計算機圖形學(xué)：分形幾何(雙語教學(xué))

Regularobjects’representation:Euclidean-geometrymethods.Irregularobjects’representation:Fractal-geometrymethods.

Review:

1Chapter8

FractalGeometry

分形幾何28.1whatarefractals

somepicturesandanimationfilms3Definitionsoffractals1.B.B.Mandelbrot

(In1982)AfractalisbydefinitionasetforwhichtheHausdorff-Besicovitchdimensionstrictlyexceedsthetopologicaldimension.

強調(diào)維數(shù)不是整數(shù)，是分?jǐn)?shù)，又稱分?jǐn)?shù)維

4Kochcurvesimilaritydimensionis1.26

567middlethirdCantorset

similaritydimension:0.68

8Sierpinskitrianglesimilaritydimension:1.58

910Definitionsoffractals1.B.B.Mandelbrot

(In1982)AfractalisbydefinitionasetforwhichtheHausdorff-Besicovitchdimensionstrictlyexceedsthetopologicaldimension.

強調(diào)維數(shù)不是整數(shù)，是分?jǐn)?shù)，又稱分?jǐn)?shù)維2.B.B.Mandelbrot

(In1986)

Afractalisshapemadeofpartssimilartothewholeinsomeway.

強調(diào)局部與整體自相似性11peanon=1n=2n=3n=412138.2FractalPropertiesFhasafinestructure,iedetailonarbitrarilysmallscales.Fhastooirregulartobedescribedintraditionalgeometricallanguage,bothlocallyandglobally.OftenFhassomeformofself-similarity,perhapsapproximateorstatistical.Usually,thefractaldimensionofFisgreaterthanitstopologicaldimension.InmostcasesofinterestofFisdefinedinaverysimpleway,perhapsrecursively.（遞歸迭代）148.3FractalDimension15Fractalsimilaritydimension:⑴thestraight-linesegmentscalenumberlength(r)(N)1/2211/331………1/nn11=N·r116⑵square(s=1)scalenumberarea(r)(N)(s)1/2411/391………1/nn211=N·r2

17⑶acube(v=1)scalenumbervolume(r)(N)(s)1/22311/3331………1/nn311=N·r318r—scalingfactorN—thenumberofsubparts

N·rD=1D=㏒N/㏒(1/r)1920·initiator—startwithagivengeometricshape8.4GeometricConstructionof

DeterministicSelf-SimilarFractals·generator—

subpartsoftheinitiatorarereplacedwithapattern21

Basicidea:constructionofthevonkoch

eachsegmentin(1)isreplacedbyanexactcopyoftheentirefigure,shrunkbyafactorof3.Thesameprocessisappliedtothesegmentsin(2)togeneratethosein(3).

22①②③④600-1200600(xs

,ys)Angle:>0counterclockwisedirection<0clockwisedirection23globalvariables:int

th;currentvalueoffloatx,y;x,ycoordinatesfloatd;thelengthofeachsegmentd=L/mn

m:等分?jǐn)?shù)24VoidGenerate–koch(n)//n:recursivedepth{if(n=0){x+=d*cos(th*3.14159/180)y+=d*sin(th*3.14159/180)lineto(x,y);return;}Generate–koch(n-1);

th+=60;Generate–koch(n-1);

th-=120;Generate–koch(n-1);

th+=60;Generate–koch(n-1);}25n=0d=Lth=0x=0y=0Generate–koch(0)x=d,y=0(0,0)(L,0)26n=1Generate-koch(1)Generate-koch(0)th+=60;Generate-koch(0)th-=1200Generate-koch(0)th+=60Generate-koch(0)27n=2Generate-koch(2)n=1Generate-koch(1)n=0Generate-koch(0)th=0x=0y=0d=L/32x=0+dcosth=dy=0+dsinth=0linetoth+=60Generate-koch(0)th=60x=dy=0x=d+dcosthy=0+dsinthlinetoth-=1200

Generate-koch(0)th=-600

x=d+dcos600y=dsin600x=x+dcosthy=y+dsinthlinetoth+=600Generate-koch(0)th=0x=x+dy=y+0lineto………28th+=850Generate-koch(1)n=0Generate-koch(0)th=600x=x+dcosthy=y+dsinthlinetoth+=600Generate-koch(0)th=1200x=x+dcosthy=y+dsinthlinetoth-=1200Generate-koch(0)th=0x=x+dy=y+0linetoth+=600Generate-koch(0)th=600x=x+dcosthy=y+dsinthlineto………n=2Generate-koch(2)29th-=1700Generate-koch(1)n=0Generate-koch(0)th+=600Generate-koch(0)th=0x=x+dy=y+0linetoth+=1200Generate-koch(0)th=-1200x=x+dcosthy=y+dsinthlinetoth+=600Generate-koch(0)th=-600x=x+dcosthy=y+dsinthlinetoth=-600x=x+dcosthy=y+dsinthlineto………n=2Generate-koch(2)30Generate-koch(1)n=0Generate-koch(0)th+=600Generate-koch(0)th=600x=x+dcosthy=y+dsinthlinetoth-=1200Generate-koch(0)th=-600x=x+dcosthy=y+dsinthlinetoth+=600Generate-koch(0)th=0x=x+dy=y+0linetoth=0x=x+dy=y+0linetoth+=600n=2Generate-koch(2)313233otherkindsofKoch①②③④⑤⑥⑦⑧⑨D=㏒N/㏒(1/r)=㏒9/㏒3=2①②③④⑤⑥⑦⑧D=㏒8/㏒4=1.534peanon=1n=2n=3n=4356QuestionsMapplottingbasedonfractalcurves363738種植果樹的山坡（韓云萍）39(a)(b)果實和果樹的構(gòu)造（韓云萍）401967年，美國《科學(xué)》雜志提出一個問題：英國海岸線有多長？

Mandelbrot對此問題的回答是：海岸線長度可以認(rèn)為是不確定的。

對此問題的分析:

如從高空飛行的飛機往下測量，測得的海岸線長度為x1。當(dāng)從低空飛行的飛機測得的海岸線長度為x2，