asymptote curve i

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Asymptote Curve Bagian Pertama Hirwanto Selasa, 27 Januari 2015 Jam 11 : 29 Daftar Isi 1 Asymptote Curve Bagian Pertama 1 1.1 Folium of Descartes .............................. 1 1.2 Klein Bottle .................................. 7 1.3 Sierpinsksi Sponge .............................. 9 1.4 Calabi - Yau .................................. 12 1.5 Teapot ..................................... 14 2 Kompilasi Dokumen 21 1 Asymptote Curve Bagian Pertama Pada tutorial ini kita hanya menjelaskan aplikasi Asymptote untuk bidang yang le- bih mendalam sehingga diharapkan dapat memberikan hasil yang maksimal ketika pada penerapan sederhana. Semua dokumen ini diambil dari forum maupun situs Asymptote. 1.1 Folium of Descartes Dalam bidang Geometri, Folium of Descartes adalah kurva aljabar yang didefinisikan oleh persamaan berikut ini : x 3 + y 3 - 3axy =0 Untuk membentuk loop nya dalam kuadran pertama dengan tittik ganda di titik asal dan asymptote. Kesimetriaanynya kira -kira y = x. Nama Folium berasal dari bahasa latin yang berarti leaf atau daun. Kurva di namaka Folium of Descartes berdasar pada penemunya dan gambar perangak Albani di tahun 1966. Untuk dapat menggambar Folium of Descartes. 1

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Asymptote CurveBagian Pertama

Hirwanto

Selasa, 27 Januari 2015Jam 11 : 29

Daftar Isi

1 Asymptote Curve Bagian Pertama 11.1 Folium of Descartes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Klein Bottle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Sierpinsksi Sponge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Calabi - Yau . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Teapot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Kompilasi Dokumen 21

1 Asymptote Curve Bagian Pertama

Pada tutorial ini kita hanya menjelaskan aplikasi Asymptote untuk bidang yang le-bih mendalam sehingga diharapkan dapat memberikan hasil yang maksimal ketika padapenerapan sederhana. Semua dokumen ini diambil dari forum maupun situs Asymptote.

1.1 Folium of Descartes

Dalam bidang Geometri, Folium of Descartes adalah kurva aljabar yang didefinisikanoleh persamaan berikut ini :

x3 + y3 − 3axy = 0

Untuk membentuk loop nya dalam kuadran pertama dengan tittik ganda di titik asaldan asymptote.

Kesimetriaanynya kira -kira y = x. Nama Folium berasal dari bahasa latin yangberarti leaf atau daun. Kurva di namaka Folium of Descartes berdasar pada penemunyadan gambar perangak Albani di tahun 1966.

Untuk dapat menggambar Folium of Descartes.

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Gambar 1: Folium of Descartes

Ada sedikitnya ada 3 cara dalam menggambarkan yaitu sebagai berikut :

1. Menggunakan Asymptote yang kita sisipkan didokumen LATEX.

2. Menggunakan PSTricks yang kita sisipkan didokumen LATEX.

3. Menggunakan MetaPost yang kita sisipkan didokumen LATEX.

Namun kita hanya akan membahas cara pertama saja.

%======================================% Dokumen ini dibuat/diedit oleh% Nama : Hirwanto

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% Email : [email protected]% Tanggal : 27 Januari 2015, 4 : 17 AM% Kompilasi : Default(PDFTeXify) + Add Ons Asymptote for WinEdt% : Juat compile your dokumen with PDFTeXiFy.%======================================\documentclass{article}\usepackage{asymptote}\begin{document}\begin{center}% Salin Kode Asymptote 1\end{center}\begin{center}% Salin Kode Asymptote 2\end{center}\end{document}

Untuk kode Asymptote 1

\begin{asy}// file fod.asy//// to get fod.pdf, run ‘asy −f pdf fod.asy‘//size(8cm);import graph;import fontsize;defaultpen(fontsize(9pt));

texpreamble(”\usepackage{lmodern}”);

pen curvepen=darkblue+0.8bp;pen linepen=darkred+0.8bp;pen fillpen=orange+opacity(0.5);

realxmin=−20, xmax=−xmin,ymin=−20, ymax=−ymin;

xaxis(xmin,xmax,RightTicks(Step=10,step=5,OmitTick(0)));yaxis(ymin,ymax, LeftTicks(Step=10,step=5,OmitTick(0)));

real a=10;

real r(real t){return 3∗a∗sin(t)∗cos(t)/(sin(t)ˆ3+cos(t)ˆ3);};

real tmin=−0.16pi, tmax=pi/2−tmin;

3

guideloop=polargraph(r,0,pi/2)−−cycle,curve=polargraph(r,tmin,tmax);

fill(loop, fillpen);draw(curve,curvepen);

pairp=point(curve,0),q=point(curve,length(curve));

draw((p.x,−p.x−a)−−(−q.y−a,q.y),linepen);\end{asy}

Untuk kode Asymptote 2

\begin{asy}// file fodsp.asy//// to get fodsp.pdf, run ‘asy −f pdf fodsp.asy‘//size(8cm);import graph;import fontsize; defaultpen(fontsize(9pt));texpreamble(”\usepackage{lmodern}”);

pen[] fillpen={red, orange, yellow, green, lightblue, blue, darkblue};

realxmin=0, xmax=20,ymin=0, ymax=20;

xaxis(xmin,xmax,RightTicks(Step=10,step=5));yaxis(ymin,ymax, LeftTicks(Step=10,step=5));

real ra(real t, real a){return 3∗a∗sin(t)∗cos(t)/(sin(t)ˆ3+cos(t)ˆ3);};real r(real);guide loop;

real a, a0=10, da=1;int n=fillpen.length;

real t; pair p;

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Gambar 2: Folium of Descartes dalam bentuk Sederhana

a=a0;for(int i=0;i<n;++i){

r=new real(real t){return ra(t,a);};loop =polargraph(r,0,pi/2)−−cycle;filldraw(loop, 0.7fillpen[i]+0.3white,fillpen[i]);t=atan(2ˆ(1/3));p=r(t)∗(cos(t),sin(t));unfill(circle(p,0.7));label(”$”+string(a)+”$”,p);a−=da;}

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label(”$r(\theta)=\displaystyle”+”\frac{3 a \sin\theta\cos\theta}{\sinˆ3\theta+\cosˆ3\theta}$, ”+”$\theta=[0,\frac\pi2]$, ”+”$a=”+string(a0−(n−1)∗da)+”$−−$”+string(a0)+”$”,((xmin+xmax)/2,ymax),S);

shipout(bbox(paleyellow,Fill));\end{asy}

Gambar 3: Folium of Descartes lebih Cantik

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Gambar 4: Klein Bottle

1.2 Klein Bottle

Klein Bottle pertama kali di temukan oleh Ahli Matematika dari Jerman dan padaabad 19 juga (tahun 1882), yang bernama Felix Klein. Klein bottle ini benar-benartidak mempunyai batas (no boundary), karena sangat menyatu dan tidak bisa dibedakanmana bagian luar dan mana bagian dalam.

Sebuah Klein bottle dibentuk dengan menggabungkan dua sisi sebuah lembaran untukmembentuk silinder, kemudian ujung silinder melingkari melalui dirinya sendiri dengansedemikian rupa sehingga bagian dalam (hijau) dan luar (putih) dari silinder bergabung.Jika kita mencoba memasukkan air kedalam lubang botol tersebut, maka air akan terjebakdi bagian ’dalam’ botol

Untuk kode Asymptote 3 :

\begin{asy}import graph3;

size(469pt);

viewportmargin=0;

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currentprojection=perspective(camera=(25.0851928432063,−30.3337528952473,19.3728775115443),up=Z,target=(−0.590622314050054,0.692357205025578,−0.627122488455679),zoom=1,autoadjust=false);

triple f(pair t) {real u=t.x;real v=t.y;real r=2−cos(u);real x=3∗cos(u)∗(1+sin(u))+r∗cos(v)∗(u < pi ? cos(u) : −1);real y=8∗sin(u)+(u < pi ? r∗sin(u)∗cos(v) : 0);real z=r∗sin(v);return (x,y,z);}

surface s=surface(f,(0,0),(2pi,2pi),8,8,Spline);draw(s,lightolive+white,”bottle”,render(merge=true));

string lo=”$\displaystyle u\in[0,\pi]: \cases{x=3\cos u(1+\sin u)+(2−\cos u)\cos u\cosv,\cr

y=8\sin u+(2−\cos u)\sin u\cos v,\crz=(2−\cos u)\sin v.\cr}$”;

string hi=”$\displaystyle u\in[\pi,2\pi]:\\\cases{x=3\cos u(1+\sin u)−(2−\cos u)\cosv,\cr

y=8\sin u,\crz=(2−\cos u)\sin v.\cr}$”;

real h=0.0125;

begingroup3(”parametrization”);draw(surface(xscale(−0.38)∗yscale(−0.18)∗lo,s,0,1.7,h,bottom=false),

”[0,pi]”);draw(surface(xscale(0.26)∗yscale(0.1)∗rotate(90)∗hi,s,4.9,1.4,h,bottom=false),

”[pi,2pi]”);endgroup3();

begingroup3(”boundary”);draw(s.uequals(0),blue+dashed);draw(s.uequals(pi),blue+dashed);endgroup3();

add(new void(frame f, transform3 t, picture pic, projection P) {draw(f,invert(box(min(f,P),max(f,P)),P),”frame”);

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});\end{asy}

1.3 Sierpinsksi Sponge

Gambar 5: Sierpinski Sponge

Didalam ilmu matematika, dikenal Menger sponge. Menger sponge merupakan ku-rva fraktal dan perumuman dari himpunan cantor dan karpet Sierpinski. Kurva ini kali

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pertama dijelaskan oleh Karl Menger pada tahun 1926 dalam pembelajaran konsep ru-ang topologi. Menger sponge secara simultan mempunyai bidang area tak hingga danvolumenya bernilai nol. Hebat bukan ?

Untuk kode Asymptote 4

\begin{asy}size(200);import palette;import three;

currentprojection=orthographic(1,1,1);

triple[] M={(−1,−1,−1),(0,−1,−1),(1,−1,−1),(1,0,−1),(1,1,−1),(0,1,−1),(−1,1,−1),(−1,0,−1),(−1,−1,0),(1,−1,0),(1,1,0),(−1,1,0),(−1,−1,1),(0,−1,1),(1,−1,1),(1,0,1),(1,1,1),(0,1,1),(−1,1,1),(−1,0,1)};

surface[] Squares={surface((1,−1,−1)−−(1,1,−1)−−(1,1,1)−−(1,−1,1)−−cycle),surface((−1,−1,−1)−−(−1,1,−1)−−(−1,1,1)−−(−1,−1,1)−−cycle),surface((1,1,−1)−−(−1,1,−1)−−(−1,1,1)−−(1,1,1)−−cycle),surface((1,−1,−1)−−(−1,−1,−1)−−(−1,−1,1)−−(1,−1,1)−−cycle),surface((1,−1,1)−−(1,1,1)−−(−1,1,1)−−(−1,−1,1)−−cycle),surface((1,−1,−1)−−(1,1,−1)−−(−1,1,−1)−−(−1,−1,−1)−−cycle),};

int[][] SquaresPoints={{2,3,4,10,16,15,14,9},{0,7,6,11,18,19,12,8},{4,5,6,11,18,17,16,10},{2,1,0,8,12,13,14,9},{12,13,14,15,16,17,18,19},{0,1,2,3,4,5,6,7}};

int[][] index={{0,2,4},{0,1},{1,2,4},{2,3},{1,3,4},{0,1},{0,3,4},{2,3},{4,5},{4,5},{4,5},{4,5},{0,2,5},{0,1},{1,2,5},{2,3},{1,3,5},{0,1},{0,3,5},{2,3}};

int[] Sponge0=array(n=6,value=1);

int[] eraseFaces(int n, int[] Sponge0) {

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int[] temp=copy(Sponge0);for(int k : index[n]) {

temp[k]=0;}return temp;}

int[][] Sponge1=new int[20][];for(int n=0; n < 20; ++n) {

Sponge1[n]=eraseFaces(n,Sponge0);}

int[][] eraseFaces(int n, int[][] Sponge1) {int[][] temp=copy(Sponge1);for(int k : index[n])

for(int n1 : SquaresPoints[k])temp[n1][k]=0;

return temp;}

int[][][] Sponge2=new int[20][][];for(int n=0; n < 20; ++n)

Sponge2[n]=eraseFaces(n,Sponge1);

int[][][] eraseFaces(int n, int[][][] Sponge2) {int[][][] temp=copy(Sponge2);for(int k : index[n])

for(int n2: SquaresPoints[k])for(int n1: SquaresPoints[k])

temp[n2][n1][k]=0;return temp;}

int[][][][] Sponge3=new int[20][][][];for(int n=0; n < 20; ++n)

Sponge3[n]=eraseFaces(n,Sponge2);

surface s3;real u=2/3;for(int n3=0; n3 < 20; ++n3) {

surface s2;for(int n2=0; n2 < 20; ++n2) {

surface s1;for(int n1=0; n1 < 20; ++n1) {

for(int k=0; k < 6; ++k){transform3 T=scale3(u)∗shift(M[n1])∗scale3(0.5);

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if(Sponge3[n3][n2][n1][k] > 0) {s1.append(T∗Squares[k]);}}}transform3 T=scale3(u)∗shift(M[n2])∗scale3(0.5);s2.append(T∗s1);}transform3 T=scale3(u)∗shift(M[n3])∗scale3(0.5);s3.append(T∗s2);}s3.colors(palette(s3.map(abs),Rainbow()));draw(s3);

\end{asy}

1.4 Calabi - Yau

Manifolde Calabi -Yu atau terkenal dengan ruang Calabi -Yau merupakan tipe kasuskhusus Manifold yang menjelaskan cabang matematika seperti geometri aljabar. SifatCalabi -Yau seperti Ricci flatnes juga aplikasi fisika teoritis . Khususnya dalam teorisuperstring. Untuk kode Asymptote 5

\begin{asy}[width=10cm,height=10cm]import graph3;

size3(200);currentprojection=orthographic(3,3,2);currentlight=light(8,10,2);

int k1, k2, n = 5;real alpha = 0.3∗pi;

// cross section of the quintic 6D Calabi−Yau manifoldtriple cy(pair z) {

pair z1, z2;

if(z==(0,0)) {z1 = exp(2∗pi∗I∗k1/n);z2 = 0;} else {

z1 = exp(2∗pi∗I∗k1/n)∗exp(log(cos(I∗z))∗2/n);z2 = exp(2∗pi∗I∗k2/n)∗exp(log(−I∗sin(I∗z))∗2/n);}

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Gambar 6: Calabi -Yau

return (z2.x, cos(alpha)∗z1.y + sin(alpha)∗z2.y, z1.x);

}

for(k1=0; k1<n; ++k1) {for(k2=0; k2<n; ++k2) {

surface s = surface(cy,(−1,0),(1,0.5∗pi),20,20);draw(s,yellow+orange);}}

\end{asy}

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1.5 Teapot

Gambar 7: Calabi -Yau

Teapot dapat diartikan sebagai ceret yang biasa digunakan untuk minum tea / terka-dang juga hanya air mineral biasa.

Untuk kode Asymptote 6

\begin{asy}import three;

size(20cm);

currentprojection=perspective(250,−250,250);currentlight=Viewport;

triple[][][] Q={{{(39.68504,0,68.0315),(37.91339,0,71.75197),(40.74803,0,71.75197),(42.51969,0,68.0315)},{(39.68504,−22.22362,68.0315),(37.91339,−21.2315,71.75197),(40.74803,−22.8189,71.75197),(42.51969,−23.81102,68.0315)},

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{(22.22362,−39.68504,68.0315),(21.2315,−37.91339,71.75197),(22.8189,−40.74803,71.75197),(23.81102,−42.51969,68.0315)},{(0,−39.68504,68.0315),(0,−37.91339,71.75197),(0,−40.74803,71.75197),(0,−42.51969,68.0315)}

},{{(0,−39.68504,68.0315),(0,−37.91339,71.75197),(0,−40.74803,71.75197),(0,−42.51969,68.0315)},{(−22.22362,−39.68504,68.0315),(−21.2315,−37.91339,71.75197),(−22.8189,−40.74803,71.75197),(−23.81102,−42.51969,68.0315)},{(−39.68504,−22.22362,68.0315),(−37.91339,−21.2315,71.75197),(−40.74803,−22.8189,71.75197),(−42.51969,−23.81102,68.0315)},{(−39.68504,0,68.0315),(−37.91339,0,71.75197),(−40.74803,0,71.75197),(−42.51969,0,68.0315)}

},{{(−39.68504,0,68.0315),(−37.91339,0,71.75197),(−40.74803,0,71.75197),(−42.51969,0,68.0315)},{(−39.68504,22.22362,68.0315),(−37.91339,21.2315,71.75197),(−40.74803,22.8189,71.75197),(−42.51969,23.81102,68.0315)},{(−22.22362,39.68504,68.0315),(−21.2315,37.91339,71.75197),(−22.8189,40.74803,71.75197),(−23.81102,42.51969,68.0315)},{(0,39.68504,68.0315),(0,37.91339,71.75197),(0,40.74803,71.75197),(0,42.51969,68.0315)}

},{{(0,39.68504,68.0315),(0,37.91339,71.75197),(0,40.74803,71.75197),(0,42.51969,68.0315)},{(22.22362,39.68504,68.0315),(21.2315,37.91339,71.75197),(22.8189,40.74803,71.75197),(23.81102,42.51969,68.0315)},{(39.68504,22.22362,68.0315),(37.91339,21.2315,71.75197),(40.74803,22.8189,71.75197),(42.51969,23.81102,68.0315)},{(39.68504,0,68.0315),(37.91339,0,71.75197),(40.74803,0,71.75197),(42.51969,0,68.0315)}

},{{(42.51969,0,68.0315),(49.60629,0,53.1496),(56.69291,0,38.26771),(56.69291,0,25.51181)},{(42.51969,−23.81102,68.0315),(49.60629,−27.77952,53.1496),(56.69291,−31.74803,38.26771),(56.69291,−31.74803,25.51181)},{(23.81102,−42.51969,68.0315),(27.77952,−49.60629,53.1496),(31.74803,−56.69291,38.26771),(31.74803,−56.69291,25.51181)},{(0,−42.51969,68.0315),(0,−49.60629,53.1496),(0,−56.69291,38.26771),(0,−56.69291,25.51181)}

},{{(0,−42.51969,68.0315),(0,−49.60629,53.1496),(0,−56.69291,38.26771),(0,−56.69291,25.51181)},{(−23.81102,−42.51969,68.0315),(−27.77952,−49.60629,53.1496),(−31.74803,−56.69291,38.26771),(−31.74803,−56.69291,25.51181)},

15

{(−42.51969,−23.81102,68.0315),(−49.60629,−27.77952,53.1496),(−56.69291,−31.74803,38.26771),(−56.69291,−31.74803,25.51181)},{(−42.51969,0,68.0315),(−49.60629,0,53.1496),(−56.69291,0,38.26771),(−56.69291,0,25.51181)}

},{{(−42.51969,0,68.0315),(−49.60629,0,53.1496),(−56.69291,0,38.26771),(−56.69291,0,25.51181)},{(−42.51969,23.81102,68.0315),(−49.60629,27.77952,53.1496),(−56.69291,31.74803,38.26771),(−56.69291,31.74803,25.51181)},{(−23.81102,42.51969,68.0315),(−27.77952,49.60629,53.1496),(−31.74803,56.69291,38.26771),(−31.74803,56.69291,25.51181)},{(0,42.51969,68.0315),(0,49.60629,53.1496),(0,56.69291,38.26771),(0,56.69291,25.51181)}

},{{(0,42.51969,68.0315),(0,49.60629,53.1496),(0,56.69291,38.26771),(0,56.69291,25.51181)},{(23.81102,42.51969,68.0315),(27.77952,49.60629,53.1496),(31.74803,56.69291,38.26771),(31.74803,56.69291,25.51181)},{(42.51969,23.81102,68.0315),(49.60629,27.77952,53.1496),(56.69291,31.74803,38.26771),(56.69291,31.74803,25.51181)},{(42.51969,0,68.0315),(49.60629,0,53.1496),(56.69291,0,38.26771),(56.69291,0,25.51181)}

},{{(56.69291,0,25.51181),(56.69291,0,12.7559),(42.51969,0,6.377957),(42.51969,0,4.251961)},{(56.69291,−31.74803,25.51181),(56.69291,−31.74803,12.7559),(42.51969,−23.81102,6.377957),(42.51969,−23.81102,4.251961)},{(31.74803,−56.69291,25.51181),(31.74803,−56.69291,12.7559),(23.81102,−42.51969,6.377957),(23.81102,−42.51969,4.251961)},{(0,−56.69291,25.51181),(0,−56.69291,12.7559),(0,−42.51969,6.377957),(0,−42.51969,4.251961)}

},{{(0,−56.69291,25.51181),(0,−56.69291,12.7559),(0,−42.51969,6.377957),(0,−42.51969,4.251961)},{(−31.74803,−56.69291,25.51181),(−31.74803,−56.69291,12.7559),(−23.81102,−42.51969,6.377957),(−23.81102,−42.51969,4.251961)},{(−56.69291,−31.74803,25.51181),(−56.69291,−31.74803,12.7559),(−42.51969,−23.81102,6.377957),(−42.51969,−23.81102,4.251961)},{(−56.69291,0,25.51181),(−56.69291,0,12.7559),(−42.51969,0,6.377957),(−42.51969,0,4.251961)}

},{{(−56.69291,0,25.51181),(−56.69291,0,12.7559),(−42.51969,0,6.377957),(−42.51969,0,4.251961)},{(−56.69291,31.74803,25.51181),(−56.69291,31.74803,12.7559),(−42.51969,23.81102,6.377957),(−42.51969,23.81102,4.251961)},

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{(−31.74803,56.69291,25.51181),(−31.74803,56.69291,12.7559),(−23.81102,42.51969,6.377957),(−23.81102,42.51969,4.251961)},{(0,56.69291,25.51181),(0,56.69291,12.7559),(0,42.51969,6.377957),(0,42.51969,4.251961)}

},{{(0,56.69291,25.51181),(0,56.69291,12.7559),(0,42.51969,6.377957),(0,42.51969,4.251961)},{(31.74803,56.69291,25.51181),(31.74803,56.69291,12.7559),(23.81102,42.51969,6.377957),(23.81102,42.51969,4.251961)},{(56.69291,31.74803,25.51181),(56.69291,31.74803,12.7559),(42.51969,23.81102,6.377957),(42.51969,23.81102,4.251961)},{(56.69291,0,25.51181),(56.69291,0,12.7559),(42.51969,0,6.377957),(42.51969,0,4.251961)}

},{{(−45.35433,0,57.40157),(−65.19685,0,57.40157),(−76.53543,0,57.40157),(−76.53543,0,51.02362)},{(−45.35433,−8.503932,57.40157),(−65.19685,−8.503932,57.40157),(−76.53543,−8.503932,57.40157),(−76.53543,−8.503932,51.02362)},{(−42.51969,−8.503932,63.77952),(−70.86614,−8.503932,63.77952),(−85.03937,−8.503932,63.77952),(−85.03937,−8.503932,51.02362)},{(−42.51969,0,63.77952),(−70.86614,0,63.77952),(−85.03937,0,63.77952),(−85.03937,0,51.02362)}

},{{(−42.51969,0,63.77952),(−70.86614,0,63.77952),(−85.03937,0,63.77952),(−85.03937,0,51.02362)},{(−42.51969,8.503932,63.77952),(−70.86614,8.503932,63.77952),(−85.03937,8.503932,63.77952),(−85.03937,8.503932,51.02362)},{(−45.35433,8.503932,57.40157),(−65.19685,8.503932,57.40157),(−76.53543,8.503932,57.40157),(−76.53543,8.503932,51.02362)},{(−45.35433,0,57.40157),(−65.19685,0,57.40157),(−76.53543,0,57.40157),(−76.53543,0,51.02362)}

},{{(−76.53543,0,51.02362),(−76.53543,0,44.64566),(−70.86614,0,31.88976),(−56.69291,0,25.51181)},{(−76.53543,−8.503932,51.02362),(−76.53543,−8.503932,44.64566),(−70.86614,−8.503932,31.88976),(−56.69291,−8.503932,25.51181)},{(−85.03937,−8.503932,51.02362),(−85.03937,−8.503932,38.26771),(−75.11811,−8.503932,26.5748),(−53.85826,−8.503932,17.00787)},{(−85.03937,0,51.02362),(−85.03937,0,38.26771),(−75.11811,0,26.5748),(−53.85826,0,17.00787)}

},{{(−85.03937,0,51.02362),(−85.03937,0,38.26771),(−75.11811,0,26.5748),(−53.85826,0,17.00787)},{(−85.03937,8.503932,51.02362),(−85.03937,8.503932,38.26771),(−75.11811,8.503932,26.5748),(−53.85826,8.503932,17.00787)},

17

{(−76.53543,8.503932,51.02362),(−76.53543,8.503932,44.64566),(−70.86614,8.503932,31.88976),(−56.69291,8.503932,25.51181)},{(−76.53543,0,51.02362),(−76.53543,0,44.64566),(−70.86614,0,31.88976),(−56.69291,0,25.51181)}

},{{(48.18897,0,40.3937),(73.70078,0,40.3937),(65.19685,0,59.52755),(76.53543,0,68.0315)},{(48.18897,−18.70866,40.3937),(73.70078,−18.70866,40.3937),(65.19685,−7.086619,59.52755),(76.53543,−7.086619,68.0315)},{(48.18897,−18.70866,17.00787),(87.87401,−18.70866,23.38582),(68.0315,−7.086619,57.40157),(93.5433,−7.086619,68.0315)},{(48.18897,0,17.00787),(87.87401,0,23.38582),(68.0315,0,57.40157),(93.5433,0,68.0315)}

},{{(48.18897,0,17.00787),(87.87401,0,23.38582),(68.0315,0,57.40157),(93.5433,0,68.0315)},{(48.18897,18.70866,17.00787),(87.87401,18.70866,23.38582),(68.0315,7.086619,57.40157),(93.5433,7.086619,68.0315)},{(48.18897,18.70866,40.3937),(73.70078,18.70866,40.3937),(65.19685,7.086619,59.52755),(76.53543,7.086619,68.0315)},{(48.18897,0,40.3937),(73.70078,0,40.3937),(65.19685,0,59.52755),(76.53543,0,68.0315)}

},{{(76.53543,0,68.0315),(79.37007,0,70.15748),(82.20472,0,70.15748),(79.37007,0,68.0315)},{(76.53543,−7.086619,68.0315),(79.37007,−7.086619,70.15748),(82.20472,−4.251961,70.15748),(79.37007,−4.251961,68.0315)},{(93.5433,−7.086619,68.0315),(99.92125,−7.086619,70.68897),(97.79527,−4.251961,71.22047),(90.70866,−4.251961,68.0315)},{(93.5433,0,68.0315),(99.92125,0,70.68897),(97.79527,0,71.22047),(90.70866,0,68.0315)}

},{{(93.5433,0,68.0315),(99.92125,0,70.68897),(97.79527,0,71.22047),(90.70866,0,68.0315)},{(93.5433,7.086619,68.0315),(99.92125,7.086619,70.68897),(97.79527,4.251961,71.22047),(90.70866,4.251961,68.0315)},{(76.53543,7.086619,68.0315),(79.37007,7.086619,70.15748),(82.20472,4.251961,70.15748),(79.37007,4.251961,68.0315)},{(76.53543,0,68.0315),(79.37007,0,70.15748),(82.20472,0,70.15748),(79.37007,0,68.0315)}

},{{(0,0,89.29133),(22.67716,0,89.29133),(0,0,80.7874),(5.669294,0,76.53543)},{(0,0,89.29133),(22.67716,−12.7559,89.29133),(0,0,80.7874),(5.669294,−3.174809,76.53543)},{(0,0,89.29133),(12.7559,−22.67716,89.29133),(0,0,80.7874),(3.174809,−5.669294,76.53543)},

18

{(0,0,89.29133),(0,−22.67716,89.29133),(0,0,80.7874),(0,−5.669294,76.53543)}},{{(0,0,89.29133),(0,−22.67716,89.29133),(0,0,80.7874),(0,−5.669294,76.53543)},{(0,0,89.29133),(−12.7559,−22.67716,89.29133),(0,0,80.7874),(−3.174809,−5.669294,76.53543)},{(0,0,89.29133),(−22.67716,−12.7559,89.29133),(0,0,80.7874),(−5.669294,−3.174809,76.53543)},{(0,0,89.29133),(−22.67716,0,89.29133),(0,0,80.7874),(−5.669294,0,76.53543)}},{{(0,0,89.29133),(−22.67716,0,89.29133),(0,0,80.7874),(−5.669294,0,76.53543)},{(0,0,89.29133),(−22.67716,12.7559,89.29133),(0,0,80.7874),(−5.669294,3.174809,76.53543)},{(0,0,89.29133),(−12.7559,22.67716,89.29133),(0,0,80.7874),(−3.174809,5.669294,76.53543)},{(0,0,89.29133),(0,22.67716,89.29133),(0,0,80.7874),(0,5.669294,76.53543)}},{{(0,0,89.29133),(0,22.67716,89.29133),(0,0,80.7874),(0,5.669294,76.53543)},{(0,0,89.29133),(12.7559,22.67716,89.29133),(0,0,80.7874),(3.174809,5.669294,76.53543)},{(0,0,89.29133),(22.67716,12.7559,89.29133),(0,0,80.7874),(5.669294,3.174809,76.53543)},{(0,0,89.29133),(22.67716,0,89.29133),(0,0,80.7874),(5.669294,0,76.53543)}},{{(5.669294,0,76.53543),(11.33858,0,72.28346),(36.85039,0,72.28346),(36.85039,0,68.0315)},{(5.669294,−3.174809,76.53543),(11.33858,−6.349609,72.28346),(36.85039,−20.63622,72.28346),(36.85039,−20.63622,68.0315)},{(3.174809,−5.669294,76.53543),(6.349609,−11.33858,72.28346),(20.63622,−36.85039,72.28346),(20.63622,−36.85039,68.0315)},{(0,−5.669294,76.53543),(0,−11.33858,72.28346),(0,−36.85039,72.28346),(0,−36.85039,68.0315)}

},{{(0,−5.669294,76.53543),(0,−11.33858,72.28346),(0,−36.85039,72.28346),(0,−36.85039,68.0315)},{(−3.174809,−5.669294,76.53543),(−6.349609,−11.33858,72.28346),(−20.63622,−36.85039,72.28346),(−20.63622,−36.85039,68.0315)},{(−5.669294,−3.174809,76.53543),(−11.33858,−6.349609,72.28346),(−36.85039,−20.63622,72.28346),(−36.85039,−20.63622,68.0315)},{(−5.669294,0,76.53543),(−11.33858,0,72.28346),(−36.85039,0,72.28346),(−36.85039,0,68.0315)},

},{{(−5.669294,0,76.53543),(−11.33858,0,72.28346),(−36.85039,0,72.28346),(−36.85039,0,68.0315)},{(−5.669294,3.174809,76.53543),(−11.33858,6.349609,72.28346),(−36.85039,20.63622,72.28346),(−36.85039,20.63622,68.0315)},

19

{(−3.174809,5.669294,76.53543),(−6.349609,11.33858,72.28346),(−20.63622,36.85039,72.28346),(−20.63622,36.85039,68.0315)},{(0,5.669294,76.53543),(0,11.33858,72.28346),(0,36.85039,72.28346),(0,36.85039,68.0315)}

},{{(0,5.669294,76.53543),(0,11.33858,72.28346),(0,36.85039,72.28346),(0,36.85039,68.0315)},{(3.174809,5.669294,76.53543),(6.349609,11.33858,72.28346),(20.63622,36.85039,72.28346),(20.63622,36.85039,68.0315)},{(5.669294,3.174809,76.53543),(11.33858,6.349609,72.28346),(36.85039,20.63622,72.28346),(36.85039,20.63622,68.0315)},{(5.669294,0,76.53543),(11.33858,0,72.28346),(36.85039,0,72.28346),(36.85039,0,68.0315)},

},{{(0,0,0),(40.3937,0,0),(42.51969,0,2.12598),(42.51969,0,4.251961)},{(0,0,0),(40.3937,22.62047,0),(42.51969,23.81102,2.12598),(42.51969,23.81102,4.251961)},{(0,0,0),(22.62047,40.3937,0),(23.81102,42.51969,2.12598),(23.81102,42.51969,4.251961)},{(0,0,0),(0,40.3937,0),(0,42.51969,2.12598),(0,42.51969,4.251961)}},{{(0,0,0),(0,40.3937,0),(0,42.51969,2.12598),(0,42.51969,4.251961)},{(0,0,0),(−22.62047,40.3937,0),(−23.81102,42.51969,2.12598),(−23.81102,42.51969,4.251961)},{(0,0,0),(−40.3937,22.62047,0),(−42.51969,23.81102,2.12598),(−42.51969,23.81102,4.251961)},{(0,0,0),(−40.3937,0,0),(−42.51969,0,2.12598),(−42.51969,0,4.251961)}},{{(0,0,0),(−40.3937,0,0),(−42.51969,0,2.12598),(−42.51969,0,4.251961)},{(0,0,0),(−40.3937,−22.62047,0),(−42.51969,−23.81102,2.12598),(−42.51969,−23.81102,4.251961)},{(0,0,0),(−22.62047,−40.3937,0),(−23.81102,−42.51969,2.12598),(−23.81102,−42.51969,4.251961)},{(0,0,0),(0,−40.3937,0),(0,−42.51969,2.12598),(0,−42.51969,4.251961)}},{{(0,0,0),(0,−40.3937,0),(0,−42.51969,2.12598),(0,−42.51969,4.251961)},{(0,0,0),(22.62047,−40.3937,0),(23.81102,−42.51969,2.12598),(23.81102,−42.51969,4.251961)},{(0,0,0),(40.3937,−22.62047,0),(42.51969,−23.81102,2.12598),(42.51969,−23.81102,4.251961)},{(0,0,0),(40.3937,0,0),(42.51969,0,2.12598),(42.51969,0,4.251961)}}};

draw(surface(Q),blue,render(compression=Low));

20

\end{asy}

2 Kompilasi Dokumen

Untuk melakukan kompilasi dokumen yang memuat Asymptote ikuti langkah berikutini :

1. Unduh program Asymptote di situs http://asymptote.sourceforge.net/.

2. Instal program tersebut sampai selesai.

3. Unduh program Ghostscript, hal ini berguna untuk konversi file dari postscriptke pdf di situs http://www.ghostscript.com/GPL Ghostscript 9.10.html. Unduhversi 9.10 dikarenakan untuk versi 9.15 terdapat bug yang memungkinkan tidaktampilnya hasil dari Asymptote.

4. Untuk dapat menjalankan Asymptote dengan MikTEX atau menyisipkan file Asymp-tote di LATEX ikuti petunjuk http://tex.stackexchange.com/questions/83035/using-asymptote-with-miktex.

5. Setelah kedua file instalasi terinstal secara benar maka lakukan instalasi pluginsAsymptote di WinEdt dan untuk memudahkan kita dalam bekerja dengan Asympo-te nantinya. Unduh pluginsnya http://www.winedt.org/Config/menus/Asymptote.php

6. Setelah semua selesai di pasang saat menggunakan Asymptote , salin semua file di-atas menjadi satu bentuk dokumen LATEX kemudian dengan editor LATEX , WinEdtmaka klik PDFTEXify dan tunggu hingga selesai. Jika proses berjalan baik makaakan menghasilkan kedua gambar diatas.

21