Interpretation of projective space

[A_B]

Hi,

I am am currently taking a second course in geometry, the first part of the course concerns projective geometry, and I feel I'm not getting the picture. I would like to know what the motivation for the development of projective geometry is. What picture you guys have in your head of projective space. Any of your own thoughts, or references to articles or books are much appreciated.

Thanks
Alex

 

[mathwonk]

well basically we are helpless with spaces that are not compact. so we want all spaces to be hausdorff and compact., so when we have a space like k^n that is not compact, we try to compactify it. then we can take limits and decide afterwards when the limit is in the original space.

so projective space P^n = k^n union P^(n-1), is the simplest and nicest compact space containing affine space k^n.
also it occurs naturally in many contexts. e.g. suppose we want to examine roots of polynomials of degree n. If we try to take limits we find that sometimes the lead coefficient goes to zero and we obtain a lower degree polynomial.

thus we want to consider polynomials of degree at most n. Then they they the same roots if they are scalar multiples of each other, and too many roots if the polynomial is zero. So we are led to consider non zero polynomials of degree at most n, and to equate those that are scalar multiples of each other. this is precisely projective space of dimension n, i.e. (k^(n+1) - {0})/k*.

I.e. the natural compactification of the set of roots of polynomials of degree n, is the projective space of dimension n.

 

[A_B]

So ,more like algebra, projective geometry tries to study structures that arise naturally in many problems, rather than to study an "interpretable geometry"? So really there's not much hope form me to try and visualize a projective space?

 

[quasar987]

There are many ways to visualize real projective space (RPⁿ).

1) You can see RPⁿ as the space of all the lines through the origin in Rⁿ⁺¹

2) You can see RPⁿ as the n-sphere in Rⁿ⁺¹ but with antipodal points identified. This is rather abstract since it means that a point in RPⁿ is like two points in Sⁿ. And when you move one of those points in Sⁿ, the other one follows along so as to remain antipodal to it.
The relation with this representation of RPⁿ and the first one is that any line in Rⁿ⁺¹ intersects the n-sphere in two antipodal point. And conversely, to any pair of antipodal point on the n-sphere, there is a unique line passing through them. This yields a bijection (and even a homeomorphism) between the space of all lines and the abstract space of antipodal points in the sphere.

3) You can see RPⁿ as the closed unit disk Dⁿ in Rⁿ but again, with antipodal points of its boundary (which is the (n-1)-sphere) identified. This model is related to the second one by taking the sphere of model #2 and removing the southern hemisphere, but keeping the equator (by doing so we don't lose any point since each point in the southern hemisphere simultaneously exists as its antipodal point in the northern hemisphere!). After this, just flatten down the northern hemisphere to a disk (like you would flatten a hat on a table with your fist!). Then you get a disk with the points in the boundary identified with their antipodal counterpart.

I like this third model best for visualization because, for instance, it allows you to view RP² as one of these old video games in which you control a spaceship and such that whenever you fly into the side (boundary) of the screen, you reappear at the opposite (antipodal) side of the screen.

Similarly, you can visualize RP³ as a 3-d space like the one we live in, but with the peculiarity that whenever you go walk in a straight line for a certain distance, you come back to where you started.

There is also a property that is common to all the even-dimensional projective spaces that if you are an n-dimensional being living in RPⁿ (where n is even) and you set out for a journey in a straight line, when you come back to your starting point, you will arrive as a mirror image of yourself! For instance, if you are a 2-dimensional being, you will come back and your left hand will be where your right hand was when you set out the journey. Redo the journey and you come back as your old self. This is an illustration of the phenomenon that RPⁿ is orientable if and only if n is odd.

 

[chiro]

Thanks for all your replies, this has helped me greatly as well :)