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

#### 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.

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
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.

• Siknature

#### Siknature

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.

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

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.

#### Siknature

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 ?