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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 Ɛ 0, and in time complexity k(Ɛ). O(n), where k(Ɛ) is the length difference between the path used for initialization and the minimum-length path, divided by Ɛ. A run-time diagram also illustrates this linear-time behavior of the implemented ESP algorithm.