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ROTOID
Closed normal ruled rotoid with a horopter
curve as bore
Other name: generalized helicoid.
Therese Eveillau's website. Alain Esculier's website. 
For a spine curve
with current point
and curvilinear abscissa s, parametrization of the rotoid with generatrix :
where and , the angle y is the angle of torsion of the spine curve (see the notations). h is the reduced shift of the rotoid and 2ph its shift. 
The word rotoid refers to any surface generated
by a regular screw of a curve (the generatrix) around a fixed curve (the
"spine curve, or "bore" of the rotoid).
The intersection between the rotoid and a tube with same
spine curve is a union of solenoids
with reduced shift h.
When h is positive, the rotoid is said to be righthanded,
and lefthanded otherwise.
When the spine curve is linear, we get the helicoids.
When the generatrix is a line, we get the ruled rotoids, with similar definitions as those of the ruled helicoids.
The Möbius surface is a closed normal ruled rotoid with a circle as its spine curve.
When the center of a regular polygon moves along a curve,
perpendicularly to it, with a regular torsion motion, the sides of the
polygon trace a surface that we will call "rotoidal prism". Each "face"
of this "prism" is a portion of open normal ruled rotoid.
For the case of a linear spine curve, see at ruled
helicoid.
For the case of a circular spine curve, a (convex) polygon
with n sides, and a torsion of k nths of a turn for
one revolution, we get a surface composed of d = GCD(n, k)
"faces" (after a revolution, the side #x of the polygon connects
with the side #x+k modulo n). Moreover, there are also d
"edges", which form a toroidal
link of type (k, n): each component turns n/d
times around the axis, and winds k/d times around the torus.
In particular, if n and k are coprime,
the rotoidal prism has only one "face" and only one edge.
Examples for n = 3:
k = 1: one face, one untied edge  k = 2: one face, one edge tied as a trefoil knot  k = 3: three faces (fake Möbius strips with 2 halftwists), three edges (Villarceau circles of the torus, fake Borromean rings)  k = 4: one face, one edge, toroidal knot of the type (4,3), prime knot with 8 crossings. 




Examples for n = 4:
k = 1: one face, one untied edge  k = 2: two faces, two untied edges, forming a Salomon knot  k = 3: one face, on edge, toroidal knot of type (3,4), prime knot with 8 crossings.  k = 4: 4 faces (fake Möbius strips with 4 halftwists), 4 edges (Villarceau circles) 




Examples for n = 6:
k = 1: one face, one untied edge  k = 2: two faces, two untied edges  k = 3: three faces, three edges, toroidal link of type (3,6).  k = 4: two faces, two untied edges, toroidal link of type (4,6). 




Opposite, a view of the case n = 4, k =
2; if the tracing square is contracted into a segment line, (in other words,
if we stick two edges together), we get a Möbius
strip.
Therefore, this solid can also be seen as a Möbius strip cut in thick cardboard, the length of the strip being equal to the width of the cardboard. 
Generalization to the case where k is a rational
number p/q.
The linear section of the resulting surface is composed of q polygons with n sides, i.e. a polygram with symbol {qn/q}. Opposite, the case n = 3, p =1, q =2. 
Game: find the values of n and k in the
sculptures below:




Work photographed in the Mentoring Museum of Tours  Work by JeanDaniel Huyghe  Work by Christophe Chini 
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© Robert FERRÉOL 2017