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<li>We introduce BijectiveRemesh, a robust algorithm for maintaining a continuous, bijective mapping across complex remeshing sequences on both 2D triangle surfaces and 3D tetrahedral meshes.</li>
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<li>Unlike traditional data transfer methods that rely on interpolation or projection, our approach constructs a mathematically rigorous composite map by chaining local bijective atlases defined for each primitive operation.</li>
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<li>For 2D triangle meshes, we introduce a Shared Scaffold structure that enforces global bijectivity through local orientation preservation, extending successive self-parameterization to edge splits, edge swaps, and vertex smoothing.</li>
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<li>For 3D tetrahedral meshes, we generalize the local atlas construction using Steinitz's Theorem and Maxwell-Cremona lifting to ensure valid embeddings.</li>
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<li>This enables exact tracking of geometric entities—points, curves, and surfaces—across remeshing, with applications from texture transfer to volumetric simulations.</li>
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