Abstract
Using concepts from integral geometry, we propose a definition for a local coarse-grained curvature tensor that is well-defined on polygonal surfaces. This coarse-grained curvature tensor shows fast convergence to the curvature tensor of smooth surfaces, capturing with accuracy not only the principal curvatures but also the principal directions of curvature. Thanks to the additivity of the integrated curvature tensor, coarse-graining procedures can be implemented to compute it over arbitrary patches of polygons. When computed for a closed surface, the integrated curvature tensor is identical to a rank-2 Minkowski tensor. We also provide an algorithm to extend an existing C++ package, that can be used to compute efficiently local curvature tensors on triangulated surfaces.
Highlights
Using concepts from integral geometry, we propose a definition for a local coarsegrained curvature tensor that is well-defined on polygonal surfaces
The definition (12) of the triangle-based integrated curvature tensor has several appealing properties: (i) owing to the parallel body construction, it is well-defined for triangulated surfaces, (ii) it is additive, and can be used for coarse-graining using Eq (13), (iii) it displays robust convergence when a triangulation approximates a smooth surface, as we show
Parallel body construction is often used in the context of integral geometry, and is at the heart of the definition and evaluation of Minkowski functionals on polygonal manifolds [12–14, 22]
Summary
Biophysics and soft matter physics have provided many examples where the geometry of the system plays a major role to understand the structural and mechanical properties of materials such as colloids, liquid crystals, membranes, cells and tissues [1–6]. Minkowski functionals are defined locally on open or closed surfaces and can take scalar or tensorial values [12–14] They possess several properties that make them especially appealing: they are additive, continuous, motion covariant (invariant for the scalars) and span the space of scalar and tensor-valued valuations on convex shapes [23–27]. For the reasons stated above, the Minkowski scalars have proven extremely robust to evaluate the local curvature of surfaces [29, 30], while the rank-2 Minkowski tensors have been used to analyze anisotropy in a wide range of phenomena ranging from the shape of neuronal cells in the brain [31] to the shape of galaxies in the universe [32] Their robustness permits to use them on cellular and discretized structures, and a coarse-grained evaluation of Minkowski tensors can even be obtained for pixelated images and polygonal surfaces defined on grids [22]. We illustrate this procedure on a few surfaces and compute principal curvatures and directions of principal curvature of the coarse-grained curvature tensor for these surfaces
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