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 : .