Riemannian metrics

Riemannian metric & unit mesh

A Riemanian metric is a symmetric positive definite (SPD) tensor field $\mathcal{M}(\mathbf{x})$, defined everywhere in the computational domain, such that the length of an edge $\mathbf{y}-\mathbf{x}$ is given by $${\ell }_{\mathcal{M}}(\mathbf{y}-\mathbf{x})=\Vert \mathbf{y}-\mathbf{x}{\Vert }_{\mathcal{M}}={\int }_0^1\sqrt{(\mathbf{y}-\mathbf{x}{)}^T\mathcal{M}((1-t)\mathbf{x}+t\mathbf{y})(\mathbf{y}-\mathbf{x})}dt$$

This field can then be used to define the size and orientations of mesh elements through the generation of a (quasi) unit mesh, i.e. a mesh for which all edges $\mathbf{e}$ have a unit length: $${\ell }_{\mathcal{M}}(\mathbf{e})\approx 1$$

Local sizes and directions

The metric is a SPD tensor: it can be characterized locally by its eigenvalues $(s_1,\cdots ,s_d)$ and an orthonormal basis of eigenvectors $({\mathbf{v}}_1,\cdots ,{\mathbf{v}}_d)$. The edges $${\mathbf{e}}_i=h_i{\mathbf{v}}_i$$ with a local size $h_i=1/\sqrt{s_i}$ are unit edges in metric space (if $\mathcal{M}$ is constant over ${\mathbf{e}}_i$ ).

In order to generate an isotropic mesh with local size $h(x)$, the metric should be $\mathcal{M}(x)=diag(1/h^2(x),\cdots ,1/h^2(x))$

Metric complexity

The physical volume of and equilateral element with unit edges in the (uniform) metric space (i.e. ideal elements) in $d$ dimensions is $$\frac{v_{\Delta }^{(d)}}{\sqrt{\det (\mathcal{M})}}\equiv v_{\Delta }^{(d)}\mathcal{V}(\mathcal{M})$$ where $\mathcal{V}(\mathcal{M})$ is the volume associated with metric $\mathcal{M}$ and $v_{\Delta }^{(d)}=\frac{\sqrt{d+1}}{d!\sqrt{2^d}}$.

The number of ideal elements per unit physical volume is thefore $1/\left(v_{\Delta }^{(d)}\mathcal{V}(\mathcal{M})\right)$ so the ideal elements to fill a domain $\Omega$ is $$\frac{1}{v_{\Delta }^{(d)}}{\int }_{\Omega }\sqrt{\det (\mathcal{M})}dx$$

The complexity is $$\mathcal{C}(\mathcal{M})={\int }_{\Omega }\sqrt{\det (\mathcal{M})}dx$$

Discrete metric field

In the context of remeshing, only a discrete metric field is known. A metric ${\mathcal{M}}_i$ is defined at each vertex ${\mathbf{x}}_i$ of the input mesh.

Metric interpolation

Logarithmic interpolation is used to interpolate between metrics: $$\mathcal{M}(t)=\exp \{(1-t)\log ({\mathcal{M}}_0)+t\log ({\mathcal{M}}_1)\}$$ as it verifies a maximum principle, i.e. if $\det ({\mathcal{M}}_0)\le \det ({\mathcal{M}}_1)$ then $\det ({\mathcal{M}}_0)\le \det (\mathcal{M}(t))\le \det ({\mathcal{M}}_1)$ for $0\le t\le 1$

When interpolating a discrete metric field at $\mathbf{x}=\sum {\alpha }_i{\mathbf{x}}_i$ is approximated as $$\mathcal{M}(\mathbf{x})=\exp \left(\sum {\alpha }_i\log ({\mathcal{M}}_i)\right)$$ which is consistent with the assumption of geometric progression of sizes.

Why not linear interpolation?

Linear interpolation between ${\mathcal{M}}_0$ and ${\mathcal{M}}_1$ is given by $$\mathcal{M}(t)=(1-t){\mathcal{M}}_0+t{\mathcal{M}}_1$$ and does not respect the maximum principle:

Edge length in metric space

In order to compute the length of an edge ${\mathbf{e}}_{i,j}={\mathbf{x}}_j-{\mathbf{x}}_i$, logarithmic interpolation is considered along the edge leading to: $$l_{\mathcal{M}}(e_{i,j})=l_i\frac{a-1}{a\ln (a)}$$ with $l_k=\sqrt{e_{i,j}^T{\mathcal{M}}_k{e}_{i,j}}$ and $a=l_j/l_i$.

Element quality in metric space

The quality of an element in metric space is given by $$q(K)={\beta }_d\frac{|K{|}_{\mathcal{M}}^{2/d}}{{\sum }_e||e|{|}_{\mathcal{M}}^2}$$ where the normalization factor ${\beta }_d$ is a such that $q=1$ of equilateral elements:

• ${\beta }_2=1/(6\sqrt{2})$
• ${\beta }_3=\sqrt{3}/4$

The same metric is used to compute all the edge lengths: among the metrics defined at all the vertices of element $K$, the one with minimum volume is used in the formula above

Invalid elements have a quality $q<0$