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Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation. Each segment is equipped with a user defined uncertainty parameter in the normal direction. This yields a collection of thin probabilistic geometric primitives that retain the geometrz of the underlying curve while extending it beyond the idealized deterministic one dimensional formulation.\n  For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty. Combining the segment wise components with appr","title":"A Geometric Gaussian Mixture Representation of Plane Curves","url":"https://arxiv.org/abs/2606.06505","vendor":"arxiv_cs_ai"},"summary":"arXiv:2606.06505v1 Announce Type: cross \nAbstract: We introduce a user defined probabilistic polygonal representation for plane curves. Given a curve, we select vertices on the curve and connect consecutive vertices by line segments to obtain a polygonal approximation. Each segment is equipped with a user defined uncertainty parameter in the normal direction. This yields a collection of thin probabilistic geometric primitives that retain the geometrz of the underlying curve while extending it beyond the idealized deterministic one dimensional formulation.\n  For each segment, we define a Random Variable that is uniform distributed in the tangent direction of the segment and Gaussian distributed in the normal direction of the segment. By matching the first and the second central moments, this construction induces a Gaussian component whose mean lies at the segment midpoint and whose covariance encodes both tangential and normal uncertainty. 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