1. What is formal calculation?

1.1 A demonstration with Maple: Villarceau circles

Here's a classic geometry problem that illustrates what Maple can do.

Consider in a plane of space a circle Γ of center ω and radius r and a straight line Δ located at a distance R > r from ω. By rotating this circle around the Δ axis, we obtain the well-known $\mathcal{T}$ surface known as a torus. Let then be an orthonormal reference frame Oxyz, where O is the projection of ω onto Δ and whose axes are such that Ox = Oω and Oz = Δ. Horizontal planes and planes containing Δ generally intersect $\mathcal{T}$...