1 Construct the portion of the cylinder x^2+y^2=1 lying between the planes z=0 and z=1+x.

> restart:

> x:=cos(t):

> y:=sin(t):

> z1:=-1:z2:=x:

> z:=(1-s)*z1+s*z2:

> plot3d([x,y,z],s=0..1,t=0..2*Pi,scaling=constrained);

2 Describe how two pipes of the same size are joined perpendicularly.

> restart:

> x:=cos(t):

> y:=sin(t):

> z1:=-1.5:z2:=x:

> z:=(1-s)*z1+s*z2:

> a:=plot3d([x,y,z],s=0..1,t=0..2*Pi,scaling=constrained):

> with(plottools):

> b:=reflect(a,[[0,0,0],[1,0,1],[2,-1,2]]):

> with(plots):

> display(a,b);

3 Construct two pipes joined as thus:

> restart:

> x:=cos(t):

> y:=sin(t):

> z1:=-1.5:z2:=x:

> z:=(1-s)*z1+s*z2:

> a:=plot3d([x,y,z],s=0..1,t=Pi/2..3*Pi/2,scaling=constrained):

> with(plottools):

> b:=reflect(a,[[0,0,0],[0,1,1],[0,-2,2]]):

> with(plots):

> c:=display(a,b):

> d:=rotate(c,0,Pi/2,0):

> e:=rotate(c,0,Pi,0):

> f:=rotate(c,0,3*Pi/2,0):

> display(c,d,e,f);

4 Describe how three pipes are symmetrically joined.

> restart:

> x:=cos(t):

> y:=sin(t):

> z1:=x/sqrt(3):

> z:=(1-s)*z1+s*(-1.5):

> a:=plot3d([x,y,z],s=0..1,t=Pi/2..3*Pi/2,scaling=constrained):

> with(plottools):

> b:=reflect(a,[[0,1,0],[0,0,1],[0,1,1]]):

> with(plots):

Warning, the name changecoords has been redefined

> d:=rotate(c,2*Pi/3,[[0,1,0],[0,-1,0]]):

> e:=rotate(c,4*Pi/3,[[0,1,0],[0,-1,0]]):

> display(c,d,e);

5 Wrap the graph of y = cos 2x around a cylinder.

> restart:

> x:=cos(t):y:=sin(t):

> z1:=-1:z2:=cos(2*t):z:=(1-s)*z1+s*z2:

> plot3d([x,y,z],s=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50]);

6 Wrap the graph of y = cos 3x around a cylinder.

> restart:

> x:=cos(t):y:=sin(t):

> z1:=-1:z2:=cos(3*t):z:=(1-s)*z1+s*z2:

> plot3d([x,y,z],s=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50]);

7 Construct the graph of z = x^2 - y^2 above the unit circle.

> restart:

> x:=r*cos(t):y:=r*sin(t):z:=x^2-y^2:

> plot3d([x,y,z],r=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50]);

8 Construct the monkey's saddle given by z = x^3 - 3xy^2 above the unit circle.

> restart:

> x:=r*cos(t):y:=r*sin(t):z:=evalc(Re(expand(x+I*y)^3)):

> plot3d([x,y,z],r=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50]);

9 Construct this model:

> restart:

> x:=r*cos(t):y:=r*sin(t):z:=x^2-y^2:

> a:=plot3d([x,y,z],r=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50],color=green):

> x:=cos(t):y:=sin(t):

> z1:=-1:z2:=cos(2*t):z:=(1-s)*z1+s*z2:

> b:=plot3d([x,y,z],s=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50],color=red):

> with(plots):

> display(a,b);

10 Construct this model:

> restart:

> x:=r*cos(t):y:=r*sin(t):z:=evalc(Re(expand(x+I*y)^3)):

> a:=plot3d([x,y,z],r=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50],color=yellow):

> x:=cos(t):y:=sin(t):

> z1:=-1:z2:=cos(3*t):z:=(1-s)*z1+s*z2:

> b:=plot3d([x,y,z],s=0..1,t=0..2*Pi,scaling=constrained,grid=[5,50],color=red):

> with(plots):

Warning, the name changecoords has been redefined

>