Dimension of the metric of a projection of a sphere

In summary, The dimension of the metric of a projection of a sphere depends on the type of projection being used, with some having a two-dimensional metric and others having a three-dimensional metric. The dimension is determined by the number of variables needed to describe the projection, and it is used to measure distances and angles on the projected map. Common metrics include the Euclidean and Loxodromic metrics, and the dimension can affect the accuracy of a projection, with a higher-dimensional metric being more complex but potentially preserving more features of the sphere.
  • #1
jk22
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Let ##(x_1,x_2,x_3)=\vec{r}(\theta,\phi)## the parametrization of a usual sphere.

If we consider a projection in two dimension ##(a,b)=\vec{f}(x_1,x_2,x_3)##

Then I don't understand how to use the metric, since it is ##g_{ij}=\langle \frac{\partial\vec{f}}{\partial x_i}|\frac{\partial\vec{f}}{\partial x_j}\rangle## which is a 3x3 matrix but we have only two coordinates ##a,b## in the projection.
 
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  • #2
AFAIK, you use the pullback metric from the polar coordinates. This is the way I have seen it done.
 

What is the dimension of the metric of a projection of a sphere?

The dimension of the metric of a projection of a sphere depends on the type of projection being used. Some common projections, such as the Mercator projection, have a two-dimensional metric. Other projections, like the Robinson projection, have a three-dimensional metric.

How is the dimension of the metric of a projection of a sphere determined?

The dimension of the metric of a projection of a sphere is determined by the number of variables needed to describe the projection. For example, a two-dimensional metric would require two variables, while a three-dimensional metric would require three variables.

What is the purpose of the metric in a projection of a sphere?

The metric in a projection of a sphere is used to measure distances and angles on the projected map. It helps to preserve the shape and size of the features on the sphere as much as possible.

What are some common metrics used in projections of a sphere?

Some common metrics used in projections of a sphere include the Euclidean metric, which preserves angles but not distances, and the Loxodromic metric, which preserves distances but not angles. Other metrics, such as the conformal metric, aim to preserve both angles and distances.

How does the dimension of the metric affect the accuracy of a projection of a sphere?

The dimension of the metric can affect the accuracy of a projection of a sphere. In general, a higher-dimensional metric can preserve more features of the sphere, but it can also be more computationally complex. This means that a two-dimensional metric may be more suitable for simpler projections, while a three-dimensional metric may be necessary for more complex projections.

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