ΠΠΊΡΠΈΠ²ΠΈΡΡΠΉΡΠ΅ ΡΡΠΎΡ ΠΊΠΎΠ΄ ΠΏΠ΅ΡΠ΅Π΄ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ Π»ΡΠ±ΡΡ ΡΡΠ΅Π΅ΠΊ Linked Sage Cells
xxxxxxxxxxvar('x,y,z,R,p,q,r,t1,t2')eq1=[x^2+y^2+z^2-R^2==0, [x==R*cos(t1)*sin(t2), y==R*sin(t1)*sin(t2), z==R*cos(t2)], x^2/p^2+y^2/q^2+z^2/r^2==1]sol1=[solve([eq1[0],R==4],R,z), solve([eq1[1][0],eq1[1][1],eq1[1][2],R==3],R,x,y,z), solve([eq1[2],p==1,q==2,r==3],p,q,r,z)]for s in sol1: pretty_print(s)xxxxxxxxxxvar('x,y,z,t1,t2'); s=(-4,4)colors=['#3636ff','#ff3636','#ff36ff']f1=[x^2+y^2+z^2-4^2,(3*cos(t1)*sin(t2), 3*sin(t1)*sin(t2),3*cos(t2)), x^2/1^2+y^2/2^2+z^2/3^2-1]s1='<left><font color="%s">$(x,y,z) = %s$</font></left>'s2='<left><font color="%s">$%s = 0$</font></left>'pretty_print(html(s1%(colors[1],latex(f1[1]))))[pretty_print(html(s2%(colors[i],latex(f1[i])))) for i in [0,2]](implicit_plot3d(f1[0],s,s,s,opacity=.3,color=colors[0])+\ parametric_plot3d(f1[1],(0,2*pi),(0,pi),opacity=.5,color=colors[1])+\ implicit_plot3d(f1[2],s,s,s,plot_points=20, color=((z).function(x,y,z),colormaps.bwr)))\.show(frame=False)xxxxxxxxxxfor i in [0..10]: print(sorted(colormaps)[7*i:7*(i+1)])ΠΠΊΡΠΈΠ²ΠΈΡΡΠΉΡΠ΅ ΡΡΠΎΡ ΠΊΠΎΠ΄ ΠΏΠ΅ΡΠ΅Π΄ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ Π»ΡΠ±ΡΡ ΡΡΠ΅Π΅ΠΊ Linked Python Cells
xxxxxxxxxximport numpy,sympy,pylab,warningssympy.init_printing(use_unicode=True)from mpl_toolkits.mplot3d import Axes3Dfrom matplotlib.lines import Line2Dwarnings.simplefilter('ignore')x,y,z,R,p,q,r,t1,t2=\sympy.symbols('x,y,z,R,p,q,r,t1,t2')eq1=[x^2+y^2+z^2-R^2, [x-R*cos(t1)*sin(t2), y-R*sin(t1)*sin(t2), z-R*cos(t2)], x^2/p^2+y^2/q^2+z^2/r^2-1]sol1=[sympy.solve([eq1[0],R-4],R,z), sympy.solve([eq1[1][0],eq1[1][1],eq1[1][2],R-3],R,x,y,z), sympy.solve([eq1[2],p-1,q-2,r-3],p,q,r,z)]for s in sol1: display(s); print('\n')xxxxxxxxxxangle=numpy.linspace(0,2*numpy.pi,64)T1,T2=numpy.meshgrid(angle,angle)X=numpy.cos(T1)*numpy.sin(T2)Y=numpy.sin(T1)*numpy.sin(T2)Z=numpy.cos(T2)labels=['$x^2+y^2+z^2=4^2$', '$x=3cos(t_1)*sin(t_2)$'+\ '\n$y=3sin(t_1)*sin(t_2)$'+\ '\n$z=3cos(t_2)$', '$x^2/1^2+y^2/2^2+z^2/3^2=1$']colors=['#3636ff','#ff3636','#ff36ff']fig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')ax.plot_surface(4*X,4*Y,4*Z,color=colors[0],alpha=.1)ax.plot_surface(3*X,3*Y,3*Z,color=colors[1],alpha=.1)ax.plot_surface(1*X,2*Y,3*Z,cmap=pylab.cm.bwr,linewidth=1, antialiased=True,rstride=1,cstride=1)fake2Dlines=[Line2D([0],[0],linestyle='none',c=c,marker='o') for c in colors]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()ΠΠΊΡΠΈΠ²ΠΈΡΡΠΉΡΠ΅ ΡΡΠΎΡ ΠΊΠΎΠ΄ ΠΏΠ΅ΡΠ΅Π΄ ΠΈΡΠΏΠΎΠ»ΡΠ·ΠΎΠ²Π°Π½ΠΈΠ΅ΠΌ Π»ΡΠ±ΡΡ ΡΡΠ΅Π΅ΠΊ Linked R Cells
xxxxxxxxxx%%rX1<-Y1<-seq(from=-3.2,to=3.2,by=.9)sphere1<-function(R,X,Y){ func<-function(x,y){(-x^2-y^2+R^2)^.5} Z<-outer(X,Y,func); Z[is.na(Z)]<-0; list(Z,-Z)}T1<-seq(from=0,to=2*pi,length.out=12) T2<-seq(from=0,to=pi,length.out=12)sphere2<-function(R,t1,t2){ list(R*cos(t1)*sin(t2),R*sin(t1)*sin(t2),R*cos(t2))}X3<-seq(from=-.9,to=.9,by=.22); Y3<-2*X3ellipsoid1<-function(p,q,r,X,Y){ func<-function(x,y){r*(-x^2/p^2-y^2/q^2+1)^.5} Z<-outer(X,Y,func); Z[is.na(Z)]<-0; list(Z,-Z)}Z1<-sphere1(4,X1,Y1); XYZ2<-sphere2(3,T1,T2) Z3<-ellipsoid1(1,2,3,X3,Y3)dZ1=data.frame(Z1[1]) rownames(dZ1)<-X1; colnames(dZ1)<-Y1; dZ1xxxxxxxxxxvar('x,y,z,p,q,r')eq2=[x^2/p^2+y^2/q^2-z^2/r^2==-1, x^2/p^2+y^2/q^2-z^2/r^2==1, x^2/p^2+y^2/q^2-z^2/r^2==0]sol2=[solve([eq2[0],p==1,q==2,r==3],p,q,r,z), solve([eq2[1],p==3,q==4,r==5],p,q,r,z), solve([eq2[2],p==2,q==3,r==4],p,q,r,z)]for s in sol2: pretty_print(s)xxxxxxxxxxvar('x,y,z'); s=(-7,7); ops=[.5,.7,.3]colors=['#ff36ff','#ff3636','#3636ff'] f2=[x^2/1^2+y^2/2^2-z^2/3^2+1, x^2/3^2+y^2/4^2-z^2/5^2-1, x^2/2^2+y^2/3^2-z^2/4^2]st='<left><font color="%s">$%s = 0$</font></left>'[pretty_print(html(st%(colors[i],latex(f2[i])))) for i in [0..2]]sum([implicit_plot3d(f2[i],s,s,s,opacity=ops[i],mesh=True, color=colors[i],plot_points=20) for i in [0..2]]).show(frame=False)xxxxxxxxxxx,y,z,p,q,r=sympy.symbols('x,y,z,p,q,r')eq2=[x^2/p^2+y^2/q^2-z^2/r^2+1, x^2/p^2+y^2/q^2-z^2/r^2-1, x^2/p^2+y^2/q^2-z^2/r^2]sol2=[sympy.solve([eq2[0],p-1,q-2,r-3],p,q,r,z), sympy.solve([eq2[1],p-3,q-4,r-5],p,q,r,z), sympy.solve([eq2[2],p-2,q-3,r-4],p,q,r,z)]for s in sol2: display(s); print('\n')xxxxxxxxxxX=numpy.linspace(-7,7,128); X,Y=numpy.meshgrid(X,X) def f1(x,y): return 3/2*numpy.sqrt(4*x**2+y**2+4)def f2(x,y): return 5/12*numpy.sqrt(16*x**2+9*y**2-144)def f3(x,y): return 2/3*numpy.sqrt(9*x**2+4*y**2)Z=[f1(X,Y),f2(X,Y),f3(X,Y),-f1(X,Y),-f2(X,Y),-f3(X,Y)]for z in Z: z[z<-7]=numpy.NaN; z[z>7]=numpy.NaNfig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')colors=2*['#ff36ff','#ff3636','#3636ff']alphas=2*[.6,.4,.2][ax.plot_wireframe(X,Y,Z[i],color=colors[i],alpha=alphas[i]) for i in range(6)]labels=['$x^2/1^2+y^2/2^2-z^2/3^2=-1$', '$x^2/3^2+y^2/4^2-z^2/5^2=1$', '$x^2/2^2+y^2/3^2-z^2/4^2=0$']fake2Dlines=[Line2D([0],[0],linestyle='none',c=c,marker='o') for c in colors[:3]]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()xxxxxxxxxx%%rX1<-seq(from=-.9,to=.9,by=.22); Y1<-2*X1hyperboloid1<-function(p,q,r,X,Y){ func<-function(x,y){r*(x^2/p^2+y^2/q^2+1)^.5} Z<-outer(X,Y,func); list(Z,-Z)}X2<-seq(from=-5,to=5,by=1.1); Y2<-4*X2/3hyperboloid2<-function(p,q,r,X,Y){ func<-function(x,y){r*(x^2/p^2+y^2/q^2-1)^.5} Z<-outer(X,Y,func); Z[is.na(Z)]<-0; list(Z,-Z)}X3<-seq(from=-1.9,to=1.9,by=.5); Y3<-3*X3/2cone1<-function(p,q,r,X,Y){ func<-function(x,y){r*(x^2/p^2+y^2/q^2)^.5} Z<-outer(X,Y,func); list(Z,-Z)}Z1<-hyperboloid1(1,2,3,X1,Y1)Z2<-hyperboloid2(3,4,5,X2,Y2)Z3<-cone1(2,3,4,X3,Y3)dZ3=data.frame(Z3[1])rownames(dZ3)<-X3; colnames(dZ3)<-Y3; dZ3xxxxxxxxxxvar('x,y,z,p,q')eq3=[x^2/p^2+y^2/q^2==2*z, x^2/p^2-y^2/q^2==2*z, x^2/p^2+y^2/q^2-1==0, x^2/p^2-y^2/q^2-1==0, y^2==2*p*x]sol3=[solve([eq3[0],p==1,q==2],p,q,z), solve([eq3[1],p==2,q==3],p,q,z), solve([eq3[2],p==3,q==4],p,q,y,z), solve([eq3[3],p==2,q==3],p,q,y,z), solve([eq3[4],p==2],p,y,z)]for s in sol3: pretty_print(s)xxxxxxxxxxvar('x,y,z'); s=(-8,8); colors=['#ff3636','#3636ff']f31=[x^2/1^2+y^2/2^2-2*z,x^2/2^2-y^2/3^2-2*z]st='<left><font color="%s">$%s = 0$</font></left>'[pretty_print(html(st%(colors[i],latex(f31[i])))) for i in [0,1]]sum([implicit_plot3d(f31[i],s,s,s,opacity=.7,mesh=True, plot_points=20,color=colors[i]) for i in [0,1]]).show()xxxxxxxxxxvar('x,y,z'); s=(-9,9)colors=['#36ff36','#36ffff','#ff36ff']f32=[x^2/3^3+y^2/4^2-1,x^2/2^3-y^2/3^2-1,y^2-2*2*x]st='<left><font color="%s">$%s = 0$</font></left>'[pretty_print(html(st%(colors[i],latex(f32[i])))) for i in [0..2]]sum([implicit_plot3d(f32[i],s,s,s,opacity=.7,mesh=True, plot_points=20,color=colors[i]) for i in [0..2]]).show()xxxxxxxxxxx,y,z,p,q=sympy.symbols('x,y,z,p,q')eq3=[x^2/p^2+y^2/q^2-2*z, x^2/p^2-y^2/q^2-2*z, x^2/p^2+y^2/q^2-1, x^2/p^2-y^2/q^2-1, y^2-2*p*x]sol3=[sympy.solve([eq3[0],p-1,q-2],p,q,z), sympy.solve([eq3[1],p-2,q-3],p,q,z), sympy.solve([eq3[2],p-3,q-4],p,q,y,z), sympy.solve([eq3[3],p-2,q-3],p,q,y,z), sympy.solve([eq3[4],p-2],p,y,z)]for s in sol3: display(s); print('\n')xxxxxxxxxxX=numpy.linspace(-8,8,128)X,Y=numpy.meshgrid(X,X) def f1(x,y): return 1/2*x**2+1/8*y**2def f2(x,y): return 1/8*x**2-1/18*y**2Z=[f1(X,Y),f2(X,Y)]colors=['#ff3636','#3636ff']; alphas=[.8,.4]for z in Z: z[z<-8]=numpy.NaN; z[z>8]=numpy.NaNfig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')labels=['$x^2/1^2+y^2/2^2=2*z$', '$x^2/2^2-y^2/3^2=2*z$'][ax.plot_wireframe(X,Y,Z[i],color=colors[i],alpha=alphas[i]) for i in range(2)]fake2Dlines=[Line2D([0],[0],linestyle='none',c=c,marker='o') for c in colors]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()xxxxxxxxxxX=numpy.linspace(-9,9,128)X,Z=numpy.meshgrid(X,X) def f3(x): return 4/3*(-x**2+9)**.5def f4(x): return 3/2*(x**2-4)**.5def f5(x): return -2*(x)**.5Y=[f3(X),f4(X),f5(X),-f3(X),-f4(X),-f5(X)] colors=2*['#36ff36','#36ffff','#ff36ff']alphas=2*[.5,.7,.3]for y in Y: y[y<-9]=numpy.NaN; y[y>9]=numpy.NaNfig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')labels=['$x^2/3^2+y^2/4^2=1$', '$x^2/2^2-y^2/3^2=1$', '$y^2=2*2*x$'][ax.plot_wireframe(X,Y[i],Z,\ color=colors[i],alpha=alphas[i]) for i in range(6)]fake2Dlines=\[Line2D([0],[0],linestyle='none',c=c,marker='o') for c in colors[:3]]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()xxxxxxxxxx%%rX1<-Y1<-X2<-Y2<-seq(from=-9,to=9,by=1)X3<-Z3<-X4<-Z4<-Y5<-Z5<-seq(from=-5,to=5,by=.1)paraboloid1<-function(p,q,X,Y){ outer(X,Y,function(x,y){(x^2/p^2+y^2/q^2)/2})}paraboloid2<-function(p,q,X,Y){ outer(X,Y,function(x,y){(x^2/p^2-y^2/q^2)/2})}Z1<-paraboloid1(1,2,X1,Y1)Z2<-paraboloid2(2,3,X2,Y2)xxxxxxxxxx%%rcylinder1<-function(p,q,X,Z){ for (x in X){for (z in Z){ y<-q*(1-x^2/p^2)^.5 v<-c(x,y,z,x,-y,z) if (sum(is.na(v))==0){ write.table(matrix(v,nrow=2,ncol=3,byrow=TRUE), file ='cylinder1.csv',append=TRUE, quote=FALSE,col.names=FALSE,row.names=FALSE)}}}}cylinder2<-function(p,q,X,Z){ for (x in X){for (z in Z){ y<-q*(x^2/p^2-1)^.5 v<-c(x,y,z,x,-y,z) if (sum(is.na(v))==0){ write.table(matrix(v,nrow=2,ncol=3,byrow=TRUE), file='cylinder2.csv',append=TRUE, quote=FALSE,col.names=FALSE,row.names=FALSE)}}}}cylinder3<-function(p,Y,Z){ for (y in Y){for (z in Z){ x<-y^2/2/p; v<-c(x,y,z) write.table(matrix(v,nrow=1,ncol=3,byrow=TRUE), file='cylinder3.csv',append=TRUE, quote=FALSE,col.names=FALSE,row.names=FALSE)}}}cylinder1(3,4,X3,Z3)cylinder2(2,3,X4,Z4)cylinder3(2,Y5,Z5)'files are created'xxxxxxxxxximport pylabfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib.lines import Line2Dfiles=['cylinder1.csv','cylinder2.csv','cylinder3.csv']colors=['#36ff36','#36ffff','#ff36ff']labels=['$x^2/3^2+y^2/4^2=1$', '$x^2/2^2-y^2/3^2=1$', '$y^2=2*2*x$']XYZ=[numpy.array(numpy.loadtxt(files[i],\ delimiter=' ',unpack=True)) for i in range(3)]fig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')[ax.scatter(XYZ[i][0],XYZ[i][1],XYZ[i][2],s=1,\ c=colors[i],marker='1') for i in range(3)]fake2Dlines=[Line2D([0],[0],linestyle='none',\ c=c,marker='1') for c in colors]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()xxxxxxxxxxvar('x,y,z,p,q')eq4=[[x^2/p^2-y^2/q^2==0,p==1,q==2], [y^2==p^2,p==1],y^2==0]for e in eq4[:2]: pretty_print(solve(e,p,q,y))s=(-3,3); f4=[x^2/1^2-y^2/2^2,y^2-1^2]colors=['#ff3636','#ff36ff']st='<left><font color="%s">$%s = 0$</font></left>'[pretty_print(html(st%(colors[i],latex(f4[i])))) for i in [0,1]](implicit_plot3d(f4[0],(x,-3,3),(y,-3,3),(z,-3,3), color=(abs(sin(x*y*z)),colormaps.bwr))+\implicit_plot3d(f4[1],s,s,s, color=colors[1],opacity=.7)).show()xxxxxxxxxxx,y,z,p,q=sympy.symbols('x,y,z,p,q')eq4=[[x^2/p^2-y^2/q^2,p-1,q-2],[y^2-p^2,p-1],y^2]for e in eq4: display(sympy.solve(e,y)); print('\n')X=numpy.linspace(-3,3,128)X,Z=numpy.meshgrid(X,X) def f1(x,z): return 2*xdef f2(x,z): return 1Y=[f1(X,Z),-f1(X,Z),f2(X,Z),-f2(X,Z)] for y in Y[:2]: y[y<-3]=numpy.NaN; y[y>3]=numpy.NaNcolors=['#ff3636','#ff36ff']labels=['$x^2/1^2=y^2/2^2$','$y^2=1^2$']fig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')[ax.plot_surface(X,y,Z,cmap=pylab.cm.bwr, linewidth=1,antialiased=True, rstride=3,cstride=3) for y in Y[:2]][ax.plot_wireframe(X,y,Z,color=colors[1],alpha=.3) for y in Y[2:]]fake2Dlines=[Line2D([0],[0],linestyle='none',c=c,marker='o') for c in colors]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()xxxxxxxxxx%%rX<-Z<-seq(from=-3,to=3,by=.1)planes1<-function(p,q,X,Z){ for (x in X){for (z in Z){ y<-x*q/p if (min(X)<=y&&y<=max(X)){ v<-c(x,y,z,x,-y,z) v<-replace(v,is.na(v),0) write.table(matrix(v,nrow=2,ncol=3,byrow=TRUE), file='planes1.csv',append=TRUE,quote=FALSE, col.names=FALSE,row.names=FALSE)}}}}planes2<-function(p,X,Z){ for (x in X){for (z in Z){ v<-c(x,p,z,x,-p,z) write.table(matrix(v,nrow=2,ncol=3,byrow=TRUE), file='planes2.csv',append=TRUE, quote=FALSE,col.names=FALSE,row.names=FALSE)}}}planes1(1,2,X,Z); planes2(1,X,Z)'files are created'xxxxxxxxxximport pylabfrom mpl_toolkits.mplot3d import Axes3Dfrom matplotlib.lines import Line2DX1,Y1,Z1=\numpy.loadtxt('planes1.csv',delimiter=' ',unpack=True)X2,Y2,Z2=\numpy.loadtxt('planes2.csv',delimiter=' ',unpack=True)fig=pylab.figure(figsize=(7,7))ax=fig.add_subplot(111,projection='3d')ax.scatter(X1,Y1,Z1,s=2,c=Z1,cmap=pylab.cm.bwr,marker='1')ax.scatter(X2,Y2,Z2,s=2,c='#ff36ff',marker='1')labels=['$x^2/1^2=y^2/2^2$','$y^2=1^2$']colors=['#ff3636','#ff36ff']fake2Dlines=[Line2D([0],[0],linestyle='none',c=c,marker='1') for c in colors]ax.legend(fake2Dlines,labels,loc=int(9)); pylab.show()xxxxxxxxxxβxxxxxxxxxximport numpy,sympy,pylabxxxxxxxxxx%%rxxxxxxxxxxβ