Geodesic
A Note on the Construction of a Set of Fundamental Circuits on a Surface of Genus p
Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 13-21

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On an orientable surface of genus p, a set of 2p fundamental circuits can be selected which all pass through a single point A. After cutting along the 2p circuits, the surface can be unfolded into a flat region bounded by a 4p-gon so that: the set of vertices corresponds to the one point A; and the 2p pairs of edges to the 2p fundamental circuits; and the interior of the polygon to the remainder of the surface. If the edges of the polygon are directed, the 2 edges which correspond to one fundamental circuit will be directed in opposite sense, since the surface is orientable [1]. The sequence and direction of the edges is the same as the sequence of the fundamental circuits. 
Buda, R. G. de. A Note on the Construction of a Set of Fundamental Circuits on a Surface of Genus p. Canadian mathematical bulletin, Tome 4 (1961) no. 1, pp. 13-21. doi: 10.4153/CMB-1961-003-1
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