Two curves joining a point A to a point B, placed in a uniform gravitational field are said to be synodal if, when two massive points are dropped from A with no initial speed, both the points arrive simultaneously at B, whether they follow one curve or the other.
 

Bonati posed and solved in 1780 the problem of determining a curve starting from O such that, at every instant, the time taken to arrive at the point M along this curve is equal to the time a massive point starting from O would take to arrive at M along the line (OM); in other words, of determining a curve that would be synodal with all its secants passing by one of its points. The answer is a half-lemniscate of Bernoulli.
 

See also at synchronous curve, isochronous curve of Huygens, isochronous curve of Leibniz, isochronous curve of Varignon, brachistochrone and tautochronous curve.
 

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© Robert FERRÉOL  2017