By a linear transformation, a conic known by its implicit equation can be put in a reduced form such as that of a unit circle $x^2+y^2=1$. The latter can be represented by rational parametric equations,
$$\begin{cases}x=\dfrac{2t}{t^2+1},\\y=\dfrac{t^2-1}{t^2+1}.\end{cases}$$
Coming back to the original equation, we obtain two other rational expressions with quadratic numerator and denominator.
My question is: is there a direct way to turn a general conic to rational parametric equations of the second degree without resorting to centering and reduction of the conic ?
Update:
An easy solution is obtained by a simple change of variable such as $y=z+mx$. By substitution, the quadratic terms become
$$ax^2+2bxy+cy^2=ax^2+2bx(z+mx)+c(z+mx)^2$$ where the coefficient of $x^2$ is $a+2bm+cm^2$ and can be canceled (when there is a real solution). Then $x$ can be expressed as a rational expression in $z$, and so can $y$.
Easy, but ugly, and still requiring the roots of a quadratic equation :)