Converting parametric equations to implicit form when isolating t is difficult

[Gagle The Terrible]

I would like to convert these parametric equations into a single
f(x,y) = 0 function.

X(t) = t^2 + t + 1
Y(t) = t^2 - t +1

In fact, what stops me is the imaginary roots of the parametric polynomials.

Is there a way to get around the seemingly impossible explicit solving of the quadratics to isolate either X or Y ?

 

[0rthodontist]

Note that x - t = y + t (or just subtract y(t) from x(t)).

 

[d_leet]

I would like to convert these parametric equations into a single
f(x,y) = 0 function.

X(t) = t^2 + t + 1
Y(t) = t^2 - t +1

In fact, what stops me is the imaginary roots of the parametric polynomials.

Is there a way to get around the seemingly impossible explicit solving of the quadratics to isolate either X or Y ?

First note that X - Y = 2t, but also notice that these are both quadratic equations so you can solve one of them or the other to find t in terms of X or Y and then substitute this into X - Y = 2t or the equation for the other variable.

 

[0rthodontist]

You just want to substitute t = (x - y) / 2. There's no need to solve any quadratics.

 

[d_leet]

You just want to substitute t = (x - y) / 2. There's no need to solve any quadratics.

Darn, I guess I should have seen that since I did notice that x - y = 2t.