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Geodesic Nets - Construction and Existence
Nguyen, Duc Toan
Nguyen, Duc Toan
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2025-05-19
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Geodesic nets are types of graphs in Riemannian manifolds where each edge is a geodesic segment. We present an algorithm for constructing approximate geodesic nets connecting any given number of points in the Euclidean plane. One important object used in the construction of geodesic nets is a balanced vertex, where the sum of unit tangent vectors along adjacent edges is zero. We prove the existence of a balanced vertex of a triangle (with three unbalanced vertices) on a general two-dimensional Riemannian surface when all angles measure less than 2?/3, if the length of the sides of the triangle is not too large. This property is a generalization for the existence of the Fermat point of a planar triangle.
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Mathematics