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CONSTANT ANGULAR ACCELERATION CURVE

Curve studied by Mikhail Gaichenkov in 2008.

 
Differential equation  :  (a > 0), so , so , after rotation.
Polar equation :  where f is the unique solution on  of , see the sequence A202407 from OEIS for this development.

 
Cette courbe est la trajectoire d'un mouvement à accélération angulaire constante non nulle. 

Plus précisément, si une droite tourne uniformément autour d'un de ses points, un point de cette droite se meut de sorte à avoir un mouvement uniformément accéléré sur sa trajectoire.
This curve is the trajectory of a movement with constant non-zero angular acceleration.

More precisely, if a line turns uniformly around one of its points, a point of this line moves so as to have a uniformly accelerated movement on its trajectory.


 
For ; the curve follows the la Galileo spiral:, in green opposite.
For  ; the curve becomes asymptote to the Archimedean spiral:, in green opposite.

Note that the curve at zero angular acceleration (i.e. at constant angular speed) is the circle.
 
 
We can also consider a curve drawn by a point moving on a line in uniform translation perpendicular to its direction with a uniformly accelerated movement on its trajectory:

 
Differential equation :  (a > 0), so , so , after translation, or .
Cartesian parametrization : .
Cartesian equation : , asymptote curve : .
Curvilinear abscissa : .
Radius of curvature : .
Intrinsic equation 1 : .

 
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© Robert FERRÉOL  2020