是在原平面基礎(chǔ)上沿著z軸連續(xù)4次上凸變形得到的.圖5(c)通過2次變形得到十字架模型.在圖5(d)中,對(duì)平面進(jìn)行4次疊加變形,用h33控制每次伸縮的長(zhǎng)度,h13控制其搖擺的角度.圖5(e)則展示了對(duì)平面連續(xù)5次伸縮得到類似于peaks函數(shù)曲面效果.在圖5(f)中我們同時(shí)控制h33,h11,使得變形沿z軸上凸且沿x軸正負(fù)方向拉伸,得到類似扇子的形狀.上述各圖表明,通過調(diào)控各參數(shù),變形操作可以在點(diǎn)、線和區(qū)域達(dá)到峰值,將變形復(fù)合和疊加,可以靈活地控制和調(diào)整變形曲面的形狀,得到豐富多彩的效果.

Fig. 5　Complex deformation effects of the plane.圖5　平面上復(fù)雜變形效果

Fig. 6　Complex deformation effects of cylindrical surface.圖6　圓柱面的復(fù)雜變形效果

Fig. 7　Complex deformation effects of paraboloid.圖7　拋物面的復(fù)雜變形效果

4結(jié)論

本文針對(duì)參數(shù)曲面的自由變形,構(gòu)造了分片的雙變量多項(xiàng)式類型的伸縮函數(shù),考察了它在變形區(qū)域的邊界及內(nèi)部各個(gè)矩形區(qū)域交界處的偏導(dǎo)值,給出了一種交互性良好、幾何意義明確的新型參數(shù)曲面變形算法.該算法無需依賴任何輔助工具,不僅可以在單點(diǎn)處達(dá)到峰值,也可以在線、矩形區(qū)域上達(dá)到峰值,且表達(dá)式中的每個(gè)參數(shù)都具備各自明顯的幾何屬性.對(duì)于任意參數(shù)形式的曲面,我們可以通過調(diào)控各個(gè)變形參數(shù)得到期望的變形效果.

本文所給出的參數(shù)曲面自由變形方法,本質(zhì)上是通過建立二元的伸縮函數(shù),借助于伸縮因子和變形矩陣來實(shí)現(xiàn)參數(shù)曲面的自由變形,盡管具有交互簡(jiǎn)單、幾何意義直觀等特點(diǎn),但變形效果也受伸縮函數(shù)的約束,無法實(shí)現(xiàn)現(xiàn)有自由變形算法中一些效果,如基于體的變形方法中可實(shí)現(xiàn)的任意拓?fù)涞淖冃巍⒍喾直媛首冃蝃10],基于曲面的變形方法中可實(shí)現(xiàn)的基于細(xì)分曲面的多分辨率變形[21]、基于掃掠曲面的多層次的聯(lián)動(dòng)變形[22]等.

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Zhang Li, born in 1976. PhD and professor in Hefei University of Technology. Member of China Computer Federation. Her main research interests include computer-aided geometric design & computer graphics.

Ge Xianyu, born in 1991.Master candidate in Hefei University of Technology. His main research interests include computer-aided geometric design & computer graphics (shuxuegxy@126.com).

Tan Jieqing, born in 1962. Professor and PhD supervisor in Hefei University of Technology. His main research interests include non-linear numerical approximation theory, computer-aided geometric design and computer graphics. (jieqingtan@hfut.edu.cn).

附錄A.

正文定義1中二元伸縮函數(shù)g(x,y)性質(zhì)4和性質(zhì)5的證明.

證明.

1) 性質(zhì)4證明.由正文圖1可知,支撐區(qū)域D的邊界是由x=x0±r2,y=y0±s2, 4條線段圍成,按照二元伸縮函數(shù)在各個(gè)區(qū)域上的表達(dá)式,將D的邊界如圖A1所示分成3類:虛線段、點(diǎn)劃線段和點(diǎn)線段.

Fig. A1　Classified support area and peak region.圖A1　分類后的支撐區(qū)域和峰值區(qū)域圖

(1) 求解虛線段處的偏導(dǎo)值

二元伸縮函數(shù)g(x,y)在x=x0±r2作為區(qū)域D2邊界處的偏導(dǎo)值,即圖A1中虛線段.

①x=x0+r2時(shí),記:

L1={(x,y)|x=x0+r2,y0-s1≤y≤y0+s1},

由二元伸縮函數(shù)的定義知,(x,y)∈L1時(shí),

g(x,y)=g1(x)=

g(x,y)實(shí)際上是關(guān)于x的一元函數(shù),不妨記為g1(x),它關(guān)于y的偏導(dǎo)數(shù)值為0.從而當(dāng)l≠0時(shí),

0≤k+l=i≤n-1,

l=0時(shí),相當(dāng)于對(duì)函數(shù)g1(x)求解在x=x0+r2處直到n-1階的導(dǎo)數(shù)值,即:

(x-x0-2r1+r2)n-k,

其中:

c=Cki(-1)i-kn(n-1)…

(n-i+k+1)n(n-1)…(n-k+1),

由于0≤i≤n-1,且n-i+k≥1,代入邊界信息,有:

②x=x0-r2時(shí),記:

L2={(x,y)|x=x0-r2,y0-s1≤y≤y0+s1},

由二元伸縮函數(shù)的定義知,當(dāng)(x,y)∈L2時(shí),

g(x,y)=g2(x)=

g(x,y)實(shí)際上是關(guān)于x的一元函數(shù),不妨記為g2(x),它關(guān)于y的偏導(dǎo)數(shù)值為零,即當(dāng)l≠0時(shí),

l=0時(shí),求g2(x)在x=x0-r2處直到n-1階的導(dǎo)數(shù)值,類似于g1(x)的證明,我們有:

(2) 求解點(diǎn)劃線段處的偏導(dǎo)值.

從式(1)可以看出,g(x,y)在區(qū)域D2是x的單變量函數(shù),而在D3上恰好是y的單變量函數(shù),輪換變量x和y,上述對(duì)于D2邊界處偏導(dǎo)值的證明同樣適用于D3的邊界,從而可證明g(x,y)在點(diǎn)劃線段上偏導(dǎo)值為0.

(3) 求解點(diǎn)線段處的偏導(dǎo)值

圖A1中點(diǎn)線段為區(qū)域D4的邊界.由二元伸縮函數(shù)的對(duì)稱性,我們僅需考慮右上角的一部分,記:

L3={(x,y)|x=x0+r2,y0+s1≤y≤y0+s2};

L4={(x,y)|x0+r1≤x≤x0+r2,y=y0+s2};

當(dāng)(x,y)∈L3,

g(x,y)=g1(x)×g1(y)=

g(x,y)為變量可分離的二元伸縮函數(shù),因此有:

類似本證明步驟①②中關(guān)于x或者關(guān)于y的偏導(dǎo)值求解過程,可得:

從而:

這樣整個(gè)邊界?D的偏導(dǎo)值即證.

2) 同理可以證明性質(zhì)5.

證畢.

Free Form Deformation Method of Parametric Surfaces on Rectangular Region

Zhang Li1, Ge Xianyu1, and Tan Jieqing1,2

1(SchoolofMathematics,HefeiUniversityofTechnology,Hefei230009)2(SchoolofComputerandInformation,HefeiUniversityofTechnology,Hefei230009)

AbstractAccording to free form deformation of parametric surfaces, a new method based on extension function is proposed. It is made by piecewise polynomials and defined on rectangular region. Based on these, the new extension factor we constructed not only possesses perfect properties such as symmetry, single peak, linear peak and region peak, but also holds some parameters which have obvious geometric meanings. In real applications, the extension factor is very suitable for interactive design due to these properties and extraordinary parameters. Furthermore, the new extension factor is easy to construct and convenient to control because of its simple polynomial forms. Applying the deformation matrix constructed by the new extension factor to the original surfaces’ equations, plentiful deformation surfaces can be achieved. Deformation effects such as shearing, concave & convex, expand & contract and changing of deformation center can be obtained. With the help of superimposing, more complex deformation effects can be realized. Lots of numerical experiments are given at the end of paper which illustrate different kinds of deformation effects and plentiful adjusting effects of different parameters.

Key wordsparametric surface; extension factor; free form deformation; rectangular region; deformation effects

收稿日期：2014-11-28；修回日期：2015-08-11

基金項(xiàng)目：國(guó)家自然科學(xué)基金-廣東聯(lián)合基金重點(diǎn)項(xiàng)目(U1135003)；國(guó)家自然科學(xué)基金項(xiàng)目(61472466,61100126)；安徽省自然科學(xué)基金項(xiàng)目(1508085QF116)；中央高校基本科研業(yè)務(wù)費(fèi)專項(xiàng)資金項(xiàng)目(JZ2015HGXJ0175)；中國(guó)博士后科學(xué)基金項(xiàng)目(2015M571926)

中圖法分類號(hào)TP391.41

This work was supported by the National Natural Science Foundation of China-Guangdong Joint Foundation Key Project (U1135003), the National Natural Science Foundation of China (61472466,61100126), the Natural Science Fundation of Anhui Province of China (1508085QF116), and the Fundamental Research Funds for the Central Universities (JZ2015HGXJ0175), and the China Postdoctoral Science Foundation (2015M571926).