I would now like to give a first definition of the turbicone.
All the cones of the second degree, rolling on each other do not have the particularity of defining closed spaces. Here is a first approach to define those who have this ability:
The following video illustrates this definition:
https://www.econologie.info/share/partag ... L4YqsL.wmv
1) It should be noted that the center of a circle of radius R which rests on the three planes of a right-angled trihedron remains at a constant distance D from the origin of this trihedron.
D = R root (2)
(The demonstration is easy because by performing a projection of this circle on one of the planes of the trihedron we recognize the figure explaining Monge's circle. We can therefore use the formulas associated with it)
We can easily verify this point by placing a bicycle wheel in the corner of a room, with a CD and a shoe box ...
The beginning of the video shows a circle in different positions.
2) The video shows in gray a little transparent, the sphere of radius D on which the center of the circle evolves.
It would also have been possible to look at things in another way: by fixing the circle and by moving the trihedron so that its three planes are constantly supported on its circumference. It is the origin which would then have described a sphere of radius D, having the same center as that of the circle.
3) It is then necessary to freeze the angle under which we see the circle from the origin. This is shown on the video by the appearance of a triangle having its vertex at the origin and retaining a fixed position relative to the circle, (in a plane perpendicular to that of the circle).
We note that despite this constraint, the circle still has many possible positions for which it remains in contact with the three planes. In fact, it now describes a curve on the gray spherical surface. (this curve is not shown on the video)
4) Since the origin is now at a fixed position with respect to the circle, it can be used as the vertex of a non-deformable cone having the circle as a directing curve.
The video shows this cone in green. It also shows that a second circle identical to the first could have given the same cone.
It is very important to note that the contact points of the circle on the planes have given rise to contact lines of the cone on the plans. And so that the space of the trihedron outside the cone is separated into three parts delimited by these contact lines.
5) This step consists in truncating the cone obtained by two concentric spheres to finally define the solid part "Turbicone".
The video shows only the outer spherical part which evolves in the trihedron.
6) Now having our initial turbicone, we must use all the symmetries of the trihedron to obtain the other seven that will compose the volumetric system.
The video also presents certain elements on the spherical parts whose use will be defined later.
7) Finally, you must completely remove the trihedron to finally see the system of the eight turbicones in action.
The planes having disappeared, the turbicones remain in contact and define spaces whose volumetric variations are particularly interesting.
In fact, the triad is not completely gone. It just went virtual. We can regenerate it at any time because it is formed by the planes tangent to the turbicones according to their contact lines.