The relationship between geocentric latitude ψ and reduced latitude β is which of the following?

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Multiple Choice

The relationship between geocentric latitude ψ and reduced latitude β is which of the following?

In an oblate Earth model, the reduced latitude β is defined via a parametric form that maps the ellipsoid to an auxiliary sphere of radius a. A convenient representation is x = a cos β cos λ, y = a cos β sin λ, z = b sin β, where a is the equatorial radius and b is the polar radius. Geocentric latitude ψ is the angle between the equatorial plane and the line from the Earth's center to the point, so tan ψ = z / sqrt(x^2 + y^2). From the parametric form, sqrt(x^2 + y^2) = a cos β and z = b sin β. Therefore tan ψ = (b sin β) / (a cos β) = (b / a) tan β. So the relationship is tan ψ = (b / a) tan β. The other forms don’t follow from this coordinate setup.

In an oblate Earth model, the reduced latitude β is defined via a parametric form that maps the ellipsoid to an auxiliary sphere of radius a. A convenient representation is

x = a cos β cos λ, y = a cos β sin λ, z = b sin β,

where a is the equatorial radius and b is the polar radius.

Geocentric latitude ψ is the angle between the equatorial plane and the line from the Earth's center to the point, so tan ψ = z / sqrt(x^2 + y^2). From the parametric form, sqrt(x^2 + y^2) = a cos β and z = b sin β. Therefore

tan ψ = (b sin β) / (a cos β) = (b / a) tan β.

So the relationship is tan ψ = (b / a) tan β. The other forms don’t follow from this coordinate setup.

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