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I added another comment to my own answer, but it was the sixth, so it has been hidden by the display software; perhaps you didn't notice it, Casebash:

And the latitudinals (did I get that right?) are trivial, as Triathematician explained.

Paraphrasing Triathematician's answer: the nearest point from P = (θ, φ) to the line of latitude given by θ = θ0 θ0 is simply the point (θ0, θ0, φ).

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I added another comment to my own answer, but it was the sixth, so it has been hidden by the display software; perhaps you didn't notice it, Casebash:

And the latitudinals (did I get that right?) are trivial, as Triathematician explained.

Paraphrasing Triathematician's answer: the nearest point from P = (θ, φ) to the line of latitude given by θ = θ0 is simply the point (θ0, φ).