Mathematical Experiment 7

 Construct this structure by joining four surfaces each of which is a part of the
 paraboloid:

 做法：

  > y:=r*cos(t);
  > z:=r*sin(t);
  > x:=y^2+z^2;
  > p1:=plot3d([x,y,z],r=abs(cos(t))..1,t=0..2*Pi,scaling=constrained,
                        color=red,grid=[31,101]):
  > with(plots):
  > x:=r*cos(t);
  > z:=r*sin(t);
  > y:=x^2+z^2;
  > p2:=plot3d([x,y,z],r=abs(cos(t))..1,t=0..2*Pi,scaling=constrained,
                        color=green,grid=[31,101]):
  > p3:=display(p1,p2):
  > with(plottools):
  > p4:=reflect(p3,[0,0,0]):
  > display(p3,p4);
　

　

 Construct this structure by joining four surfaces each of which is a part of the
 hemisphere:

 做法：

  > z:=cos(phi);
  > r:=sin(phi);
  > x:=r*cos(theta);
  > y:=r*sin(theta);
  > solve(x+z=1,phi);
  > s:=arctan(2*cos(theta)/(1+cos(theta)^2),-(cos(theta)^2-1)/(1+cos(theta)^2));
  > p1:=plot3d([x,y,z],theta=0..2*Pi,phi=abs(s)..Pi/2,scaling=constrained,
                        grid=[101,31]):
  > with(plottools):
  > with(plots):
  > q1:=translate(p1,0,0,-1):
  > q2:=rotate(q1,0,Pi/2,0):
  > q3:=rotate(q1,0,Pi,0):
  > q4:=rotate(q1,0,3*Pi/2,0):
  > display(q1,q2,q3,q4);
　

 Construct three mutually tangential cylinders of the same size situated like
 this:

 做法：

  > with(plots):
  > p1:=tubeplot([1,-1,t],t=-1..1,grid=[5,51]):
  > with(plottools):
  > p2:=rotate(p1,2*Pi/3,[[0,0,0],[1,1,1]]):
  > p3:=rotate(p1,4*Pi/3,[[0,0,0],[1,1,1]]):
  > display(p1,p2,p3);
 Arrange three identical mutually tangential cones like this:

 做法：

  > with(plots):
  > p1:=tubeplot([t,0,0],t=-1..1,radius=t,grid=[5,51]):
  > with(plottools):
  > p2:=rotate(p1,0,Pi/2,0):
  > p3:=rotate(p1,0,0,Pi/2):
  > display(p1,p2,p3);
　
 Arrange four identical mutually tangential half-cones like this:

 做法：

  > with(plots):
  > p1:=tubeplot([-t,-t,-t],t=0..1,radius=sqrt(6)*t,grid=[5,51],
                           scaling=constrained,color=red):
  > p2:=tubeplot([-t,t,t],t=0..1,radius=sqrt(6)*t,grid=[5,51],
                           scaling=constrained,color=green):
  > p3:=tubeplot([t,-t,t],t=0..1,radius=sqrt(6)*t,grid=[5,51],
                           scaling=constrained,color=blue):
  > p4:=tubeplot([t,t,-t],t=0..1,radius=sqrt(6)*t,grid=[5,51],
                           scaling=constrained,color=yellow):
  > display(p1,p2,p3,p4);
 Arrange eight identical half-cones each tangent to the other three like this:

 做法：

  > with(plots):
  > p1:=tubeplot([t,t,t],t=0..1,radius=sqrt(3/2)*t,grid=[5,51],
                           scaling=constrained,color=red):
  > p2:=tubeplot([-t,t,t],t=0..1,radius=sqrt(3/2)*t,grid=[5,51],
                           scaling=constrained,color=green):
  > p3:=tubeplot([t,-t,t],t=0..1,radius=sqrt(3/2)*t,grid=[5,51],
                           scaling=constrained,color=blue):
  > p4:=tubeplot([-t,-t,t],t=0..1,radius=sqrt(3/2)*t,grid=[5,51],
                           scaling=constrained,color=yellow):
  > with(plottools):
  > q1:=display(p1,p2,p3,p4):
  > q2:=reflect(q1,[0,0,0]):
  > display(q1,q2);
Construct the solid bounded by the parabolic cylinder

z = 4 - y2

and the elliptic paraboloid

z = x2 + 3y2.

 做法：

  > x:=r*2*cos(t);
  > y:=r*sin(t);
  > z:=4-y^2;
  > p1:=plot3d([x,y,z],r=0..1,t=0..2*Pi,scaling=constrained,
                        color=red,grid=[11,101]):
  > z:=x^2+3*y^2;
  > p2:=plot3d([x,y,z],r=0..1,t=0..2*Pi,scaling=constrained,
                        color=green,grid=[11,101]):
  > with(plots):
  > display(p1,p2);