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GALILEAN SPIRAL
Curve studied by Galileo in 1636.
Galileo (15641642): 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.
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 ! 
approximated by the green circle. 
The Galilean spiral has a cuspidal point at O only when a = 0 

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 2019