4
$\begingroup$

This is a pretty elementary concerning the terminology commonly used in Desargues' Theorem from plane geometry (or really, projective geometry).

At least in some representative cases, I totally buy the terminology of two triangles being "perspective from a point," at least in the sense that if I visualize the triangle furthest from the perspective point as the base of a tetrahedron (with the perspective point as a vertex), then the closer of the two triangles is a cross-section of that tetrahedron. Alternatively, I can make it mesh reasonably well in my head with the point of view of taking a triangle in an "object plane" and projecting it on to a "target plane" (or in the language of art, to see a triangle in the distance and to correctly paint a picture of it on my canvas.)

I have no similar picture for the notion of two triangles being "perspective from a line" used in the theorem. Can someone clue me in here? Is there any link to the "painting on a canvas" point of view, or is it just something like the formal dual notion to being perspective from a point?

Thanks for your help.

1 Answers 1

1

I've always assumed the "perspective from a line" usage arose simply as the dual of being "perspective from a point".

For the point, it makes sense, because from the point's perspective (i.e. from its point of view, from where it is located), the two things look the same.

But for a line, it makes no sense.

It is reasonable to say the way line A "looks" to line B is given by the point of intersection. But the way a line sees an object as a bunch of points doesn't seem to have anything to do with any traditional notion of perspective.

  • 0
    The metaphor of touch ... interesting! What if we tweak just a bit more? A point on a line identifies where one *touches*, say, a violin string to produce a given note. So, lines through the same point of a third line represent the same note on that third line. Maybe the proper sensory metaphor isn't sight or touch ... but hearing. :)2011-10-04