Stereographic projection preserves angles but not distances. I've been lead to believe that it isn't possible to go from a sphere to a plane preserving distances between points, but is it possible to do so with a segment of a sphere?
For context, I'm planning on using locations in the United States as inputs, and so I only would need to project the area of a sphere corresponding to the continental United States and Alaska (I'd likely just eyeball the numbers for Hawaii given it's distance from the rest of the states). Is there a way to preserve distance in this transformation of a subset of the sphere?
It's not possible to do it even for a segment of a sphere. The reason is essentially the same: a segment of the sphere is intrinsically curved, just as the whole sphere is. If the segment is small relative to the whole sphere, the distortion will be relatively small; the approximation gets worse as the segment grows. This is why local maps give approximately correct distances. For a region as large as the continental United States, the distortion will be substantial no matter what you do.
For exact distances, the best you can do is the various equidistant projections. The azimuthal equidistant projection preserves the distances along any diameter of the map. In particular, this means that the distance from the center point $P$ to any other point is correct.
In the equidistant conic projection, distances are correct along two particular parallels.