The process of turning symbols into a mathematical sculpture has never been easier! This is what you need:

- a copy of Maple V release 5.
- a web browser configured with a VRML plugin such as the Cosmo Player.

Building virtual 3D models creates many fruitful learning activities. We find the following examples particularly exciting.

It takes seven lines of Maple commands to create such an object:

> a:=3;b:=7;c:=sqrt(a^2+b^2);> r:=c+a*cos(t);> x:=r*cos(s);> y:=r*sin(s);> z:=a*sin(t);> u:=solve(b*z=a*x,s);> plot3d([x,y,z], s=-u..u, t=0..2*Pi, scaling=constrained,
grid=[30,80]); |

Assuming the symbol h stands for the process of drawing the object:

> h:=plot3d([x,y,z], s=-u..u, t=0..2*Pi, scaling=constrained,
grid=[30,80]): |

The conversion of the geometric object into the required VRML file is
now straightforward.

>vrml(h,"d:/vrml/villarceaux.wrl", background_color=white); |

As a related question, one might ask if a half-torus can have a boundary consisting of two disjoint interlocking circles? The answer: yes, just choose a Villarceaux circle of any torus as a generator of the half-torus and rotate it by 180 degrees about the axis.

> a:=3;b:=7;c:=sqrt(a^2+b^2);> x1:=b*cos(t);> y1:=a+c*sin(t);> z:=a*cos(t);> x:=x1*cos(u)-y1*sin(u);> y:=x1*sin(u)+y1*cos(u);> plot3d([x,y,z],u=0..Pi,t=0..2*Pi,scaling=constrained); |

> r:=3+cos(t);> x:=r*cos(s);> y:=r*sin(s);> z:=sin(t);> s:=2/5*t;> tubeplot([x,y,z], t=0..10*Pi, radius=0.6,scaling=constrained,
grid=[100,10]); |

generates a torus knot as in this illustration

Find the volume of the region common to the interior of the cylinders
x ^{2} + y^{2} = a^{2} and x^{2} + z^{2}
= a^{2}. |

The solid looks like this:

> u:=[cos(t),sin(t),sin(t)];> v:=[cos(t),sin(t),-sin(t)];> w:=expand((1-s)*u+s*v);> plot3d(w, s=0..1, t=0..2*Pi, scaling=constrained, color=green); |

while the second part is obtained by rotating the first:

> plot3d([w[1],-w[3],w[2]],s=0..1, t=0..2*Pi, scaling=constrained,
color=yellow); |

Realizing the parametric description of the solid is the most difficult part in solving the original problem. If we examine how the surfaces are represented symbolically, we find exactly what is need: the cross-sections perpendicular to the x-axis consist precisely of squares with u, v, -u and -v as corners. This serves as an example in which a clear visual understanding of the problem leading to a clear analytical solution.

Here is another similar challenging calculus problem:

Find the volume of the region common to the interior of the intersection
of the cylinder (x - 0.5)^{2} + y^{2} = 0.25 and the sphere
x^{2} + y^{2} + z^{2} =1. |

Again, the vaxed question is: "What does the solid region looks like?"
Since a part of the boundary lie on the unit sphere, the attention is thus
turned to the spherical coordinates. The Maple V commands used to describe
the region is in fact a direct translation of the related mathematical
formula:

> r:=cos(s);> x:=r*cos(t);> y:=r*sin(t);> z:=sin(s);> a1:=plot3d([x,y,z],t=-s..s,s=0..Pi/2,scaling=constrained,
color=red,grid=[100,10]):> a2:=plot3d([x,y,-z],t=-s..s,s=0..Pi/2,scaling=constrained,
color=red,grid=[100,10]):> t:=s;> b:=plot3d([x,y,(1-2*u)*z],u=0..1,s=0..Pi, scaling=constrained,
color=red,grid=[10,100]):> display(a1,a2,b); |

Platonic solids and related geometric objects

Subsets of the sphere

Ruled surfaces

Orthogonal grids on a cone