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

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

Originally Posted by Siknature

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.

Originally Posted by Opalg

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?

Originally Posted by Siknature

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.

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 ?