# Charts of a Torus

1. Dec 3, 2012

### mcafej

1. The problem statement, all variables and given/known data
Charts on the torus T.
Let S1 be the unit circle, and for each value 0 ≤ θ < 2π, let P(θ) be the point on the circle
at angle θ.
Let S1×S1 be the Cartesian product of two circles. The elements of S1×S1
are P(φ), P(θ), where 0 ≤ φ, θ < 2π.Let P(φ, θ) be the point on the torus at angles φ and θ, i.e., P(φ, θ) = ((R + r cos(φ)) cos(θ),(R + r cos(φ)) sin(θ), r sin(φ)).
Deﬁne a map δ : T → S1 × S1
by δ(P(φ, θ)) = (P(φ), P(θ)).
Let MφN and MφS be the charts on the ﬁrst S1
and let MθN and MθSbe the charts on the second S1.
Putting the ﬁrst charts together with the second charts, we get 4 charts on S1 × S1.

1) Use the 4 charts on S1XS1 to define 4 charts on T.

2) For each of the 4 charts on T, describe points on T that are not on the chart

2. Relavent information
MS from the circle minus the south pole S to the x-axis that take a point P on the circle to the intersection of the line from the south pole (0, −1) through P with the x-axis, and the second MN from the circle minus the north pole N to the x-axis that take a point P on the circle to the intersection of the line from the north pole (0, 1) through P with the x-axis.

3. The attempt at a solution

Ok, so I still don't understand how charts work. I missed the class when the professor talked about charts, and I've looked them up online, but I don't understand how to define charts and in particular, I don't understand how to define these charts. To be completely honest, I have no clue where to even start with this. My "charts" for M seem to have nothing at all to do with the taurus, and I have no clue how to relate them (let alone relating 2 sets of charts relating M to M and M x M to T). If somebody could maybe explain this to me or at lease give me a push in the right direction, it would be GREATLY appreciated. Also, if you could explain how to show a chart (I'm guessing he doesn't literally mean a chart of all possible values, but I don't know what charts look like)