1

Thus projective geometry deals with triangles, quadrangles, and so on, but not with right-angled triangles, paralleograms, and so on.
-Projective Geometry, Coxeter pg 3.

The first question of the section is:

Which of the following figures belongto projective geometry:

The book states that both choices (ii) an isosceles triangle and (iii) a triangle and its medians, are not in projective geometry.

A bit confused about why exactly. Is it beccause projective geometry has no unit of measurement?

yiyi
  • 7,352
  • 2
    Roughly speaking yes. More precisely, projective geometry is the study of properties invariant under projective transformations. A projective transformation need not preserve equality of distances. – André Nicolas Dec 09 '13 at 07:50
  • 2
    Put the triangles in a plane parallel to the floor and 1 meter above it. Put a lamp 2 meters on top of the middle of a side that is not the basis of the first triangle. Check the measurements of the shadow of the triangle on the floor. – OR. Dec 09 '13 at 07:54
  • Keep the lamp there put now the second triangle, and start slanting the plane. – OR. Dec 09 '13 at 07:57
  • @abc what does "Basis of the first triangle" mean? I found http://www.cfm.brown.edu/crunch/nektar/Thesis.Html/node29.html#SECTION00641100000000000000 but I don't understand it. – yiyi Dec 09 '13 at 08:30
  • @AndréNicolas know of any good introduction papers on this subject? – yiyi Dec 09 '13 at 08:31
  • 1
    No. The standard books are tend to be quite good. – André Nicolas Dec 09 '13 at 08:36
  • @AndréNicolas and ABC as well: both your comments sound more like answers, and I'd like to see them posted as such. While I prefer Anré and his precise formulation, upvotes on the comments show that others might prefer the intuition ABC communicates. I'd love to see both posted as answers, so that people can vote. I wouldn't have much to add to either. At least not while staying close to the original question, that is. – MvG Dec 09 '13 at 14:11
  • @MvG: Within the context of the question, I prefer ABC's comment, which says roughly the same thing that I did, but in a more striking, less pedantic way. Indeed it would be a good answer. – André Nicolas Dec 09 '13 at 17:14
  • @abc I really like your answer, and I believe that I understand what you posted, could you post this with a picture so I get a better idea, because I am confused by basis means in this context. – yiyi Dec 10 '13 at 00:32
  • Bad English. It should have been base, not basis. People call base of an isosceles triangle to the third side that is not necessarily equal to another side. I am too lazy now for pictures. The idea is this (http://upload.wikimedia.org/wikipedia/commons/4/4c/Perspective_projection_of_triangle_ABC_on_plane_%CE%A0_from_point_S.svg) Play moving with the source of light and planes. Take into account any triangle is the shadow of any other triangle. For the medians put the light so that the projection of one of the vertices goes to infinity. – OR. Dec 10 '13 at 02:43
  • @abc Such a clever way to describe the answer. – yiyi Dec 10 '13 at 06:34

0 Answers0