Affine to Projective Plane
- Thread starter tdu1993
- Start date
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,089
I haven't touched this in a long time, and your terminology is a little different from what I find in searching to refresh my memory. (Maybe that's why you didn't find resources -- though I found what I did starting with your phrase.) Language varies.
I think you are talking about this:
Finite geometry - Wikipedia
Your "3rd degree affine plane" would then be what they call an "affine plane of order 3"; and I would not say that the projective plane "has been completed in this plane", but that the affine plane has been completed to form a projective plane (that is, the projective plane is the projective completion of the affine plane, which is a subset of it):
Affine plane (incidence geometry) - Wikipedia
An affine plane can be obtained from any projective plane by removing a line and all the points on it, and conversely any affine plane can be used to construct a projective plane by adding a line at infinity, each of whose points is that point at infinity where an equivalence class of parallel lines meets.
To complete the affine plane, you need to add a line at infinity. What this means is that for each set of parallel lines in your plane, you need to create a new point where they intersect. That is, extend each of the set of lines to make them meet at a new "point at infinity". Then join all these points at infinity, to form a new line at infinity.
First of all, I'm sorry for the wrong pronunciation. Unfortunately my English is not very good. 
I want to complete such an example. (projective completion of the affine plane)
The points and lines of this plane must be clearly stated. I don't know how to do this.
I can't think of this 3D example. Is there a ready made visual for this? Or it could be a video for the method.
I really thank you for your help. Thank you so much.
Yeah. Like you said. I'm talking about this.I think you are talking about this:
Finite geometry - Wikipedia
en.wikipedia.org
Your "3rd degree affine plane" would then be what they call an "affine plane of order 3"; and I would not say that the projective plane "has been completed in this plane", but that the affine plane has been completed to form a projective plane (that is, the projective plane is the projective completion of the affine plane, which is a subset of it):
I want to complete such an example. (projective completion of the affine plane)
The points and lines of this plane must be clearly stated. I don't know how to do this.
It's easy to think of what you are saying here for a square.
I can't think of this 3D example. Is there a ready made visual for this? Or it could be a video for the method. I really thank you for your help.
Thank you so much.
Dr.Peterson
Elite Member
- Joined
- Nov 12, 2017
- Messages
- 16,089
Please make an attempt, so I can see if you are thinking wrong at some point.
Can you identify three lines in your picture that are parallel? Then extend them so that they meet in one point. Then we can talk about the next set, and the next ...
I'm not sure what you mean in calling it 3D. It's a "plane", though it doesn't embed in the Euclidean plane, in the sense that some lines are said not to cross, though they would have to when drawn on paper; that's why we distinguish those apparent crossings (by dots) where we consider it to actually do so, producing points. You don't have to think of the lines as actually bending outside of the physical plane in order to understand it.
I also don't know why you say, "The points and lines of this plane must be clearly stated. I don't know how to do this." Your picture shows 9 point and 12 lines; isn't it clear what they are? It would appear that when you said "I can draw ...", you really meant that you could find a picture of it, but perhaps don't fully understand it. So tell me what you do understand about it, so we can increase your understanding.
Can you identify three lines in your picture that are parallel? Then extend them so that they meet in one point. Then we can talk about the next set, and the next ...
I'm not sure what you mean in calling it 3D. It's a "plane", though it doesn't embed in the Euclidean plane, in the sense that some lines are said not to cross, though they would have to when drawn on paper; that's why we distinguish those apparent crossings (by dots) where we consider it to actually do so, producing points. You don't have to think of the lines as actually bending outside of the physical plane in order to understand it.
I also don't know why you say, "The points and lines of this plane must be clearly stated. I don't know how to do this." Your picture shows 9 point and 12 lines; isn't it clear what they are? It would appear that when you said "I can draw ...", you really meant that you could find a picture of it, but perhaps don't fully understand it. So tell me what you do understand about it, so we can increase your understanding.