User:Michael Retriever/Earth shape

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[edit] The Earth's shape

This is the Earth:



The Earth can be represented as a geometric shape within a Cartesian space. For what we can see in the picture, the Earth looks similar to a sphere. To define sphere-looking shapes in a Cartesian space, the Cartesian space must have three dimensions, i.e. it has to use three axes: x, y and z. Also, it must use at least one radius for each axis (three radii in total). This can be seen in the equation of a sphere, most denoted as


{x^2 \over r^2}+{y^2 \over r^2}+{z^2 \over r^2}=1


We will now draw the three axes over the picture of the Earth:



There is fairly precise data about the Earth's radii. This data is showed in the Wikipedia article for Earth:



We know two radii: the polar radius and the equatorial radius (and the mean between both). We will call the equatorial radius a, and the polar radius b. We'll have to use one more radius (or one radius twice) in order to be coherent with what we said before.


The polar radius is the one which goes from the center of the Earth to those points which fulfill the characteristic of being polar. There are two poles on Earth: one to the north, and one to the south. Both poles fulfill the characteristic of being polar. Points need a unidimensional Cartesian space to be represented, a set of one single coordinate (1). We attach radius b to the y axis, because that's the axis which marks both poles.



The equatorial radius is the one which goes from the center of the Earth to those points which fulfill the characteristic of being equatorial. All the points that cut the Earth in two (northern hemisphere and southern hemisphere) are equatorial. That is, in fact, because the equator is not a finite number of points, but rather a circumference. Circumferences need a a bidimensional Cartesian space to be represented, a set of two coordinates (1, 2). We attach radius a to the x and z axes, because those are the axes which mark the equator. Furthermore, you can check that the top view is indeed a picture of the circumference called equator.



We are left with the picture



and the equation


{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over a^2}=1


which fit the description of an oblate spheroid (ellipsoid).

[edit] Equivalent equations for the Earth's shape

The equation we've used for the Earth until now is


{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over a^2}=1Image:Isometric 2.png


where the angular coordinate 0º 0º would be equivalent to the Cartesian coordinate (x, y, z) = (0, 0, a)


The thing about Cartesian spaces is that they can use any letter for any axis, as long as all axes stay positive in the first quadrant. Thus, we would also be able to work with


{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over a^2}=1Image:Isometric 6.png


where the angular coordinate 0º 0º would be equivalent to the Cartesian coordinate (x, y, z) = (a, 0, 0)


or also with


{x^2 \over a^2}+{z^2 \over b^2}+{y^2 \over a^2}=1;\,\,{x^2 \over a^2}+{y^2 \over a^2}+{z^2 \over b^2}=1Image:Isometric 5.png


where the angular coordinate 0º 0º would be equivalent to the Cartesian coordinate (x, y, z) = (0, a, 0)


All this moving-around can get confusing, specially for people who aren't used to Cartesian spaces or Cartesian coordinates. That's the reason why it is most common amongst mathematicians to use always the array


{x^2 \over a^2}+{y^2 \over a^2}+{z^2 \over b^2}=1Image:Isometric 1.png


where the angular coordinate 0º 0º would be equivalent to the Cartesian coordinate (x, y, z) = (a, 0, 0)
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