Below are four sets of six points each on the first quadrant of the plane. They all lay on a circle with center on the origin and with a radius equal 1.
The first set was generated using the parametric equation of the circle $[(x, y) = (cosθ, sinθ)]$, taking equally spaced "θ"s. The second using $[(x, y) = (x,√(1-x^2 ))]$, taking equally spaced "x"s. The third using $[(x, y) = (√(1-y^2 ), y)]$ taking equally spaced "y"s. And the fourth using $[(x, y) = ((1 - t^2) / (1 + t^2), 2 t / (1 + t^2))]$ taking equally spaced "t"s. They are:
On "θ" - ((1.,0.),(0.951,0.309),(0.809,0.588),(0.588,0.809),(0.309,0.951),(0.,1.))
On "x" - ((1.,0.),(0.8,0.6),(0.6,0.8),(0.4,0.917),(0.2,0.98),(0.,1.))
On "y" - ((1.,0.),(0.98,0.2),(0.917,0.4),(0.8,0.6),(0.6,0.8),(0.,1.))
On "t" - ((1.,0.),(0.923,0.385),(0.724,0.69),(0.471,0.882),(0.22,0.976),(0.,1.))
They are all plotted in the left graph and it can clearly be seen that they lay on the circle.
Each set of those points was used to generate an interpolating polynomial, on "x":
From "θ" - $ 1. -8.77181 x+63.1158 x^2-160.288 x^3+169.631 x^4-64.6871 x^5$
From "x" - $ 1. -0.251314 x+2.13091 x^2-9.2879 x^3+13.3399 x^4-6.93156 x^5$
From "y" - $ 1. -363.724 x+1833.61 x^2-3424.16 x^3+2809.95 x^4-856.671 x^5$
From "t" - $ 1. -1.5556 x+13.7816 x^2-43.4582 x^3+53.6775 x^4-23.4453 x^5$
They are plotted on the right graph. As can clearly be seen only the data that has equally spaced "x"s produces an acceptable curve when compared to the circle. All other 3 have different spaced "x"s and vary wildly between the original 6 points.
It is very well known that passing thru "n" points we get only one unique polynomial of degree "n - 1". For each set above we got a different polynomial, with coefficients varying a lot among each other.
Quite a few sets of points were found that have unequally spaced "x"s and produce very acceptable interpolating polynomials but those were found by trial and error, experimenting, and probably it would be difficult to find a generating function for those sets. Other parametric equations of different functions were tested also and showed similar results. As the number of interpolating points gets higher in the interval the fitting gets worse if "x" is unequally spaced. It can be observed in the equations above that the greater the absolute value of the coefficients of the polynomial the worse is the fitting. This is a rule showed in various tests made.
Why the best fit seems to require equally spaced "x"s? Is it due to the fact that equally spaced "x"s appear to produce the shortest length for the polynomial in the interpolating interval?