We propose a constructive, geometry-based method for approximating indicator functions of planar regions using two-layer sigmoidal neural networks. Unlike ReLU or tropical constructions that yield piecewise-linear boundaries, our approach produces smooth and interpretable decision regions within the standard multi-layer perceptron (MLP) format with sigmoidal activations. The networks approximate compact subsets and their unions, forming prescribed planar boundaries at initialization while remaining compatible with standard training. This framework offers a practical geometric realization of the Universal Approximation Theorem within the classical sigmoidal MLP format.

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