2732.sagews

The first picture is of the solid formed by intersecting three cylinders; the second shows the solid along with the original cylinders. You can view these in 3D in a variety of ways. Right-click on the picture and go to style->stereographic and pick a method. Most people will find it easiest to use the cross-eyed method. If you haven't done this before, you may find this page useful:  http://www.starosta.com/3dshowcase/ihelp.html

# Useful definitions
(x,y,u,t,a,b)=var("x y u t a b")
minimum(a,b)=((a+b)-abs(b-a))/2
x=cos(u); y=sin(u)
v=t*minimum(sqrt(1-x^2),sqrt(1-y^2))

redsurf1=parametric_plot3d([x,v,y],(u,0,pi/2),(t,-1,1),color='red',aspect_ratio=(1,1,1),plot_points=(100,10))
redsurf2=parametric_plot3d([x,v,y],(u,pi/2,pi),(t,-1,1),color='red',aspect_ratio=(1,1,1),plot_points=(100,10))
redsurf3=parametric_plot3d([x,v,y],(u,pi,3*pi/2),(t,-1,1),color='red',aspect_ratio=(1,1,1),plot_points=(100,10))
redsurf4=parametric_plot3d([x,v,y],(u,3*pi/2,2*pi),(t,-1,1),color='red',aspect_ratio=(1,1,1),plot_points=(100,10))
bluesurf1=parametric_plot3d([x,y,v],(u,0,pi/2),(t,-1,1),color='blue',aspect_ratio=(1,1,1),plot_points=(100,10))
bluesurf2=parametric_plot3d([x,y,v],(u,pi/2,pi),(t,-1,1),color='blue',aspect_ratio=(1,1,1),plot_points=(100,10))
bluesurf3=parametric_plot3d([x,y,v],(u,pi,3*pi/2),(t,-1,1),color='blue',aspect_ratio=(1,1,1),plot_points=(100,10))
bluesurf4=parametric_plot3d([x,y,v],(u,3*pi/2,2*pi),(t,-1,1),color='blue',aspect_ratio=(1,1,1),plot_points=(100,10))
yellowsurf1=parametric_plot3d([v,x,y],(u,0,pi/2),(t,-1,1),color='yellow',aspect_ratio=(1,1,1),plot_points=(100,10))
yellowsurf2=parametric_plot3d([v,x,y],(u,pi/2,pi),(t,-1,1),color='yellow',aspect_ratio=(1,1,1),plot_points=(100,10))
yellowsurf3=parametric_plot3d([v,x,y],(u,pi,3*pi/2),(t,-1,1),color='yellow',aspect_ratio=(1,1,1),plot_points=(100,10))
yellowsurf4=parametric_plot3d([v,x,y],(u,3*pi/2,2*pi),(t,-1,1),color='yellow',aspect_ratio=(1,1,1),plot_points=(100,10))

redsurf1+redsurf2+redsurf3+redsurf4+bluesurf1+bluesurf2+bluesurf3+bluesurf4+yellowsurf1+yellowsurf2+yellowsurf3+yellowsurf4

(u,v)=var("u v")
yellowcyl=parametric_plot3d([u,cos(v),sin(v)],(u,-1.5,1.5),(v,0,2*pi),color='yellow',opacity=0.1,plot_points=(100,100))
redcyl=parametric_plot3d([cos(v),u,sin(v)],(u,-1.5,1.5),(v,0,2*pi),color='red',opacity=0.1,plot_points=(100,100))
bluecyl=parametric_plot3d([cos(v),sin(v),u],(u,-1.5,1.5),(v,0,2*pi),color='blue',opacity=0.1,plot_points=(100,100))

redsurf1+redsurf2+redsurf3+redsurf4+bluesurf1+bluesurf2+bluesurf3+bluesurf4+yellowsurf1+yellowsurf2+yellowsurf3+yellowsurf4+redcyl+bluecyl+yellowcyl