On Asymptotically Optimal Meshes by Coordinate Transformation

Abstract

Summary. We study the problem of constructing asymptotically optimal meshes with respect to the gradient error of a given input function. We provide simpler proofs of previously known results and show constructively that a closed-form solution exists for them. We show how the transformational method for obtaining meshes, as is, cannot produce asymptotically optimal meshes for general inputs. We also discuss possible variations of the problem definition that may allow for some forms of optimality to be proved.

Anisotropic quadrangulation

“…As discussed in [26,3], that error has a strong dependence on the shape of the element in the physical space (in our case, the shape of the approximating quad).…”

“…An attractive alternative is to define an embedding of the surface such that the Euclidean metric on the surface for this embedding yields an approximation to the desired metric [3]. For the shape operator, the relevant embedding is the Gauss map: f (p) = n(p) ∈ R 3 , because S = ∇n T , i.e.…”

Surface remeshing in arbitrary codimensions

“…We may wish to directly map the surface through its Gauss map -mapping every point to its normal in S 2 . We show in the appendix that, for surfaces, if a uniform tesselation of the Gauss map is used as range, then in the limit each intersection-region will have aspect ratio roughly proportional to the ratio of the surface's principle curvatures, which has been shown to be the optimal aspect ratio with respect to derivative error [9,8,7]. In order to avoid dealing with a codimension-0 mapping, and to ensure that the elements in the range space are approximately uniform (of approximate the same size and unit aspect ratio), we simply embed the Gauss map in R 3 and reconstruct the transformed surface in R 3 directly, rescaling the normals so they map to the unit cube and not the unit sphere.…”

Shape Operator Metric for Surface Normal Approximation

“…However, instead of being inferred from the vertex positions, the normals of the output mesh are optimally assigned to triangles. This distinction is necessary to avoid difficulties like those described in [5,7] that can occur when triangles have large internal angles, even if they have the right limit shape and size.…”

“…However, this correspondence is not available. To analyze approximation error, the very closely-related problem of approximation of the gradient of a height field over the plane is considered [it has optimal limit aspect ratio ξ 1 /ξ 2 , where ξ i are the eigenvalues of the heigh field Hessian [4,5]]. Because both VSA and SOM only look at normals and shape operators, it is possible to naturally adapt both to the gradient approximation case by measuring distances between gradients, as opposed to normals, by computing a Hessian of the height field at each point, instead of a shape operator.…”