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MATING GEAR PROFILES (CONTINUATION)
 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