Continent to Map to Globe conversions

[Rasstik]

Hello, CG Forum.
I've not found a thread describing my problem exactly, and so I've joined to ask for assistance.

I've been given the shapes of continents and islands to eventually transfer to a tangible globe; I do not, however, know how to go about shaping them to correctly fit to a sphere. Many of the members here have already expressed and dealt with ways to warp images to sit tidily on a digital sphere, but it seems that there has been no discussion on how to preserve a certain shape and proportion when transferring that shape to a rectangle or to an "sinusoidal interrupted" map which can then be applied to an approximate globe.
I am particularly interested in how to preserve a shape around the poles.

I will provide two or more examples of the continental shapes in an upcoming post, or edit this one should no one have responded by then.