Manifold structure on set of all lines in R2

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November 11th 2007, 08:11 AM
johnbarkwith

Manifold 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????
November 11th 2007, 12:50 PM
Opalg

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.
November 11th 2007, 01:13 PM
johnbarkwith

How would one explicitly write the charts from (a,b) to L ? thanks for the help
November 12th 2007, 12:01 AM
Opalg

Quote:

Originally Posted by johnbarkwith

How would one explicitly write the charts from (a,b) to L ? thanks for the help

Fix three non-collinear points $(x_i,y_i)$ (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 $a(x-x_i)+b(y-y_i)+1=0$ . 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 $(x_i,y_i)$ ).
November 12th 2007, 02:55 AM
johnbarkwith

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?
November 12th 2007, 03:06 AM
Opalg

Quote:

Originally Posted by johnbarkwith

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?

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.
November 12th 2007, 03:15 AM
johnbarkwith

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...????
November 12th 2007, 08:07 AM
Opalg

Quote:

Originally Posted by johnbarkwith

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...????

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 $(m,c)\to(n,d)=(m^{-1},-m^{-1}c)$ with domain $\{(m,c)\in\mathbb{R}^2:m\neq0\}$ (because the line $y=mx+c$ is the same as the line $x=m^{-1}y-m^{-1}c$ ).
November 12th 2007, 09:10 AM
johnbarkwith

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....