  We propose a simple, geometrically-motivated construction of smooth random
paths in the plane. The construction is such that, with probability one, the
paths have finite curvature everywhere (and the realizations are visually
pleasing when simulated on a computer). Our construction is Markov of order 2.
We show that a simpler construction which is Markov of order 1 fails to exhibit
the desired finite curvature property.
