Does anybody know how to go about introducing manifold structure to the set X of all lines in R2 (Real 2D space) and write the charts explicitly????

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- Nov 11th 2007, 08:11 AMjohnbarkwithManifold structure on set of all lines in R2
Does anybody know how to go about introducing manifold structure to the set X of all lines in R2 (Real 2D space) and write the charts explicitly????

- Nov 11th 2007, 12:50 PMOpalg
See the Wikipedia page on the real projective plane, in particular the section on homogeneous coordinates. The essence of the construction is that a line in R^2 is given by a linear equation ax+by+c=0 (but you can multiply a, b and c by any nonzero constant and the equation will still represent the same line). So we can take the set of all lines with c≠0 and represent each line L by an equation ax+by+1=0. This yields a chart in which L is mapped to the point (a,b) in R^2.

What about charts covering lines of the form ax+by=0? Simple: just shift the origin to a point not on the line and then use the same construction as above. - Nov 11th 2007, 01:13 PMjohnbarkwith
How would one explicitly write the charts from (a,b) to L ? thanks for the help

- Nov 12th 2007, 12:01 AMOpalg
Fix three non-collinear points $\displaystyle (x_i,y_i)$ (i=1,2,3) in R^2. For each i, define the map f_i from R^2 to X (the set of lines) by letting f_i(a,b) be the line $\displaystyle a(x-x_i)+b(y-y_i)+1=0$. Each of these three maps is a homeomorphism from R^2 to a subset of X, and the three maps together cover the whole of X. (The range of f_i is the set of all lines in X that do not pass through the point $\displaystyle (x_i,y_i)$).

- Nov 12th 2007, 02:55 AMjohnbarkwith
If I used the equation y=mx+c for the equation of a line, then map (m,c) in R2 to (y=mx+c) in R2, then this takes care of all lines that are not vertical. All these lines would be of the form x=d, so I could map d in R to (x=d) in R2. So there would be two charts, mapping to the set of all real lines. would this have the same overall outcome?

- Nov 12th 2007, 03:06 AMOpalg
The first chart is fine. The second one is not. For a start, it has the wrong dimension (you can't have a two-dimensional manifold suddenly becoming one-dimensional). Also, it only gives you vertical lines, and so it excludes the possibility of a vertical line being "close" to a non-vertical one.

You can get around this by taking the second chart to be the map taking (m,c) to the line x=my+c. That map takes care of all lines that are not horizontal. So the two charts together form an atlas for the manifold. - Nov 12th 2007, 03:15 AMjohnbarkwith
Thankyou very much. so to add manifold structure to the set and explicitly write the charts, I could say the set is frully described by the equations y=mx+c and x=ny+d and the charts writen explictly are f: (m,c) to (y=mx+c) and g: (n,d) to (x=ny+d) with both codomains and domains being R2...????

- Nov 12th 2007, 08:07 AMOpalg
Domain of f and of g is R^2.

*Codomain*in both cases is X (the set of lines). The*range*of f is the set of non-vertical lines in X, and the range of g is the set of non-horizontal lines in X.

If you want to complete the specification, then the transition map from f to g is the map $\displaystyle (m,c)\to(n,d)=(m^{-1},-m^{-1}c)$ with domain $\displaystyle \{(m,c)\in\mathbb{R}^2:m\neq0\}$ (because the line $\displaystyle y=mx+c$ is the same as the line $\displaystyle x=m^{-1}y-m^{-1}c$). - Nov 12th 2007, 09:10 AMjohnbarkwith
you are obviously extreamly good with advanced geometry... Do you know how to build on this and add manifold structure to the set RP2 (real projective 2d space) I am trying to figure out how to find an atlas for this manifold consisting of 3 charts....