I've been trying to understand projective space as follows:
Consider the plane at z=1 as the 2D affine plane, and for any curve in this affine plane, let the inclusion in projective space be the set of all lines formed by a point on the curve and the origin in R3. For example, take two paralell lines in the affine plane. Then the set of all lines through a point on each respective line and the origin forms two separate planes, which intersect in a line lying in the plane z=0. This line of intersection of the planes then corresponds to the "point at infinity" at which the paralell lines intersect.
Is this the way in which one should understand projective space? Would such an inclusion work for, say, a parabola in the affine plane to show that such a curve in projective space is closed?
Thanks for any advice on such visualization and intuition.