Geometric Inidentity(page under development) 
Numerical intervals may be equal, or unequal. Since geometric intervals do not have size, they can neither be equal nor unequal. However, they may be identical, or inidentical, by way of elementary incidence. Geometric intervals that do not project each into the other are inidentical 

Degenerate and Nondegenerate ConicsA conic may be defined by five points in the plane.If any three of the five are colinear, the conic degenerates into a pair of lines. 
To see this degeneration in action,
drag any three of the five ‘points’ below into line. 

Projective  Projective 

The endpoints L_{0} and R_{0} of the blue interval i_{0} on line l are projected via some intermediate point C_{0} on the conic C by lines d_{0} and u_{0} respectively into directrices D_{0} and U_{0} respectively, which are also on the conic. The same procedure is followed for the red interval i_{1}, except that line u_{1} is chosen to project R_{1} into point U_{1} such that directrix U_{1} is coincident with directrix U_{0}. Line u_{1} also defines the intermediate point C_{1} on the conic C. Line d_{1} then projects endpoint L_{1} via intermediate point C_{1} into directrix D_{1} on the conic C. If directrices D_{0} and D_{1} do not coincide, the blue and red intervals are geometrically inidentical. Else, they are identical. 
The endpoints L_{0} and R_{0} of the blue interval i_{0} on line l are projected via some intermediate point C_{0} on the conic C by lines d_{0} and u_{0} respectively into directrices D_{0} and U_{0} respectively, which are also on the conic. The same procedure is followed for the red interval i_{1}, except that line u_{1} is chosen to project endpoint R_{1} into directrix U_{1} so that U_{1} is coincident with directrix U_{0}. Line u_{1} also defines the intermediate point C_{1} on the conic C. Line d_{1} then projects endpoint L_{1} via intermediate point C_{1} into directrix D_{1} on the conic C. If directrices D_{0} and D_{1} do not coincide, then the blue and red intervals are geometrically inidentical. Else, they are identical. 

You will see that the wording above is the same for both conics.


Recovery of IdentityHowever, in both the degenerate and nondegenerate cases,

