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MATING GEAR PROFILES
Notion studied by Euler and by Miquel
in 1838.
Other name, given by Miquel: syntrepent curves, literally "that turn together". 
Two curves and are called mating profiles associated to the centres of rotation O_{1} and O_{2} (the gear hubs) if can turn around O_{1} and around O_{2} so that the two curves always are in contact (i.e. tangent without slipping on one another).
The get the curve
when the curve ,
the hub O_{1} and the distance d
= O_{1}O_{2}
are given, one only has to determine the movement
of a plane over a fixed plane the base of which is
and a roulette of which is the circle with centre O_{1}
and radius d: the rolling curve is then the curve .
Writing
the polar coordinates of the current point on
in a frame centred on O_{1}, and, similarly,
for in
a frame centred on O_{2}, the relation
between
and is
given by:

Remark: By the Descartes theorem, the contact point between the two mating profiles is always aligned with the two hubs.
Examples:
 if
is a circle with radius a and the hub is at its centre,
is a circle with radius d  a.
 if
is linear, located at distance a from O, then the equation
of is ,
which gives:
if d > a:
with 
if d = a: 
if d < a: with (polygasteroid with e > 1) 
Case n = 1 (i.e. k = 2):
(G_{2}) is an ellipse isometric to . The ellipse is said to be isotrépente for its focus. 
Case n = 2: is an inverse of the peanut. 


Remark: if
is a hyperbola, the same property holds, but is not illustrated here, because
the hyperbola is not a closed curve. See the case n = 1 on the page
of the hyperbola.
 if
is a logarithmic spiral,
is an isometric logarithmic spiral; the logarithmic spiral is therefore
also an isotrépente curve.
This mechanism was discovered by the astronomer Ole Rømer. It is used in the "varistart" system, and allows for a progressive increase of the rotation speed of a gear. 
 if
is a cardioid:
(hub at the cuspidal point),
the parametrization of is: (d = ka). Opposite, the case k = 18/5. 
The mouth with the heart... 
if
is a circle:
(hub on the circle),
the parametrization of is: (d = ka). Opposite, the case k = 5/3. 

Schroeder
gears dating from 1867, coming from the Musée des Arts et Métiers.
The bottom curve is a limaçon of Pascal. See also this link. 
When the hub of one of the gears is at infinity, we get
a wheelroad couple.
We saw above that if we attach one of the gears and that
we make the other one roll on it, the latter's hub describes a circle.
This enables to imagine "Shaddock" vehicles like the one represented opposite...
(by Alain Esculier: see at the bottom of this
page)
For similar vehicles, but with a linear motion, see the next page. 
See also the hyperboloid gears, generalisation to space
of this planar notion, at the bottom of this
page.
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© Robert FERRÉOL , Jean LEFORT, Alain ESCULIER 2017