Charts of a Torus

In summary, charts are used to define local coordinates on a surface or manifold, and in this problem, we use maps to define charts on the torus T.
  • #1
mcafej
17
0

Homework Statement


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.

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)
 
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  • #2
I would be very grateful.

I understand that charts can be a difficult concept to grasp, especially without having attended the class where they were discussed. Allow me to explain the concept and how it applies to the given problem.

Charts are essentially coordinate systems that are used to map points on a given surface or manifold. They are used to define local coordinates on a surface, which can then be used to describe and analyze the geometry of that surface. In this case, we are dealing with the torus T, which is a 3-dimensional surface. In order to fully understand the structure of T, we need to define charts that cover the entire surface.

Now, let's look at the given problem. We have four charts on S1XS1, which is the Cartesian product of two circles. These charts are MφN, MφS, MθN, and MθS. Each of these charts maps points on the torus T to points on the Cartesian product S1XS1. In other words, they provide a way to describe points on T using coordinates on S1XS1.

To define charts on T, we need to use the maps provided in the problem. The map δ takes a point on T, given by its coordinates (φ, θ), and maps it to a point on S1XS1, given by its coordinates (P(φ), P(θ)). This means that the charts on T can be defined as the inverse of this map, i.e., they map points on S1XS1 back to points on T. These charts can be denoted as Mδφ, Mδθ, MδφN, and MδθS.

To describe points on T that are not on the chart, we need to look at the ranges of the coordinates in each chart. For example, in the chart Mδφ, points with φ < 0 or φ > 2π are not on the chart. Similarly, in the chart Mδθ, points with θ < 0 or θ > 2π are not on the chart. You can use this logic to describe points that are not on the other two charts as well.

I hope this explanation helps you understand the concept of charts and how they apply to the given problem. If you have any further questions or need clarification, please do not hesitate to ask. I am more than happy to assist you in understanding this concept further
 

1. What is a torus chart?

A torus chart is a mathematical representation of a torus, a shape that resembles a donut or inner tube. It is a 3-dimensional shape with a hole in the center, and the chart shows its cross-sections and properties.

2. How is a torus chart created?

A torus chart is created using mathematical equations and computer software. The equations define the shape and dimensions of the torus, and the software generates the visual representation of the chart.

3. What are the applications of torus charts?

Torus charts have various applications in different fields such as mathematics, physics, and engineering. They are often used to study and visualize complex geometric concepts, as well as in computer graphics and animation.

4. How is a torus chart different from other 3D charts?

A torus chart is different from other 3D charts because it represents a specific shape, the torus, which is not common in everyday objects. Other 3D charts may represent more familiar shapes such as cubes or spheres, but a torus chart is unique in its shape and properties.

5. Can torus charts be used to model real-life objects?

Yes, torus charts can be used to model real-life objects that have a torus shape, such as inner tubes, bagels, and some types of pipes. They can also be used in engineering and architecture to design structures with toroidal shapes.

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