Circle parameterized by parabolic points
A remarkable parameterization of the circle x² + y² - y = 0 exists with respect to the parabola y = x².
Points on the circle are described by coordinates: (t/(t²+1),1/(t²+1))
Points on the parabola with coordinates: (1/(t-1),1/(t-1)²) and (1/(t+1),1/(t+1)² respectively, combined in proportion '(t-1)²' : '(t+1)²', simplify to this form.
Additionally the 'weights' correspond to the height of each point above the axis, so defining centres of mutually tangent circles.
Varying 't' by 2 gives one new and one existing parabolic point and tangent to the circle, thus continuing a 'chain' of mutually tangent circles touching the x-axis.
As will be seen, the motivation for this is as a generator of points on circles giving the solutions of 'nested' circles. These are defined as circles centred on the axis and subdividing a unit interval [0,1] without overlap - in other words circles of inverse integral radius, '1/n' for all Natural numbers. A solution is then when two or more such circles (necessarily of different radii) meet rationally.