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Mesh parameterization is a key problem in digital geometry rocessing. By cutting a surface along a set of edges (a seam), one can map an arbitrary topology surface mesh to a single chart. Unfortunately, high distortion occurs when protrusions of the surface (such as fingers of a hand and horses’ legs) are flattened into a plane. This paper presents a novel skeleton-based algorithm for computing a seam on a triangulated surface. The seam produced is a full component Steiner tree in a graph constructed from the original mesh. By generating the seam so that all extremal vertices are leaves of the seam, we can obtain good parametrization with low distortion.

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Geometry processing algorithms often require the robust extraction of curvature information. We propose to achieve this with principal component analysis (PCA) of local neighborhoods, defined via spherical kernels centered on the given surface $\Phi$. Intersection of a kernel ball $B_r$ or its boundary sphere $S_r$ with the volume bounded by $\Phi$ leads to the so-called ball and sphere neighborhoods. Information obtained by PCA of these neighborhoods turns out to be more robust than PCA of the patch neighborhood $B_r \cap \Phi$ previously used. The relation of the quantities computed by PCA with the principal curvatures of $\Phi$ is revealed by an asymptotic analysis as the kernel radius $r$ tends to zero. This also allows us to define principal curvatures 'at scale $r$ in a way which is consistent with the classical setting. The advantages of the new approach are discussed in a comparison with results obtained by normal cycles and local fitting; whereas the former method somewhat lacks in robustness, the latter does not achieve a consistent behavior at features on coarse scales. As to applications, we address computing principal curves and feature extraction on multiple scales.

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