Is this a valid parametrization of the torus?

In summary, the author provides a parameterization for the torus in the form of x(u,v) = ((a + r cos u)cos v, (a + r cos u)sin v, r sin u) with the conditions 0 < u < 2*pi and 0 < v < 2*pi. Although this parameterization does not include 0 or 2*pi, it still covers most of the torus except for two intersecting circles. This is not a problem as it is not possible to cover the entire torus with a single chart.
  • #1
hideelo
91
15
I am reading "Differential Geometry of Curves and Surfaces" by Manfredo do Carmo and on page 156 he gives the following parameterization of the torus

x(u,v) = ((a + r cos u )cos v, (a + r cos u)sin v, r sin u) 0 < u < 2*pi, 0 < v < 2*pi

Doesnt this leave out some of the torus,? I know that he needs to map from an open set in R^2 and he therefore want to map from the product of two open intervals in R, but by not including either 0 or 2*pi I don't see how this is onto the torus. Am I missing something here?
 
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  • #2
Yes, strictly speaking it covers all of the torus except two intersecting circles.
 
  • #3
But then it's not covering it, why is this not a problem?
 
  • #4
You can't cover the torus in a single chart.
 

1. What is a parametrization of a torus?

A parametrization of a torus is a mathematical representation of a torus using a set of parameters. It is a way of expressing the coordinates of points on the surface of a torus in terms of one or more parameters.

2. How do you determine if a parametrization is valid for a torus?

A parametrization is considered valid for a torus if it accurately represents all points on the surface of the torus and allows for a smooth and continuous mapping of the points.

3. What makes a parametrization of a torus invalid?

A parametrization may be considered invalid if it does not accurately represent all points on the surface of the torus, if it produces discontinuities or singularities, or if it results in a non-smooth mapping of points.

4. How do you know if a parametrization is the best choice for a torus?

The best parametrization for a torus is one that is simple, efficient, and accurately represents all points on the surface of the torus. It should also allow for easy calculations and visualizations of the torus.

5. Can a parametrization of a torus be used for other shapes?

While a parametrization is specifically designed for a torus, its concept can also be applied to other shapes and surfaces. However, the specific parameters and equations used may vary depending on the shape being represented.

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