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This thesis explores reconstructing explicit scene geometry via geometry particles carrying local Lagrangian patches. We formulate a signed distance field as a weighted sum of moving basis functions and describe an optimization framework to fit target shapes in both 2D and 3D. Experiments on canonical geometry meshes show that with a modest number of particles, our approach can capture coarse geometric structures while providing intuitive control and interpretable local geometry images in a storage-efficient representation. Although these preliminary results do not yet match state-of-the-art accuracy, they highlight the promise of a particle-based, differentiable explicit representation that is suitable to inspire further work in a vast array of workflow improvements from digital sculpting to generative modeling. We conclude by discussing avenues for further research on improving particle placement, blending strategies, and interactive editing capabilities.