TITLE: Shortest irrational path
QUESTION [8 upvotes]: What is the shortest curve $\gamma$ in $\mathbb{R}^2$
  from the origin $o=(0,0)$ to a rational point $p=(a,b)$
  that (a) passes through no other rational point, and
  (b) contains no point a rational distance from both $o$ and $p$?

A rational point is one with rational coordinates.
I am wondering if one can reach $p$ via such a highly
"irrational route."
My guess is that there are curves whose lengths approach
the straight-line distance $|p|$.
Perhaps the curve should be restricted to a specific class:
$C^\infty$, analytic, elliptic, quadratic, circle arc.
(Just avoiding rational points [condition (a)] can be accomplished with a
circle arc of an appropriately chosen radius.)
This is far from my experience, and it may be that all
the variations have trivial, uninteresting answers, in
which case I apologize for the distraction.

REPLY [8 votes]: A circle passing throiugh the origin and your other point is determined by just one more point. Each rational point therefore determines a unique circle. Hence there are circles with large radius whose arcs ar arbitrarily close to the minimal distance containing no rational point, because there are uncountably many such.