GALILEAN SPIRAL 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.

 Since , we can consider that the spiral is the edge of a trapezoidal tape rolled ... Isn't there any resemblance to the tail of the chameleon?  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 .