Circle parameterized by parabolic points
- Author:
- Paul Miller
- Topic:
- Algebra, Centroid or Barycenter, Circle, Conic Sections, Constructions, Coordinates, Curve Sketching, Equations, Optimization Problems, Fractions, Integers, Intersection, Parabola, Parametric Curves, Plane Figures or Shapes, Rational Numbers, Ratios, Sequences and Series, Special Points, Tangent Line or Tangent
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.