Equation of a circle using rational fractions Why does the following equation draw a circle ?
$$\left(\frac{t^4-6t^2+1}{t^4+2t^2+1},\frac{4t-4t^3}{t^4+2t^2+1}\right),|t|\le1$$
Does it draw a perfect circle, or an approximation ? On Desmos, it looks like a perfect circle.
(Added by edit) If it is an exact equation, how does one find such an equation ? Where does it come from ?

The rational parametrization of the unit circle that’s most often seen is
$$
x=\frac{2t}{t^2+1}\,,\quad y=\frac{t^2-1}{t^2+1}\,.
$$
You can get this, as I recall, by drawing the line through $(0,-1)$ with slope $t$ and seeing where it intersects the unit circle. It’s a nice exercise.
