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Digital neuron reconstruction from 3D microscopy images is an essential technique for investigating brain connectomics and neuron morphology. Existing reconstruction frameworks use convolution-based segmentation networks to partition the neuron from noisy backgrounds before applying the tracing algorithm. The tracing results are sensitive to the raw image quality and segmentation accuracy. In this paper, we propose a novel framework for 3D neuron reconstruction. Our key idea is to use the geometric representation power of the point cloud to better explore the intrinsic structural information of neurons. Our proposed framework adopts one graph convolutional network to predict the neural skeleton points and another one to produce the connectivity of these points. We finally generate the target SWC file through the interpretation of the predicted point coordinates, radius, and connections. Evaluated on the Janelia-Fly dataset from the BigNeuron project, we show that our framework achieves competitive neuron reconstruction performance. Our geometry and topology learning of point clouds could further benefit 3D medical image analysis, such as cardiac surface reconstruction. Our code is available at https://github.com/RunkaiZhao/PointNeuron.