The relationship between geocentric latitude ψ and geodetic latitude Φ is which of the following?

• A. tan ψ = (b^2 / a^2) tan Φ
• B. tan ψ = (b / a) tan Φ
• C. tan Φ = (b^2 / a^2) tan ψ
• D. tan Φ = (b / a) tan ψ

Answer: (A)

On an oblate ellipsoid like the Earth, geodetic latitude and geocentric latitude differ because the surface is flattened at the poles. Geodetic latitude Φ is the angle between the normal to the surface and the equatorial plane, while geocentric latitude ψ is the angle between the radius vector from the center and the equatorial plane.

In the meridional plane, the ellipsoid looks like an ellipse with semi-axes a (horizontal) and b (vertical). If you work with the gradient of the ellipse equation x^2/a^2 + z^2/b^2 = 1, the normal direction has components proportional to (x/a^2, z/b^2). The slope of this normal relative to the horizontal gives tan Φ = (a^2/b^2) tan ψ. Rearranging yields tan ψ = (b^2/a^2) tan Φ.

Since b < a for Earth, the factor b^2/a^2 is less than 1, so geocentric latitude is generally smaller in magnitude than geodetic latitude. The correct relationship is tan ψ = (b^2 / a^2) tan Φ.