When the two perspectivities are not co-planar, and the lines bearing our three equivalent intervals are all taken though a single point, as above, we obtain the famous Desargues configuration, which concerns a particular relationship between two triangles, here Δ TUV and Δ T'U'V'.
The truth of Desargues Theorem is trivially obvious if the triangles
Δ TUV and
Desargues: the deep base → Five Planes
However, the illustration has two triangles in the same plane - that of the screen: it is a set of ‘shadows’ (projections), from a point (‘centre’), of spatially distributed elements. This planar case is the degenerate case. That is, given a planar case such as this, we will always be able to (re)generate a spatial construction, and a centre, that together give rise to it. This is the basis of Perspective.