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

• Nov 8th 2009, 04:35 AM
Siknature
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
• Nov 8th 2009, 11:46 AM
Opalg
Quote:

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.
• Nov 8th 2009, 12:32 PM
Siknature
Quote:

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.

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

If so, how would this be done?
• Nov 8th 2009, 12:42 PM
Opalg
Quote:

Originally Posted by Siknature