parabola in homogeneous coordinates

Question

So if I have the parabola $Y = X^2$, how do I go about representing this homogeneously? I know I can parameterize it as $F(t) = (t, t^2)$, but then what?

The reason I ask is because I have a $3\times 3$ matrix that I want to apply to this parabola but before I can do that I need to get it in the proper format.

Answer 1

1. Reformulate this to $x^2 - y = 0$
2. Homogenize to $x^2 - yz = 0$
3. Write quadratic form as a matrix: $$A = \begin{pmatrix}1 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 0 & 0\end{pmatrix}$$ A point $p$ lies on this parabola if it satisfies $p^T\cdot A\cdot p=0$.
4. Symmetrize this matrix: $$B = A+A^T = \begin{pmatrix}2 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & -1 & 0\end{pmatrix}$$
5. If you want to apply a projective transformation $M$ to all points, i.e. $p\mapsto M\cdot p$, then you can transform the matrix of your conic section to $(M^{-1})^T\cdot B\cdot M^{-1}$.