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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.  Point C is called the “perspector”, and the line of incidence of the two planes, through P, S and D, is called the “perspectrix”.

 

Desargues:  the deep base → Five Planes

desargues

However, the illustrations have two triangles actually in the same plane - that of the screen: it is a set of ‘shadows’ (projections), from a point (‘centre’), of spatially distributed elements.  This coplanar case is the degenerate case: given a coplanar case such as this, we are always able to (re)generate a spatial, non-degenerate construction, and a centre, that together give rise to it.

In fact, a “normal” eye always does this quite involuntarily (consider that a one-eyed person has a sense of depth, without seeing in 3D).  As an artist, I do this in reverse, so to say: I imagine a 3D scene, and “flatten” it on to the drawing surface – as above.  Whether or not I do so consciously, I use Desargues.
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