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. Meridians (curves of equal longitude) are traces in planes of the form y=mx.

Required:
a. Find an equation of the earth's surface.
b. What is the shape of these meridians?

Answer 1

a) x²/(6378.137)² + y²/(6378.137)² + z²/(6356.523)² = 1

b) x²/(6378.137)²/(1+m²) + z²/(6356.523)² = 1

Explanation:

a)

We know that the general equation of the ellipsoid with center at the origin is

x²/a² + y²/b² + z²/c² = 1

Since the north pole is along the z-axis and north pole is at a distance of 6356.523 km from the center, so c = 6356.523

And the distance between center to a point on the equator is 6378.137 km so we have, a = b = 6378.137

Thus, the equation of the earth's surface as used by WGS-84 is

x²/(6378.137)² + y²/(6378.137)² + z²/(6356.523)² = 1

b)

Meridians are traces in planes of the form y=mx, so the equation of paraboloid gives;

x²/(6378.137)² + m²x²/(6378.137)² + z²/(6356.523)² = 1

x²/(6378.137)²/(1+m²) + z²/(6356.523)² = 1