Traditionally the earths surface has been modeled as a spher

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.


(b) Curves of equal latitude are traces in the planes

z = k.

What is the shape of these curves?

These curves are  ---Select--- circles ellipses parabolas hyperbolas 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  ---Select--- circles ellipses parabolas hyperbolas given by the family of equations (in terms of x, z, and m)

Solution

a)

For a general equation

x 2/ a 2 + y 2 /b 2 + z 2/ c 2 = 1,

the distance from the origin to x-intercept (y, z-intercepts respectively) is a (b, c respectively). From the description above, the equation of the earth’s surface is

x 2 /(6378.137)2 + y 2/ (6378.137)2 + z 2 /(6356.523)2 = 1.

b)

z = k x 2 (6378.137)2 + y 2 (6378.137)2 + k 2 (6356.523)2 = 1

x 2/ (6378.137)2 + y 2/ (6378.137)2 = 1 k 2/ (6356.523)2

x 2 + y 2 = 1 k 2 / (6356.523)2 (6378.137)2 ; 40680631.591 1.007k 2

It is a circle

c)

y = mk x 2 / (6378.137)2 + (mx) 2 / (6378.137)2 + z 2 / (6356.523)2 = 1

(1 + m2 )x 2 / (6378.137)2 + z 2 /(6356.523)2 = 1

It is an ellipse