Question

Traditionally, the earth's surface has been modeled as a sphere, but the World Geodetic System of 1984 (WGS-84) uses an ellipsoid as a more accurate model. It places the center of the earth at the origin and the north pole on the positive z-axis. The distance from the center to the poles is 6356.523 km and the distance to a point on the equator is 6378.137 km.

(a) Find an equation of the earth's surface as used by WGS-84.
(x^{2}/(6378.137)^{2})+(y^{2}/(6378.137)^{2})+(z^{2}/(6356.523)^{2})=1 (this is the correct answer)
(b) Curves of equal latitude are traces in the planes z = k. What is the shape of these curves?
These curves are circles given by the family of equations (in terms of x, y, and k) ?????
(c) Meridians (curves of equal longitude) are traces in planes of the form y=mx. What is the shape of these meridians?
These meridians are ellipses given by the family of equations (in terms of x, z, and m) ?????

Answer 1

Answer:

Step-by-step explanation: Hi!

a) We know that the most accurate model is the ellipsoid, that is :

$\frac{x^{2} }{a^{2} } +\frac{y^{2} }{b^{2} } +\frac{z^{2} }{c^{2} } =1$  where a, b, c are positive real numbers.

We can re-write the equation considering that the distance from origin to x-intercept is a, in our case a=6378.137. Using the same reasoning for b and c, we have :

$\frac{x^{2} }{6378.137^{2} } +\frac{y^{2} }{6378.137^{2} } +\frac{z^{2} }{6356.523^{2} } =1$ (*)

b) We know that curves of equal latitude are traces in the planes $z=k$. where $k$ is a constant. Then, we can replace z in the equation (*), that is:

$\frac{x^{2} }{6378.137^{2} } +\frac{y^{2} }{6378.137^{2} } +\frac{k^{2} }{6356.523^{2} } =1\\\frac{x^{2} }{6378.137^{2} } +\frac{y^{2} }{6378.137^{2} } =1-\frac{k^{2} }{6356.523^{2} }\\\\(x^{2}+y^{2} )\frac{1}{6378.137^{2} } =1-\frac{k^{2} }{6356.523^{2} }\\x^{2} +y^{2} =(1-\frac{k^{2} }{6356.523^{2} })6378.137^{2}\\=406806331.591-1.007k^{2}$

so ,

$x^{2} +y^{2} =40680631.591 - 1.007k^{2}$

And this expression corrresponds to equation of circule.

c) We know that the meridians (curves of equal longitude) are traces in the planes of the form $y=mx$. where m is  constant Then, we can replace y in the equation (*), that is:

$\frac{x^{2} }{6378.137^{2} } +\frac{(mx)^{2} }{6378.137^{2} } +\frac{z^{2} }{6356.523^{2} } =1$

Factoring

$\frac{(1+m^{2} )x^{2} }{6368.137^{2} } +\frac{z^{2} }{6356.523^{2}} =1$

Therefore, the expression corresponds to ellipses by the family equations in terms of x,z and m.