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Desargues Mode

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
Δ T'U'V' are in different planes, as they are here: the lines connecting the vertices through C are just our three interval-lines. The Δ's are ‘shadows’ of each other with respect to ‘source’ C. Corresponding triangle lines (mutual ‘shadows’) must meet in points in the line of incidence of the triangles' planes.


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.