Minho Kim good of you to say, thank you

It took you 2 years to understand this....😅😂

geogebra

The notion of distance is not a concept of projective geometry. Projective geometry is an axiomatic system for studying points and lines. The axiom that every pair of lines meet at a point seems un-physical initially. However, one can obtain affine geometry from projective one (projective one is more fundamental). In a generic projective plane, single out one special line. Any pair of lines that meet that special line at the same point, we call "parallel". (there is one parallel "class" for each point on the special line). So, it starts out just as a labelling: You identify each parallel class with a point, and the set of all of them, with a line. But then magic happens, the new "points" and "line" satisfy exactly the same axioms and theorems as the rest of points and lines of the projective plane. So there is a kind of miracle of nature, that what we perceive as the infinity of a physical plane, behaves exactly like a line. And it is beautiful and powerful to treat it as such in mathematics.

@@clickaccept thats why in the 18th century, desargues’ theory was rejected, it was not realistic. “The noticeable return to the phenomena implicit in the method of physics and natural history during the eighteenth century, reinforced the status of Euclidean spatiality” Perez-Gomez (Architecture and the Crisis of Modern Science 1983)

1. Every line meets every other line at a point of the space. 2. If each line is incident with n points, then each point is incident with n lines, by the projective duality. 3. By (1) and (2), a given line is therefore incident with n*(n-1) other lines, which exhaust all lines of the space. 4. Add to this number, the original line, we have k=n^2-n+1 lines in total. 5. Total number of points coincides with total number of lines, again using the duality. This proof relies on the projective duality (and implicitly on transitivity of projective transformations).

You: ... the intersection of planes. Me: Ohhhhhhhhhhhhhh........

The lines at the corresponding sides of the triangles intersect, they are on intersecting planes, therefore the intesecting lines must be on a line (the intersection of the planes).

This seems to imply that the three projected rays along the corresponding vertices of any two triangles in three dimentional space will intersect at a point. Wonder how to prove it.

No, that ladder statement in incorrect! Ill still have to think around this a bit.

myName haha, well im glad it helps, maybe you might find this book cover picture enjoyable