  This paper reports about the development of two provably correct approximate
algorithms which calculate the Euclidean shortest path (ESP) within a given
cube-curve with arbitrary accuracy, defined by $\epsilon >0$, and in time
complexity $\kappa(\epsilon) \cdot {\cal O}(n)$, where $\kappa(\epsilon)$ is
the length difference between the path used for initialization and the
minimum-length path, divided by $\epsilon$. A run-time diagram also illustrates
this linear-time behavior of the implemented ESP algorithm.
