Visualization Notes: 12/13/19

Friday, December 13, 2019

Complex Functions as Exercises 2

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x,y,k=var('x,y,k')
m,n=144,3; seg=[-.5,-.48,..,.5]
xy=arg(sum(((x+I*y)^n+10^(-3)*(x+I*y))^factorial(k),k,1,3))
f(x,y)=cos(m*xy)*sin(xy)
fl=[f(x,y) for x in seg for y in seg]
list_plot(fl,size=3,axes=False,figsize=(6,6))

Complex Functions as Exercises

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z,k=var('z,k'); m,n=144,16
zn=arg(sum((z^n+10^(-3)*z)^factorial(k),k,1,4))
complex_plot(cos(m*zn)*sin(zn),(-1,1),(-1,1),
             axes=False,plot_points=200,figsize=(6,6))

Visualization of Recurrence Tables 2

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def gen(max=Infinity):
    n=1; xyn=[.1,.1]
    while n<=max:
        yield xyn; n+=1
        xn,yn=xyn[0],xyn[1]
        xyn=[.7*xn+yn,-.8+xn^2]
N=3000; XY=gen(N); XYN=[el for el in XY]
cols=[colormaps.autumn(1.*i/N)[:3] 
      for i in [1..N]]
p=polygon([[-1.5,-1],[-1.5,1.5],[1,1.5],
           [1,-1],[-1.5,-1]],
          color='silver',alpha=.05)
p+sum([point([XYN[i]],size=1,color=cols[i]) 
       for i in [1..N-1]])

Visualization of Recurrence Tables

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def gen(max=Infinity):
    n=1; xyn=[.1,.01]
    while n<=max:
        yield xyn; n+=1; xn,yn=xyn[0],xyn[1]
        xyn=[xn+.684*(xn-xn^2+yn),
             yn+.684*(yn-yn^2+xn)]
N=3000; XY=gen(N); XYN=[el for el in XY]
cools=[colormaps.cool(1.*i/N)[:3] 
       for i in [1..N]]
p=polygon([[-1,-1],[-1,3],[3,3],[3,-1],[-1,-1]],
          color='steelblue',alpha=.1)
p+sum([point([XYN[i]],size=1,color=cools[i]) 
       for i in [1..N-1]])

Random Polar Plots as Exercises 3

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var('t,i'); n=3 
tmin,tmax=[0,0,0],[8.95*pi,12.18*pi,11.77*pi]
f=[sin(1.9*t),1.5*sin(2.3*t),2*sin(1.7*t)]
sum([polar_plot(random()*f[i-1],(tmin[i-1],tmax[i-1]),
                color=(0,.07*(i+j),.15*(i+j)),thickness=.5) 
     for i in [1..n] for j in [1..n]]).show(axes=False)

Random Polar Plots as Exercises 2

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var('t,i'); tmin,tmax,n=-pi,pi,12
f1(t,i)=(randint(5,10)+.9*cos(randint(11,20)*t+pi*i/n))
f2(t,i)=(1+.1*cos(randint(25,100)*t+pi*i/n))
f3(t,i)=(1+.01*cos(randint(200,300)*t+pi*i/n))
f(t,i)=f1(t,i)*f2(t,i)*f3(t,i)*(1+sin(t+pi*i/n))
sum([polar_plot(f(t,i),(tmin,tmax),
                thickness=.5,color=hue(i/n)) 
     for i in [1..2*n]]).show(axes=False,figsize=6)

Random Polar Plots as Exercises

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var('t,i,a,b,c,d'); tmin,tmax,n=-pi,pi,4
f1(t,i,a,b,c,d)=(a+.9*cos(b*t+pi*i/n))*(1+.1*cos(c*t+pi*i/n))
f2(t,i,a,b,c,d)=(1+.05*cos(d*t+pi*i/n))*(1+sin(t+pi*i/n))
[a,b,c,d]=[randint(8,11),randint(12,24),
           randint(25,81),randint(216,256)]
sum([polar_plot(f1(t,i,a,b,c,d)*f2(t,i,a,b,c,d),(tmin,tmax),
                thickness=.3,axes=False,color=hue(i/(3*n))) 
     for i in [1..6*n]])

From Binary to Decimal Digits

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print(r.eval("""
bin2dec<-function(n) {
    sapply(strsplit(as.character(n),split=''),
           function(x) 
               sum(as.numeric(x)*2**(rev(seq_along(x)-1))))}
bin2dec(c('11100110010','10001','1010101111','1001001','10'))"""))

Surfaces' Plotting as Exercises 4

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var('u,v'); a,b,c=8,4,3
u_min,u_max,v_min,v_max=-.4,.4,-2*pi,2*pi
fx=a*cos(v)*sinh(u)-b*cos(c*v)*sinh(c*u)
fy=a*sin(v)*sinh(u)+b*sin(c*v)*sinh(c*u)
fz=cos(a*v)*cosh(a*u)
parametric_plot3d(
    [fx,fy,fz],(u_min,u_max),(v_min,v_max),plotpoints=400,
    color='crimson',opacity=.5,mesh=True,frame=False)

Surfaces' Plotting as Exercises 3

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var('u,v'); a,b,c=4,5,1
u_min,u_max,v_min,v_max=0,2*pi,0,2*pi
fx,fy,fz=u*cos(a*v),u*sin(b*v),-v*cos(c*u)
sum([parametric_plot3d(
    [fx*i,fy*j,fz],(u_min,u_max),(v_min,v_max),plotpoints=300,
    color='silver',opacity=.3,mesh=True,frame=False) 
     for i in [-1,1] for j in [-1,1]])

HTML Exercises

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html("""
<div id='img' title='An Image Example' 
style='border:15px double white; width:250px; height:200px; \
       overflow:auto; padding:5px; background-color:#ff66cc'>
<img src='https://raw.githubusercontent.com/"""+\
"""OlgaBelitskaya/data/main/flowers/00_001.png' 
     width='98%' height='95%'/>
</div>""")

Text Vectorizer

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from sklearn.feature_extraction.text import CountVectorizer
corpus=['Have you already set your goals for the New Year?', 
        'Do you want to lose ten kilos, '+\
        'run a marathon or speak fluent English?', 
        'Some experts believe that you need systems, not goals.', 
        'A system is something you do on a regular basis. ',
        'This means focusing on what you can control '+\
        '(your actions) rather than what you can&#8217;t.',
        'For example, do not focus on losing ten kilos.',
        'Focus on shopping for healthy food and '+\
        'cooking something light every day.',
        'Do not focus on the marathon.',
        'Focus on the training schedule.',
        'Invent a system to improve your English, '+\
        'one step at a time.',
        'Good luck!']
c_vectorizer=CountVectorizer(min_df=1)
corpus_features=c_vectorizer.fit_transform(corpus)
corpus_array=corpus_features.toarray().astype('int16')
c_analyzer=c_vectorizer.build_analyzer()
import pylab as pl; pl.figure(figsize=(6,5))
pl.title('Word Occurrences in Sentences',fontsize=12)
for i in range(len(corpus_array)):
    pl.scatter(range(len(corpus_array[i])),
               (corpus_array[i]*.5+i),marker='*')
pl.tight_layout(); pl.grid(); pl.show()

Fitted Lines

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A,B=1+round(random(),2),round(.01+random(),2)
N=500; X=[.1*i*random() for i in range(N)]
Y=[A*x+B+10*random()*(-1)^randint(0,1) for x in X]
var('a,b,x'); F=a*x+b
ff=find_fit(list(zip(X,Y)),F,variables=[x]) 
a,b=float(str(ff[0])[5:]),float(str(ff[1])[5:])
F=[a*x+b for x in X]
pf=points(zip(X,F),color='#ff36ff',size=5,
          legend_label='fitted line')
pr=points(zip(X,Y),color='#3636ff',marker='*',
          size=9,legend_label='real data') 
(pf+pr).show(figsize=(6,5),gridlines=True,
             title=['[A == %s,B == %s]'%(A,B),ff])

Polar Plotting 2

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@interact
def _(N=(16,4,-1)):
    p1=sum([polar_plot(log(floor(x)),(x,k*pi/2,(k+1)*pi/2),
                       color=colormaps.winter(N*k)[:3],thickness=1,
                       exclude=[1..int((k+1)*pi/2)]) for k in [1..N]])
    p2=sum([polar_plot(log(floor(x)),(x,k*pi/2,(k+1)*pi/2),
                       color=colormaps.spring(N*k)[:3],thickness=1,
                       exclude=[1..int(k*pi/2)]) for k in [1..N]])
    p1.show(figsize=(4,4)); p2.show(figsize=(4,4))

Plotting of Derivatives

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@interact
def _(N=(9,1,-1)):
    s1=r'<center><font color=%s size=5>'%'#3636ff'
    s2=r'Derivatives till the %sth Order</font></center>'
    var('x'); pretty_print(html(s1+s2%str(N)))
    f,g=exp(x)*x,1/(x+1); F,G=[f],[g]
    for i in range(N): 
        f=f.diff().factor(); g=g.diff().factor()
        F.append(f); G.append(g)
    p1=sum([plot(F[i],(-2,1),color=colormaps.jet(30*i)[:3],
                 legend_label=str(i)) for i in [0..N]])
    p2=sum([plot(G[i],(0,3),color=colormaps.jet(30*i)[:3],
                 legend_label=str(i)) for i in [0..N]])
    p1.show(title=r'$f=x \cdot e^x$',fontsize=12,
            figsize=(5,5),gridlines=True)
    p2.show(title=r'$f=\frac{1}{x+1}$',fontsize=12,
            figsize=(5,5),gridlines=True,ymin=-20,ymax=20)

Polar Plotting

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@interact
def _(N=(11,0,-1)):
    var('t'); 
    p=sum([polar_plot((.5*j+1)*sin(6*t)^2,(pi*i/6,pi*(i+1)/6),
                      color=colormaps.jet(20*i)[:3]) 
           for i in [0..N] for j in range(3)])
    show(p,figsize=6,gridlines=True,
         ymin=-2,ymax=2,xmin=-2,xmax=2)

LaTex & HTML

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%%html
<text style='color:#3636ff;'>
$\mathscr{
\begin{bmatrix}
\begin{array}{c}
G = \alpha * A * B^T \ + \ \beta * C^T * D \\ 
\hline &\;&\;&\;&\;&\;&\;&\;
\end{array} \\
\begin{array}{c}
\alpha = 2, \ \beta = -5, \\
A_{(2,3)} = \begin{pmatrix}
-3 & -3 & -6 \\ -4 & -7 & 8 \\ 
\end{pmatrix}, \ 
B_{(4,3)} = \begin{pmatrix}
2 & 8 & 7 \\ -5 & -6 & -2 \\ 
    -8 & 4 & -8 \\ 7 & 4 & -6 \\ 
\end{pmatrix}, \\
C_{(2,2)} = \begin{pmatrix}
-2 & -4 \\ 3 & -5 \\ 
\end{pmatrix}, \ 
D_{(2,4)} = \begin{pmatrix}
-2 & -3 & -1 & 3 \\ -2 & -9 & 8 & 9 \\ 
\end{pmatrix}. \\ 
\end{array} \\ 
\begin{array}{c}
\hline &\;&\;&\;&\;&\;&\;&\;&\;&\;
    &\;&\;&\;&\;&\;&\;&\; \\
G = ? 
\end{array} 
\end{bmatrix}}$
</text>

Pylab Tables

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%matplotlib inline
import pylab as pl,numpy as np
columns=['Fresh','Milk','Grocery','Frozen',
         'Detergents_Paper','Delicatessen']
rows=['Customer 1','Customer 2','Customer 3']
colors=['#3636ff','#36ff36','#ff3636']
data=[[26373,36423,22019,5154,4337,16523],
      [16165,4230,7595,201,4003,57],
      [14276,803,3045,485,100,518]]
pl.figure(figsize=(7,4))
pl.ylabel('value'); pl.xticks([])
x=np.arange(len(columns))+.2
y=np.array([0.]*len(columns))
bar_width=.5; cell_text=[]
for i in range(len(rows)):
    pl.bar(x,data[i],bar_width,bottom=y,color=colors[i])
    y=y+data[i]
    cell_text.append(['%1.0f'%(d)for d in data[i]])
pl.table(cellText=cell_text,
         rowLabels=rows,rowColours=colors,
         colLabels=columns,loc='bottom')
pl.title('Samples of the Wholesale Customers Dataset')
pl.subplots_adjust(left=.1,bottom=.05);

Plotting of Function Derivations

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var('x'); t=15
f=cos(x); g=sqrt(x)*sin(x); h=cos(sqrt(x)*sin(x))
fun=[f,f.diff(),g,g.diff(),h,h.diff()]
c=[colormaps.hsv(35*i)[:3] for i in [0..5]] 
l=[r'$f$',r'$f \prime$',r'$g$',
   r'$g \prime$',r'$h$',r'$h \prime$']
p=plot(fun,(0,t),color=c,thickness=2,fill='axis',
       fillcolor=c,fillalpha=0.1,legend_label=l)
p.show(figsize=(6,5),legend_fancybox=True,
       legend_loc=3,legend_font_size=16,gridlines=True)

3D Adaptive Plotting

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f=lambda x,y: sin(.3+x^3*y^2)*exp(.4+x/pi)+cos(.5+x^2*y^3)
c=[colormaps['terrain'](.1^2*i)[:3] for i in range(100)]
plot3d(f,(-pi,pi),(-pi,pi),color=c,
       adaptive=True,mesh=True,frame=False)

Elliptic Curves

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@interact
def _(N=(8,1,-1)):
    g=Graphics()
    for i in [1..N]:
        a1,a2,a3,a4,a6=i-5,i+5,i-1,i+1,i+3
        c=colormaps.jet(30*i)[:3]; 
        E=EllipticCurve([a1,a2,a3,a4,a6]);
        g+=plot(E,color=c,plot_points=50,
                legend_label='$%s$'%latex(E))
    g.show(gridlines=True,ymin=-30,ymax=30,figsize=(5.5,5))

Simple Exercises with HTML 2

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%%html
<style>
.text1 {color:darkred; font-size:150%; 
        text-shadow:3px 3px 3px #aaa;}
.text2 {color:darkslategray; font-size:150%; 
        text-shadow:3px 3px 3px #bbb;}
.text3 {color:steelblue; font-size:150%; 
        text-shadow:3px 3px 3px #ccc;}
</style>
<p class='text1'>
&#x1F4D5; &nbsp; $\mathfrak {Choose \ your \ style!}$</p>
<p class='text2'>
&#x1F4D3; &nbsp; $\mathbb {Choose \ your \ style!}$</p>
<p class='text3'>
&#x1F4D8; &nbsp; $\mathscr {Choose \ your \ style!}$</p>

Simple Exercises with HTML

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%%html
<p>Calculation the number of symbols in strings:</p>
<p id='script_output'>evaluate the next code cell</p>

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%%javascript
function getInteger(min,max) {
    return Math.floor(Math.random()*(max-min+1))+min;};
var string='@@@***морозИсолнцеДЕНЬчудестный***@@@'+getInteger(1,19);
document.getElementById('script_output')
        .innerHTML=string+' - '+string.length+' symbols';

Polar and Parametric Coordinate Systems 2

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@interact
def _(a=(2,4,1),b=(4,8,1),c=(4,8,1),n=(2,8,1),m=(2,8,1)):
    var('t'); r=cos(a*t)^n+sin(b*t)^m+1/c; col='#3636ff'
    string='<center><font color=%s>$r = %s$</font></center>'
    pretty_print(html(string%(col,latex(r))))
    p=parametric_plot((sin(t)*r,cos(t)*r),(0,2*pi),
                      color=col,fill=True,fillcolor=col)
    p+=plot(r,(0,2*pi),linestyle='--',color=col,gridlines=True)
    show(p,figsize=(7,7),
         xmin=-2.2-1/c,xmax=6.4+1/c,ymin=-2-1/c,ymax=2+1/c)

Transformations of Coordinate Systems

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Md=Manifold(2,'Md')
U=Md.open_subset('U'); V=Md.open_subset('V')
XU.<x,y>=U.chart(); XV.<xp,yp>=V.chart('xp:x&#8217; yp:y&#8217;')
Md.declare_union(U,V)
XU_to_XV=XU.transition_map(
    XV,(2*x/(x^2+y^2),2*y/(x^2+y^2)),
    intersection_name='W',
    restrictions1=x^2+y^2!=0,
    restrictions2=xp^2+yp^2!=0)
R3=Manifold(3,'R^3',r'\mathbb{R}^3')
XR3.<X,Y,Z>=R3.chart()
Delta1=Md.diff_map(
    R3,{(XU,XR3):[2*x/(1+x^2+y^2),2*y/(1+x^2+y^2),
                  (x^2+y^2-1)/(1+x^2+y^2)],
        (XV,XR3):[3*xp/(1+xp^2+yp^2),3*yp/(1+xp^2+yp^2),
                  (1-xp^2-yp^2)/(1+xp^2+yp^2)]},
                   name='Delta1',latex_name=r'\Delta_1')
Delta2=Md.diff_map(
    R3,{(XU, XR3):[x/(1+x^2+y^2),y/(1+x^2+y^2),
                   (x^2+y^2-1)/(1+x^2+y^2)],
        (XV, XR3):[4*xp/(1+xp^2+yp^2),4*yp/(1+xp^2+yp^2),
                   (1-xp^2-yp^2)/(1+xp^2+yp^2)]},
                   name='Delta2',latex_name=r'\Delta_2')
cols=['darkblue','#ff3636','darkred','#3636ff']
elements=[(XU,Delta1),(XV,Delta1),(XU,Delta2),(XV,Delta2)]
p=sum([elements[i][0].plot(
           chart=XR3,mapping=elements[i][1],
           number_values=50,color=cols[i],label_axes=False)
       for i in range(4)])
p.show(frame=False)

Polar and Parametric Coordinate Systems

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def tf1(i,k,t): 
    f1=(1/k[0]+sin(k[1]+k[2]*t))*(1/k[0]+cos(k[1]+k[2]*t))
    f2=(1/k[0]-sin(k[1]+k[2]*t)^3)*(1/k[1]-cos(k[1]+k[2]*t)^3)
    return [i*f1*f2*cos(t),i*f1*f2*sin(t)]
def tf2(i,k,t): 
    return [(i+cos(i*k[0]*t)+cos(k[0]*t))*cos(t)/k[1]/k[2],
            (i+cos(i*k[0]*t)+cos(k[0]*t))*sin(t)/k[1]/k[2]]
var('t'); K=[randint(2,16) for i in range(3)]
p1=sum([parametric_plot(tf1(i,K,t),(t,0,2*pi),color=hue(i/6)) 
        for i in range(1,5)])
p2=sum([parametric_plot(tf2(i,K,t),(t,0,2*pi),color=hue(i/6)) 
        for i in range(1,5)])
(p1+p2).show(title=K,axes=False,figsize=6)

Taylor Series

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@interact
def _(N=(24,1,-1)):
    var('x'); f=sin(x)*exp(1)^(-x)
    p=plot(f,-1,6,thickness=5,color='#3636ff',alpha=.5)+\
    sum([plot(f.taylor(x,0,i),-1,6,hue=cos(i/24),thickness=1,
              legend_label=str(i)) for i in [1..N]])
    ti=r'$Taylor \ Series \ f=sin \ x \cdot e^{-x}$'
    p.show(ymin=-.5,ymax=.5,figsize=(5.5,5),
           gridlines=True,title=ti,fontsize=12)

Pattern Drawing 5

function_plotting_js.ipynb

from IPython.display import HTML

HTML("""<script src='https://code.highcharts.com/highcharts.js'></script>

<div id='container2' style='height:600px; width:600px; margin:0 auto'></div>

<script>

function getinteger(min,max) {return Math.floor(Math.random()*(max-min+1))+min;};

function ar(k,a,b) {return Array(12800).fill(k).map((k,t)=>

    [k*(Math.cos(0.001*t)+Math.sin(a*0.001*t)/2-Math.cos(b*0.001*t)/6),

     k*(Math.sin(0.001*t)+Math.cos(a*0.001*t)/2-Math.sin(b*0.001*t)/6)]);};

function col(i) {var r=getinteger(i,255),g=getinteger(i,255),b=getinteger(i,255);

                 return 'rgb('+r.toString()+','+g.toString()+','+b.toString()+')';}; 

var series=[]; var i; var n=3;

var a=getinteger(7,19),b=getinteger(18,64);

for (i=1; i<n+1; i++) {series.push({name:i.toString(),color:col(i),

                                    lineWidth:0.5,data:ar(i,a,b)})};

Highcharts.chart('container2', {

    chart:{type:'line',backgroundColor:'mintcream'},

    xAxis:{title:{text:'x'}},yAxis:{title:{text:'y'}},

    title:{text:'Random Parametric Plot: a,b = '+[a,b].toString()},

    credits:{enabled:false},legend:{enabled:false},series:series});

</script>""")

Pattern Drawing 4

function_plotting_js.ipynb

from IPython.display import HTML

HTML("""<script src='https://code.highcharts.com/highcharts.js'></script>

<div id='container3' style='height:600px; width:600px; margin:0 auto'></div>

<script>

function getinteger(min,max) {return Math.floor(Math.random()*(max-min+1))+min;};

function ar(k,a,b) {return Array(6400).fill(k).map((k,t)=>

    [Math.cos(0.001*t+k*Math.PI/6)+Math.cos(a*0.001*t)/2+Math.sin((a+b)*0.001*t)/3,

     Math.sin(0.001*t+k*Math.PI/6)+Math.sin(a*0.001*t)/2+Math.cos((a+b)*0.001*t)/3]);};

function col(i) {var r=getinteger(i,255),g=getinteger(i,255),b=getinteger(i,255);

                 return 'rgb('+r.toString()+','+g.toString()+','+b.toString()+')';}; 

var series=[]; var i; var n=6;

var a=getinteger(7,15),b=getinteger(10,48);

for (i=1; i<2*n+1; i++) {series.push({name:i.toString(),color:col(i),

                                      lineWidth:0.5,data:ar(i,a,b)})};

Highcharts.chart('container3', {

    chart:{type:'line',backgroundColor:'lavender'},

    xAxis:{title:{text:'x'}},yAxis:{title:{text:'y'}},

    title:{text:'Random Parametric Plot: a,b = '+[a,b].toString()},

    credits:{enabled:false},legend:{enabled:false},series:series});

</script>""")

Pattern Drawing 3

function_plotting_js.ipynb

from IPython.display import HTML

HTML("""<script src='https://code.highcharts.com/highcharts.js'></script>

<div id='container4' style='height:600px; width:600px; margin:0 auto'></div>

<script>

function getinteger(min,max) {return Math.floor(Math.random()*(max-min+1))+min;};

function ar(k,a,b) {return Array(6400).fill(k).map((k,t)=>

    [k*(Math.cos(0.001*t)+Math.cos(a*0.001*t)/2+Math.sin((a+b)*0.001*t)/3),

     k*(Math.sin(0.001*t)+Math.sin(a*0.001*t)/2+Math.cos((a+b)*0.001*t)/3)]);};

function col(i) {var r=getinteger(i,255),g=getinteger(i,255),b=getinteger(i,255);

                 return 'rgb('+r.toString()+','+g.toString()+','+b.toString()+')';}; 

var series=[]; var i; var n=4;

for (i=1; i<n+1; i++) {

    var a=getinteger(5,15),b=getinteger(10,24);

    series.push({name:[i,a,b].toString(),color:col(i),lineWidth:0.7,data:ar(i,a,b)})};

Highcharts.chart('container4', {

    chart:{type:'line',backgroundColor:'ghostwhite'},

    xAxis:{title:{text:'x'}},yAxis:{title:{text:'y'}},

    title:{text:'Random Parametric Plot'},credits:{enabled:false},

    legend:{enabled:false},series:series});

</script>""")

Pattern Drawing 2

function_plotting_js.ipynb

from IPython.display import HTML

HTML("""<script src='https://code.highcharts.com/highcharts.js'></script>

<div id='container5' style='height:600px; width:600px; margin:0 auto'></div>

<script>

function getinteger(min,max) {return Math.floor(Math.random()*(max-min+1))+min;};

function fx(a,b,c,d,t) {

    return Math.sin(0.001*t/6)+a*Math.sin(b*0.001*t)*Math.cos(0.001*t)+c*Math.sin(b*0.001*t)};

function fy(a,b,c,d,t) {

    return Math.cos(0.001*t/6)+a*Math.sin(b*0.001*t)*Math.sin(0.001*t)+c*Math.cos(d*b*0.001*t)};

function ar(k,a,b,c,d) {return Array(b*6400).fill(k).map((k,t)=>

                               [(1+0.01*k)*fx(a,b,c,d,t),(1+0.01*k)*fy(a,b,c,d,t)]);};

function col(i) {var r=getinteger(i,255),g=getinteger(i,255),b=getinteger(i,255);

                 return 'rgb('+r.toString()+','+g.toString()+','+b.toString()+')';}; 

var series=[]; var i; var n=2;

var a=0.5+Math.random(),c=0.01*getinteger(1,99),b=getinteger(6,18),d=getinteger(3,6);

for (i=1; i<n+1; i++) {series.push({name:i.toString(),color:col(i),

                                    lineWidth:0.3/i,data:ar(i,a,b,c,d)})};

Highcharts.chart('container5', {

    chart:{type:'line',backgroundColor:'mintcream'},

    xAxis:{title:{text:'x'}},yAxis:{title:{text:'y'}},

    title:{text:'Random Parametric Plot: a,b,c,d = '+[a,b,c,d].toString()},

    credits:{enabled:false},legend:{enabled:false},series:series});

</script>""")

Pattern Drawing

function_plotting_js.ipynb

from IPython.display import HTML

HTML("""<script src='https://code.highcharts.com/highcharts.js'></script>

<div id='container6' style='height:600px; width:600px; margin:0 auto'></div>

<script>

function getinteger(min,max) {return Math.floor(Math.random()*(max-min+1))+min;};

function fx(a,b,c,t) {

    return Math.sin(0.001*t/6)+a*Math.sin(b*0.001*t)*Math.cos(0.001*t)-c*Math.sin(16*b*0.001*t)};

function fy(a,b,c,t) {

    return Math.cos(0.001*t/6)+a*Math.sin(b*0.001*t)*Math.sin(0.001*t)-c*Math.cos(16*b*0.001*t)};

function ar(k,a,b,c) {return Array(b*12800).fill(k).map((k,t)=>

                             [(1+0.01*k)*fx(a,b,c,t),(1+0.01*k)*fy(a,b,c,t)]);};

function col(i) {var r=getinteger(i,255),g=getinteger(i,255),b=getinteger(i,255);

                 return 'rgb('+r.toString()+','+g.toString()+','+b.toString()+')';}; 

var series=[]; var i; var n=2;

var a=0.5+Math.random(),c=0.001*getinteger(1,99),b=getinteger(3,12);

for (i=1; i<n+1; i++) {series.push({name:i.toString(),color:col(i),

                                    lineWidth:0.6/i,data:ar(i,a,b,c)})};

Highcharts.chart('container6', {

    chart:{type:'line',backgroundColor:'mintcream'},

    xAxis:{title:{text:'x'}},yAxis:{title:{text:'y'}},

    title:{text:'Random Parametric Plot: a,b,c = '+[a,b,c].toString()},

    credits:{enabled:false},legend:{enabled:false},series:series});

</script>""")

Patterns by Polar & Parametric Plotting

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi; t=numpy.arange(0,2*pi,.1**4)

def col(): return [numpy.append([1],numpy.random.random(2))]

def f(t): return numpy.exp(numpy.cos(t)**2+numpy.sin(t))-3*numpy.cos(4*t)

def fx(k,t): return .1*(k+1)*f(t)*numpy.cos(t)

def fy(k,t): return .1*(k+1)*f(t)*numpy.sin(t)

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('ivory')

[pylab.scatter(fx(i,t),fy(i,t),s=.1**3,c=col()) for i in range(12)]

pylab.grid(c='silver',alpha=.4); pylab.show()

"Pattern" Drawing by Functions 4

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi 

def randi(nmin,nmax): return numpy.random.randint(nmin,nmax)

def t(i): return numpy.arange((i-1)*pi/24,i*pi/24,1/10**4)

def x(a,b,i,k): return numpy.cos(t(i)+k*pi/6)+numpy.cos(a*t(i))/2+numpy.sin((a+b)*t(i))/3

def y(a,b,i,k): return numpy.sin(t(i)+k*pi/6)+numpy.sin(a*t(i))/2+numpy.cos((a+b)*t(i))/3

A,B=randi(5,15),randi(10,36) 

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('ghostwhite')

for i in range(48):

  col=[numpy.random.random(3)]

  for k in range(12): pylab.scatter(x(A,B,i,k),y(A,B,i,k),s=.1**3,c=col)

pylab.title('a=%d; b=%d'%(A,B)); pylab.grid(); pylab.show()

"Pattern" Drawing by Functions 3

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi 

def randi(nmin,nmax): return numpy.random.randint(nmin,nmax)

def x(a,b,t,k): return k*(numpy.cos(t)+numpy.cos(a*t)/2+numpy.sin((a+b)*t)/3)

def y(a,b,t,k): return k*(numpy.sin(t)+numpy.sin(a*t)/2+numpy.cos((a+b)*t)/3)

A,B=randi(5,11),randi(12,36); T=numpy.arange(0,2*pi,1/10**4)

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('ghostwhite')

for k in range(12):

    pylab.scatter(x(A,B,T,k),y(A,B,T,k),s=.1**3,c=[numpy.random.random(3)])

pylab.title('a=%d; b=%d'%(A,B)); pylab.grid(); pylab.show()

"Pattern" Drawing by Functions 2

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi 

def randi(nmin,nmax): return numpy.random.randint(nmin,nmax)

def randc(a): return numpy.random.choice(a,1)[0]

a,c=(.5+numpy.random.random())*randc([-1,1]),.001*randi(1,99)*randc([-1,1])

b=randi(3,12); t=numpy.arange(0,16*b*pi,1/(720*b))

fx=numpy.sin(t/6)+a*numpy.sin(b*t)*numpy.cos(t)-c*numpy.sin(16*b*t)

fy=numpy.cos(t/6)+a*numpy.sin(b*t)*numpy.sin(t)-c*numpy.cos(16*b*t)

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('mintcream')

pylab.scatter(fx,fy,s=.1**4,c=[numpy.random.random(3)])

pylab.title(['a=%f'%a,'b=%d'%b,'c=%f'%c]); pylab.grid(); pylab.show()

"Pattern" Drawing by Functions

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi 

def randi(nmin,nmax): return numpy.random.randint(nmin,nmax)

def randc(a): return numpy.random.choice(a,1)[0]

a,c=(.5+numpy.random.random())*randc([-1,1]),.01*randi(1,99)**randc([-1,1])

b,d=randi(6,18),randi(3,6); t=numpy.arange(0,8*b*pi,1/(720.*b))

fx=numpy.sin(t/6)+a*numpy.sin(b*t)*numpy.cos(t)+c*numpy.sin(b*t)

fy=numpy.cos(t/6)+a*numpy.sin(b*t)*numpy.sin(t)+c*numpy.cos(d*b*t)

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('lightgray')

pylab.scatter(fx,fy,s=.1**4,c=[numpy.random.random(3)])

pylab.title(['a=%f'%a,'b=%d'%b,'c=%f'%c,'d=%d'%d]); pylab.grid(); pylab.show()

Abstract "Herbal" Drawing by Functions 2

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi 

def randi(nmin,nmax): return numpy.random.randint(nmin,nmax)

n=36; m=randi(100,400); t=numpy.arange(0,2*pi,2*pi/(m*n))

a,b,c,d=randi(5,11),randi(12,24),randi(25,81),randi(216,256)

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('lightgray')

for i in range(n):

    f1=(a+.9*numpy.cos(b*t+2*pi*i/n))*(1+.1*numpy.cos(c*t+2*pi*i/n))

    f2=(1+.05*numpy.cos(d*t+2*pi*i/n))*(1+numpy.sin(t+2*pi*i/n))

    x=f1*f2*numpy.cos(t); y=f1*f2*numpy.sin(t)

    pylab.scatter(x,y,s=.01,c=[numpy.random.random(3)])

pylab.title(['a=%d'%a,'b=%d'%b,'c=%d'%c,'d=%d'%d,'m=%d'%m])

pylab.grid(c='white',alpha=.4); pylab.show()

Abstract "Herbal" Drawing by Functions

function_plotting_py.ipynb

import numpy,pylab; pi=numpy.pi; n=36; t=numpy.arange(0,2*pi,2*pi/(360.*n))

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('black')

def randi(nmin,nmax): return numpy.random.randint(nmin,nmax)

for i in range(n):

    a,b,c,d=randi(5,11),randi(12,24),randi(25,81),randi(216,256)

    f1=(a+.9*numpy.cos(b*t+2*pi*i/n))*(1+.1*numpy.cos(c*t+2*pi*i/n))

    f2=(1+.05*numpy.cos(d*t+2*pi*i/n))*(1+numpy.sin(t+2*pi*i/n))

    x=f1*f2*numpy.cos(t); y=f1*f2*numpy.sin(t)

    pylab.scatter(x,y,s=3,c=[numpy.random.random(3)])

pylab.grid(c='white',alpha=.4); pylab.show()

Random Coefficients in Trigonometrics 2

function_plotting_py.ipynb

import numpy,pylab,random; r=[-1,0,1]; th=.1+random.random()

a,b,c=random.randint(1,3),random.randint(4,6),random.randint(2,4)

d,e,f=random.randint(9,11),random.randint(14,16),random.randint(1,3)

t=numpy.arange(0,2*numpy.pi+.5,1/10**random.randint(1,3)) 

fx=-d*numpy.cos(t)-f*numpy.cos(b*t)+e*numpy.sin(a*t)

fy=-e*numpy.cos(a*t)+d*numpy.sin(t)-f*numpy.sin(b*t) 

fz=d*numpy.cos(c*t)

fig=pylab.figure(figsize=(12,12)); ax=fig.gca(projection='3d')

for i in r:

    for j in r:

        for k in r:

            col=[random.random() for l in range(3)]

            ax.plot(i*fx,j*fy,k*fz,c=col,linewidth=th)

pylab.title('a=%d; b=%d; c=%d; d=%d; e=%d; f=%d'%(a,b,c,d,e,f))

pylab.axis('off'); pylab.show()

Random Coefficients in Trigonometrics

function_plotting_py.ipynb

import numpy,pylab,random; a,b,c=random.randrange(2,12),1.5,random.randrange(1,5) 

r=numpy.arange(1,c+1); th=.1+random.random() 

t=numpy.arange(-2*b*numpy.pi,2*b*numpy.pi,5/10**random.randrange(1,3)) 

fx=(a+numpy.cos(b*t))*numpy.cos(t); fy=(a+numpy.cos(b*t))*numpy.sin(t); fz=numpy.sin(b*t)

fig=pylab.figure(figsize=(12,12)); ax=fig.gca(projection='3d')

for i in r:

    for j in r:

        for k in r:

            col=[random.random() for l in range(3)]

            ax.plot(i*fx,j*fy,k*fz,c=col,linewidth=th)

pylab.axis('off'); pylab.title('a=%d; b=%.1f; c=%d'%(a,b,c)); pylab.show()

Periods of Trigonometric Functions

function_plotting_py.ipynb

import numpy,pylab; [a,b,c]=[.01*numpy.random.randint(30,99) for i in range(3)]

d=numpy.max([a,2.*b,3.*c]); n=numpy.random.randint(5,11)

def coord(m,k,p,n):

    t=numpy.arange(0,k*numpy.pi,k*numpy.pi/(72000.*n))

    x=m*numpy.sin(p*t)*numpy.cos(t); y=m*numpy.sin(p*t)*numpy.sin(t)

    return [x,y]

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('silver')

for [x,y] in [coord(a,8.95,1.9,n),coord(b,12.18,2.3,n),coord(c,11.77,1.7,n)]:

    pylab.scatter(x,y,s=.0001,c=[numpy.random.random(3)])

pylab.title(['a=%.2f'%a,'b=%.2f'%b,'c=%.2f'%c,'d=%.2f'%d]) 

pylab.grid(c='slategray',alpha=.4); pylab.show()

Plotting Recurrence Tables 5

function_plotting_py.ipynb

import numpy,pylab

def gen(a,b,n):

    i=1; xyi=[0.,0.]

    while i<=n:

        yield xyi; i+=1; xi,yi=xyi[0],xyi[1]

        xyi=[(1+.01*a)*abs(xi)-(1+.00001*b)*yi+1,xi]

for n in [50000,100000]: 

    a,b=7,5; xy=gen(a,b,n); xyn=numpy.array([el for el in xy])      

    pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('ghostwhite')

    pylab.scatter(xyn[:,0],xyn[:,1],s=.05,c=[numpy.random.random(3)])

    pylab.grid(c='slategray',alpha=.4); pylab.show()

Plotting Recurrence Tables 4

function_plotting_py.ipynb

import numpy,pylab

def gen(a,b,n):

    i=1; xyi=[0.,0.]

    while i<=n:

        yield xyi; i+=1; xi,yi=xyi[0],xyi[1]

        xyi=[1-(.2+.01*a)*xi**2+yi,(0.9991+.0001*b)*xi]

for [a,b] in [[0,0],[-11,-19987]]:

    n=300000; xy=gen(a,b,n); xyn=numpy.array([el for el in xy])      

    pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('lightgray')

    pylab.scatter(xyn[:,0],xyn[:,1],s=.3,c=[numpy.random.random(3)])

    pylab.grid(c='slategray',alpha=.4); pylab.show()

Plotting Recurrence Tables 3

function_plotting_py.ipynb

import numpy,pylab

def gen(n):

    i=1; xyi=[.1,.1]

    while i<=n:

        yield xyi; i+=1; xi,yi=xyi[0],xyi[1]

        xyi=[.7*xi+yi,-.8+xi**2]

n=200000; xy=gen(n); xyn=numpy.array([el for el in xy])      

pylab.figure(figsize=(10,10)); ax=pylab.gca(); ax.set_facecolor('black')

pylab.scatter(xyn[:,0],xyn[:,1],s=.01,c=[numpy.random.random(3)])

pylab.grid(c='slategray',alpha=.4); pylab.show()