- #1
orion
- 93
- 2
I am confused about the procedure for finding the transition functions given an atlas. I understand the theory; it's applying it to real life examples where I have my problem. So for example, take S1 (the circle). I want to use 2 charts given by:
U1 = {α: 0 < α < 2π} φ1 = (cos α, sin α)
U2 = {β: -π < β < π} φ2 = (cos β, sin β)
Now I want to derive the transition function which is where I'm stuck. I know that α = arctan(y/x) and that β = arctan(y/x) which to me implies (and rightly so I think) that α = β on the overlap. My question is what is the transition function?
Another question I have is how are things improved using these 2 patches over 1 patch (which I know fails due to continuity of φ-1)?
Thanks in advance for any insight!
U1 = {α: 0 < α < 2π} φ1 = (cos α, sin α)
U2 = {β: -π < β < π} φ2 = (cos β, sin β)
Now I want to derive the transition function which is where I'm stuck. I know that α = arctan(y/x) and that β = arctan(y/x) which to me implies (and rightly so I think) that α = β on the overlap. My question is what is the transition function?
Another question I have is how are things improved using these 2 patches over 1 patch (which I know fails due to continuity of φ-1)?
Thanks in advance for any insight!