Conic Sections and the Projective Unity of Conics

Having defined conics in general, we might like to distinguish among those of different sizes or different eccentricities. Since distinctions among different sizes would require metric ideas (which we lack), we can't make those distinctions. Nor can we make the distinctions among eccentricities, not even so far as to distinguish positive, negative or zero eccentricity. In other words, we can't distinguish hyperbolas, ellipses, and parabolas; all conics are projectively equivalent.

This makes sense when we go back to the classical definition of a conic as a section of a cone by a plane. This construction

shows that all conics are projectively equivalent to circles; we use the vertex of the cone to project the conic back onto the base of the cone. So to showe the projective equivalence of two conics, we project them both onto the bases of their cones and then show that the two circular bases are projectively equivalent.

We may also understand the unity of conics in the projective context by comparing this with the affine context. In affine geometry, an ellipse is simply a conic which has no points in common with the line at infinity. A hyperbola is a conic which has two points in common with the line at infinity; these are the points in the directions of the two asymptotes. A parabola is a conic which has exactly one point in common with the line at infinity; it is the point in the direction of the axis of the parabola. We can develop these ideas even further by use of homogenous coordinates. However, in projective geometry we do not identify any one line as the line at infinity, and so we do not distinguish among the various affine types of conics.