I have a bunch of curves that need to be projected to this cube and stay continuous, is it possible to do in Grasshopper?
What if you would scale your continuous curve so that it fits into the cube, then create a loft between the new and the old curve and intersect that surface with the cube?
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Perfect, thank you… I am a bit stupid today 
Really? Did you try that yet? I have some ideas but without a file from you, I won’t try to explain them (because I’m not sure how well they would work until I try them myself on your geometry).
Yes, that worked indeed
Did you scale the curve in two dimensions only? If also scaled by “length”, I would expect a different result than pull/projection methods. But maybe that doesn’t matter for your purposes.
If there were a way to project to the center point it would be distorted anyway, so it’s close enough to what I need
There are other ways.