These formulas give a efficient method to calculate the intersection point of two lines. If in the previous system x or y is the side unknown, the resulting coordinates are the same. An cartesian equation of a curve is the necessary and sufficient condition for the cartesian coordinates of a point in order that the point is on the curve. A homogeneous equation is such condition for the homogeneous coordinates of the point. Since the equation is a polynomial equation, it can be written as a homogeneous polynomial equation in x,y,z.
P is the ideal point.
We'll transform this equation to a homogeneous equation. The last equation is the homogeneous equation of the parabola.