Projective Geometry Part 0
6 more posts in this thread. [Missing image file: ]
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.
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
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?