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Curve studied by Galileo in 1636.
Galileo (1564-1642): Italian physicist and astronomer.

Reduced polar equation: .
Curvilinear abscissa:  (algebraic rectification if a = 0, elliptic otherwise).
Radius of curvature for a = 0: .

The Galilean spiral is the trajectory of a point moving in a uniformly accelerated motion on a line of a plane, this line turning around one of its points.

The general polar equation of such a motion is given, with the classic kinematic notations, by:
which, after rotation, gives indeed a polar equation of the type above.
The trajectory of a body subjected to its weight (neglecting the friction, and on a portion small compared to the distance to the centre of the Earth) in the plane of the Equator considered as an inertial frame of reference is a portion of Galilean spiral. 
Galileo posed the problem of this trajectory, under the form of a search for the curve on which travels a rock falling from a tower, hence the name given to this spiral. 
He thought the trajectory was most certainly an arc of a circle. He was not far from the truth as the portion of the spiral  opposite shows!
The area of the loop of the spiral is  and that of the disk is  !
In black, the Earth, in red, the Galilean spiral,
approximated by the green circle.



The Galilean spiral has a cuspidal point at O only when a = 0

It can be obtained as a roulette of the motion associated to a parabola rolling on an Archimedean spiral.

The Galilean spiral is also a special case of isochronous curve of Varignon.

In the case a = 0, we have  as  goes to infinity: the Galilean spiral is close to the Sturm spiral, which satisfies exactly .
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© Robert FERRÉOL 2017