Post 1 (>>8266952)

I'm learning projective geometry in my spare time, and supposedly one of the best ways to learn something is to teach it to someone else, so here we are.

I will make several large posts in each thread, but with time to respond in between. 'Part 0' refers to the whole thread.

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In Prop I.4 of the Elements, Euclid uses 'the principle of superposition' to prove the SAS congruence theorem. The idea is to change a copy of the space(+objects) and then compare the two. While useful, we will only be using this to get an idea of projective geometry.

One thing we can do is see what changes preserve which properties of figures. For example, all changes to a plane that preserve Euclidean properties(length, angle, area, etc.) are composed of at most three reflections on a line.

Now consider the following 'change': Start with the xy-plane in 3d Euclidean space. Call this the object plane. True to its name, there are already figures on this plane. The change we wish to make to the copy goes in two steps:

1: A central projection to the xz-plane(image plane) with centre (0, 2, -1).

2: A parallel projection from the image plane back to the object plane with the axis as the line from (0, 1, 0) to (0, 2, -1).

The posted picture show how this change moves points around. It will also move any figures around, but what is important is to see which properties of the figures aren't changed.

Object plane before change: http://i.imgur.com/lFCanqy.png

After: http://i.imgur.com/OyCk1jO.png

Each picture is exactly the same scale. The thick black line is the X-axis, and the large blue dot is the point (0, 1, 0) in both pictures.

Which properties of figures are changed and which remain the same?



Post 2

Now let's list of some properties of figures and points and see if they are preserved. Any such property might make a good primitive notion for a new axiomatic system.

Lengths, angles, and areas: From the patterned square, clearly not.

Parallelism and perpendicularity: From the lines on the left and the square, also no.

Betweenness: Before the change, on the dotted line to the right, it is the pink point in between the green and black points. Afterwards, it is the black point that is in between the green and pink points. Thus, betweenness is not preserved.

Linearity: At first sight, it seems like lines remain lines under the change, so this seems to be a yes.

Incidence: Points on lines and curves seem to stay on the changed lines and curves, so this is also a yes.

Point-ness: Points seem to remain points under change, so point-ness seems to be preserved.

So far we have linearity, incidence and point-ness that seem to be preserved under this change. These make them good candidates for primitive notions in a system which 'respects' the change.

With what we have now, though, there is a problem with simply saying that points, lines, and incidences are preserved by the change with no exceptions. What is that problem?


Post 3

The problem just saying that lines stay as lines, points stay as points, and incidence is preserved, is with certain points where is fails. The red points in the original plane do not appear at all in the changed plane, and the white points in the changed plane do not appear in the original. Even worse, the errant points lie on the red and green dashed lines in the original and copy respectively, which are completely missing from the other plane in their own right. Call the red and green dashed lines the anti-horizon and horizon respectively.

Our resolution to this will make lean on of one property that we have so far neglected. Note that in addition to lines still looking like lines, curves are still curves. Neither of these has become a disconnected cloud of points after the change. This suggests that continuity is largely maintained, with notable exception the horizon and anti-horizon. This in turn suggests we look at points that are close to but not on the (anti-)horizon to see what is happening.

For the anti-horizon, we see that the small blue point is very close in the original plane. However, if we look at the picture of the changed plane we see no small blue point. Only when we zoom out(http://i.imgur.com/PKBIrEJ.png) can we find it. As for the horizon, note that the unmarked intersections of of the black dotted line with the lines on the right are close to the horizon. In the original plane, those intersections clearly end up well outside the frame of the picture.

From these observations, we have that:

-Points which are 'far away' in the original plane end up close to the horizon in the changed plane.

-Points which are close to the anti-horizon end up 'far way' in real life.

This suggest that to correspond to the (anti-)horizon itself, we need a line that is 'infinitely' far away in the appropriate plane. The next post will explore the implications of this ideal line.

Post 4

An interesting question about the ideal line is what points on it would be. If we look at the solid lines on the left, on the original plane their intersection lies on the anti-horizon. In the changed plane, those two lines are are parallel. Conversely, the blues on the left, which are parallel in the original plane, intersect at the horizon on the changed plane.

From this, we can reasonably conclude following:

-Parallel lines intersect on the ideal line.

Parallel lines in the Euclidean plane have the same slope, so in the Euclidean plane with an ideal line, each ideal point is incident to the lines with some particular slope. Since line slopes partition all the lines on the Euclidean plane, we can conclude:

-For any two distinct lines, there exists a point incident to both.

A real line can be uniquely specified by a real point on it and its slope. In addition, only the ideal line is incident to more than two points. Thus,

-For any two distinct points, there exists exactly one line incident to both.

This is enough for the beginnings of an axiomatic system.

Primitive terms: point, line, incidence(a relation between a line an a point).

Axiom 1: For any two distinct points, there exists at least one line incident to both.

Axiom 2: For any two distinct points, at most one distinct line is incident to both.

Axiom 3: For any two lines, there is at least one point which is incident to both.

For the next part/thread, we will refine and expand this into a suitable axiomatic system with a more rigorous approach.