Manifolds

I was asked to consider the set X of all straight lines in R^2, not necessarily through the origin.

a) I had to introduce a nature manifold structure into X, and write down all the charts explicitly, showing their domains and codomains.

For this I know,

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.

b) What is the dimension of the manifold X?

I know that its a 2-dimensional manifold, as the charts go to the space R^2.

But am struggling with the following question?

c) I need to define a natural surjection p:X -> RP^1. I need to write it explicitly using the local coordinates on X which i introducted in a) and the natural local coordinates on RP^1. And also need to show that p is a smooth.

Any help is appreciated.

Here's a hint. The space RP^1 can be identified with the set of all lines through the origin in R^2. So the natural surjection would correspond to the map that sends a line in R^2 to the line through the origin that has the same gradient. Now you have to describe that map explicitly in terms of what it does on charts.

Thanks for your hint, its been helpful.

I let U0 = {y=mx+c} and U1 = {nx+d}

I defined the natural surjection p: X -> RP^1 by send c, d terms to zero:

U0 = {y=mx+c} -> {y=mx} = [1:m]
and
U1 = {x=ny+d} -> {x=ny} = [n:1]

How can i show that p is a smooth map.

Any help is appreciated.