njdevscup,
I'm not Ozzy, but no, the runner does not have to come back to the direct line between 1B and 2B in advancing if he takes a "45* degree lead" off of 1B.
With a very few exceptions (i.e. the "running start on a retouch" and "positioning to interfere with the fielder's vision" mentioned earlier in this thread, when avoiding a tag, and running in a way that interferes with a protected fielder's opportunity to field) the runner may take virtually any path he likes in advancing on the bases.
He can't abandon his effort to run, enter dead ball territory, nor illegally run the bases in reverse order.
Other than that, it's up to him and it's legal.
JM
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