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SYNCHRONOUS CURVE

Given a family of secant lines *O* placed in a uniform
gravitational field, and massive points sliding without friction on each
of these curves, Galileo posed the problem of determining on which curves
the points had to be in order to, with no initial speed, arrive simultaneously
at *O*; the answer is a circle the bottommost point of which is *O*.

We derive from this that if all the points leave *O*
with no initial speed on lines passing by *O*, then the curve they
generate at every instant is also a circle.

If the lines are replaced by a family of curves passing
by *O*, with the condition that the points have to pass simultaneously
by *O*, then the curve they generate at every instant is called *synchronous
curve*.

Example:

If the curves are cycloids
with horizontal base and cuspidal point at *O*, the points leaving
*O* with no initial speed, then the synchronous curves are the
orthogonal
trajectories of these cycloids (Jean Bernoulli).

See also synodal
curve,
isochronous curve of Huygens,
isochronous curve of Leibniz,
isochrononous curve of Varignon,
brachistochrone
and tautochronous curve.

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© Robert FERRÉOL 2017