# Projective Plane vs. Reference Plane

I was told that the Projective Plane was also known as the Reference Plane in Projective geometry, but when I told my professor this, he freaked and told me I was completely wrong. He said that the Projective plane is the lines that go through the origin that intersect the Reference plane at a point. He said "a 'point' in the Projective plane is a line", and "even though they look like lines, they are called 'points'"...

This is word for word what he said, and he is the one grading my presentation. So I am going to believe what he says, but I still don't understand this idea and the fact the Projective plane is NOT the Reference plane in Projective geometry.

Thanks in advance!

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Dear Ellette,  Could you give some information on the course you're taking?  It might help people in formulating their answers.  Regards, – Matt E Jun 1 '11 at 1:33

## 1 Answer

I am not sure that this is what you are asking, but it seems that you are talking about two different models for the projective plane.

One can develop projective geometry from a completely synthetic viewpoint, which means that you prove theorems using basic axioms without ever referencing to what the points actually are.

Now, projective geometry can be seen in (at least) two ways. The first way is imagining the regular two dimensional euclidean plane, and adding a "point at infinity" for each set of parallel lines. This means that parallel lines will all meet at the same point at infinity (but a different point for each parallelism class). This seems to be what your teacher called the "Reference Plane" (although I have never heard that terminology, and google doesn't seem to give many results).

The second way is to call lines through the origin of $R^3$ points.

The link between the two models is easy to see, given the model of lines through the origin, you can map each line to the $R^2$ plane by taking its intersection with the plane $z=1$. The only lines that don't intersect the plane $z=1$ are those contained in the $xy-$plane, and they correspond to the points at infinity that we added in the previous model.

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Dear Vhailor, I also found it hard to find instructive uses of the term "Reference Plane" via google; the limited results I got suggest that it is more common in applied projective geometry courses (and I was surprised to find that such courses exist!).  This is one reason that I asked @Ellette to say a little more about the course she is taking.  Regards, – Matt E Jun 1 '11 at 13:14
Vhailor, thank you so much for your response.  What you said was very interesting and actually clears a lot of things up for me!  I have been researching Projective geometry for 4 days straight now and your answer, I have to say, made more sense than anything I have read about so far! hahah.  Thank you for such an extensive and clear response!  :)  and if I had a higher "reputation" I would "thumbs up" your answer.  Thank you! – Ellette Jun 2 '11 at 4:02
@Matt E.,  I am currently taking a math history course (Math 212), and I am required to teach my class about Projective and Perspective geometry for my final project.  It is actually a course required for my education major, but there are entire courses, I believe, that teach just Projective geometry!  :) – Ellette Jun 2 '11 at 4:05
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