Search one 2D distribution for point cluster most similar to another 2D distribution

Given a hand drawn constellation (2d distribution of points) and a map of all stars, how would you find the actual star distribution most similar to the drawn distribution?

If it's helpful, suppose we can define some maximum allowable threshold of distortion (e.g. a maximum Kolmogorov-Smirnov distance) and we want to find one or more distributions of stars that match the hand-drawn distribution.

I keep getting hung up on the fact that the hand-drawn constellation has no notion of scale or orientation, so we may need to rotate and scale the hand drawn points to find the globally optimum (or sufficiently similar) distribution of stars...

How would others approach this problem? No idea is too crazy!