Geometric Identity 
Geometric intervals that project each into the other are identical. 

Because a number is not a geometric element, and a geometric element is not a number,Numerical equality and geometric identity are distinct and disjoint states.By extension, inequality and inidentity are also distinct and disjoint states. We explore how we may establish whether or not


We will need the following facts:For a geometric interval to exist at all,



Projective Comparison of Intervals on Coplanar Lines 
“Perspectivity” and “Projectivity” 

A profoundly important special case, illustrated below, arises if point B_{R1} degenerates into point B_{S0}, on line s, so that intervals k_{R0} and k_{S0} “abut” – that is to say, the intervals neither separate nor overlap. In the instance above, the right end of the left interval would become ‘one’ with the left end of the right interval—which should remind us that, while elements do not have ends, intervals do. In this case, these ends are points, so these intervals are interpunctual intervals. There are also interlinear intervals, and interplanar intervals. This, together with iteration, can be used to construct a chain of identical intervals on a line, which might serve as a sort of projective gradation, or scale — but one must take great care not to construe from it that the intervals have size, least of all equal size. 
Following from the above, two points, say A_{0} and A_{1}, in a line, say u, form two intervals, say i_{0} and i_{1}, on that line (fig. 1).
Let A_{0} lie to the left of A_{1} from the observer's vantage. Then, interval i_{0}, ‘between’ A_{0} and A_{1}, lies to the right of A_{0}, and the second interval, i_{1}, ‘between’ A_{0} and A_{1}, lies to the left of A_{0}, continuing leftwards to the right of A_{1} (because lines have no ends). Similarly, let points B_{0} and B_{1}, on a second line, v, For the intervals on line u, to be geometrically identical That is, A_{0} on u must project into B_{0} on v, via line m_{0}, say, and vice versa, and A_{1} on u must project into B_{1} on v, via line m_{1}, say, and vice versa. This immediately requires that The foregoing is a perspectivity, which is a range of points centrally projected directly from one line to another, and clearly, from the above, its lines and points are necessarily coplanar. We see that whenever we have one perspectivity, we must have two. Here, we have one in point P, and another in point Q. They come in pairs. Next, consider two lines, r and s Now, let both R and S project points A_{0} & A_{1} in line r, into points B_{R0} & B_{R1}, and B_{S0 }& B_{S1}, in line s, which forms two pairs of intervals, k_{S0} & k_{S1}, and k_{R0} & k_{R1}, in line s. Since intervals k_{Rn} & k_{Sn} are “perspectivities” of a single common interval, i_{n}, they are each geometrically identical to that common interval, and are, therefore, identical to each other. A double projective comparison, such as this, of two intervals by indirection via a third, is a projectivity. 

“Lincoln” (Daniel DayLewis) on 

Projective Comparison of Skew (i.e., noncoplanar) Intervals 

Fig. 4 
If two lines, say m and n, are skew to each other, they have no condition of incidence, with the result that intervals on these lines cannot be compared by singlestage, central projection—but they can be compared by twostage, central projection to, and from, a third interval on an intermediate line, say r. That is to say, by a projectivity. This works because, while skew lines cannot be incident, the planes (here, α and β) in or on which they lie must be – in a line.And this line then serves as our
intermediate (here, r) for the two perspectivities carried by the two
incident planes. Note that line m on plane α meets line r in point G, distinct from point H, in which line n on plane β meets line r. This confirms that lines m and n are indeed skew to each other. Corresponding intervals i_{[0,1]} and k_{[0,1]} are thus geometrically identical, as they are in perspective with the same intervals, q_{[0,1]}, on line r. 

A Note on Absolute Size and Equality, with reference to Figure 4 If the intervals i_{[0,1]} and k_{[0,1]} (fig. 4) are to have absolute size and to be ‘equal’ to each other in the Euclidean sense, However,
since lines m and n
are by definition skew, only one of points G and H can so lie, We conclude that it is always possible to compare skew intervals projectively for geometric identity, but never possible to compare skew intervals projectively for numerical equality. The oftentwinned assumptions that 

We have considered identity of intervals made by pointpairs on lines. (type a, above) We would need also to consider identity of intervals made by linepairs on points, (type b) and to consider identity of intervals made by planepairs on lines. (type c)  
^{**} Perhaps surprisingly, a line in a plane has no ‘points in common’ with that plane (intuition might suggest that they have all the points of the line in common). But lines and planes are not made of points, so there are no points available here, from either the line or the plane, to be “in common”. 