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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.

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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.

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    Duality is wonderful for math, but can be brutal to metaphors. Dualized "perspective" forces us to abandon sight-based intuition. "point of view" becomes "line of view", which is not the same as "line of sight", which dualizes to "point of sight" and gets at the gist of @Matt's last paragraph: point $P$ can "see" all points on lines that pass through it; dually, line $\ell$ can "see" all lines through points that lie on it. Under perspection, all points on a given line of sight look the same to $P$; all lines through a given point of sight look the same to $\ell$. Dualized vision is weird.2011-08-21
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    @DayLateDon I like your "line of view" notion, where a line sees other lines only via the points of intersection. Let's move the metaphor from sight to touch. Just as two points can look the same to a third point, by being in the same direction (w.r.t. the 3rd point), two lines can feel the same to a third line, by being in the same place (w.r.t. the 3rd line). So points can see, while lines can feel. Going back to the original post and link, we can see that the two triangles in Desargues' theorem look the same to the "center of perspectivity", and feel the same to the "axis of perspectivity".2011-10-02
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    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