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Single-view 3D object reconstruction is a challenging fundamental problem in computer vision, largely due to the morphological diversity of objects in the natural world. In particular, high curvature regions are not always captured effectively by methods trained using only set-based loss functions, resulting in reconstructions short-circuiting the surface or cutting corners. In particular, high curvature regions are not always captured effectively by methods trained using only set-based loss functions, resulting in reconstructions short-circuiting the surface or cutting corners. To address this issue, we propose learning an image-conditioned mapping function from a canonical sampling domain to a high dimensional space where the Euclidean distance is equal to the geodesic distance on the object. The first three dimensions of a mapped sample correspond to its 3D coordinates. The additional lifted components contain information about the underlying geodesic structure. Our results show that taking advantage of these learned lifted coordinates yields better performance for estimating surface normals and generating surfaces than using point cloud reconstructions alone. Further, we find that this learned geodesic embedding space provides useful information for applications such as unsupervised object decomposition.