The relationship between geodetic latitude Φ and reduced latitude β is which of the following?

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

Answer: (B)

Latitudes on an ellipsoid can be related to a corresponding latitude on an auxiliary sphere of radius a, which makes many geodetic calculations simpler. The reduced (or parametric) latitude β is the angle on that sphere, while φ is the geodetic latitude on the ellipsoid. For a point on the ellipsoid you can write its coordinates as X = a cos φ cos λ, Y = a cos φ sin λ, Z = b sin φ. On the auxiliary sphere, with the same longitude λ, the coordinates are Xs = a cos β cos λ, Ys = a cos β sin λ, Zs = a sin β. Using the ratio of the vertical component to the horizontal distance, Z / sqrt(X^2 + Y^2) on the ellipsoid equals (b sin φ) / (a cos φ) = (b/a) tan φ, while on the sphere tan β equals Zs / sqrt(Xs^2 + Ys^2) = (a sin β) / (a cos β) = tan β. Equating these slopes for the same meridian gives tan β = (b / a) tan φ. Therefore the reduced latitude is related to the geodetic latitude by tan β = (b/a) tan φ.