How to introduce a natural manifold structure into the set of all straight lines?

Oct 2009
39
0
I have a set X which is the set of all straight lines in R^2.

These lines are not necessarily through the origin.

I am unsure of how to introduce a natural manifold structure into this set X.

I know i need to write down charts showing the domains and codomains but i don't know what the charts are.

I think defining coordinate charts on X requires considering equations that specify straight lines but im not certain.

Thanks for any help
 

Opalg

MHF Hall of Honor
Aug 2007
4,039
2,789
Leeds, UK
I have a set X which is the set of all straight lines in R^2.

These lines are not necessarily through the origin.

I am unsure of how to introduce a natural manifold structure into this set X.
A line in R^2 has an equation of the form ax + by + c = 0, where a and b must not both be 0.

If a ≠ 0 then we can divide by a and write the equation as x + b'y + c' = 0. The map taking this line to the point (b',c') in R^2 is one chart.

If b ≠ 0 then we can divide by b and write the equation as a'x + y + c' = 0. The map taking this line to the point (a',c') in R^2 is also a chart.

Those two charts together specify the manifold structure.
 
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Oct 2009
39
0
A line in R^2 has an equation of the form ax + by + c = 0, where a and b must not both be 0.

If a ≠ 0 then we can divide by a and write the equation as x + b'y + c' = 0. The map taking this line to the point (b',c') in R^2 is one chart.

If b ≠ 0 then we can divide by b and write the equation as a'x + y + c' = 0. The map taking this line to the point (a',c') in R^2 is also a chart.

Those two charts together specify the manifold structure.
Thanks you for your help.

Is it possible to work out the dimension of the manifold from this information?

If so, how would this be done?
 

Opalg

MHF Hall of Honor
Aug 2007
4,039
2,789
Leeds, UK
Thanks you for your help.

Is it possible to work out the dimension of the manifold from this information?

If so, how would this be done?
Simple. It's a 2-dimensional manifold, because the charts go to the space R^2.
 
Oct 2009
39
0
Now that i know these charts, is there a way to define a natural surjection p: X-->RP^1 (where RP^n is the real projective space) ?

How would i write it explicitly using the local coordinates on X and the local coordinates on RP^1 which are inhomogeneous coordinates ?