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I have some polygons I would like to map onto the face of a cone.

I can see from this page that I can convert the points of the polygon to cylindrical coordinates, which is almost what I want.

How do I go about modifying the formulas to work for conical coordinates?

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Presumably you already have the formulas for converting from conical to rectangular coordinates as listed on the Wikipedia page for conical coordinates. You'll need to solve for r, μ, and ν in terms of x, y, and z to get your answer. I can't see offhand the easiest way to find a general formula, but if you're trying to find it for particular values of r, μ, and ν, it shouldn't be too hard.