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R. Brahana. Systems of circuits on two-dimensional manifolds. Ann. Math. 23 (1922), 144–168. [3] M. Dehn and P. Heegard. Analysis situ. Enz. Math. Wiss. III A B 3, Leipzig (1907). [4] A. A. Markov. Insolubility of the problem of homeomorphy. In Proc. Int. Congr. , 1958, 14–21. 2 39 Searching a Triangulation Searching a Triangulation Many algorithms benefit from a convenient data structure that represents a surface by storing its triangulation. In this section, we describe such a data structure and show how to use it to determine the topological type of a surface.

We usually envision them put into three-dimensional space, sometimes with and preferably without selfintersections. Not all surfaces can be embedded in three-dimensional Euclidean space and self-intersections are unavoidable, but often they are accidental. Indeed, choosing a nice embedding of a surface in space is an interesting computational problem. We address this question for surfaces made out of triangles. 1 II Surfaces Two-dimensional Manifolds In our physical world, the use of the term surface usually implies a 3dimensional, solid shape of which this surface is the boundary.

To see this, we may again trace the closed curve, its image in R3 , and this time draw parallel curves to the left and the right on one of the two intersecting sheets. At the time we come back to where we started, the parallel curves have moved to the other sheet. There is either a clockwise or a counterclockwise rotation of the first sheet to the second that maps each curve locally to itself. If the rotation is clockwise, as seen by looking in the direction of the curve, then it is clockwise at all points of the curve.

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