Metric

Metric and Euclidean metric space

An Euclidean metric space $({\mathbb{R}}^d,\mathcal{M})$ is a finite-dimensional vector space where the dot product is defined by means of a symmetric definite positive matrix $\mathcal{M}$:

$$⟨\mathbf{u},\mathbf{v}{⟩}_{\mathcal{M}}=⟨\mathbf{u},\mathcal{M}\mathbf{v}⟩={\mathbf{u}}^{\top }\mathcal{M}\mathbf{v},\text{ for }(\mathbf{u},\mathbf{v})\in {\mathbb{R}}^d\times {\mathbb{R}}^d$$

The properties of $\mathcal{M}$ ensure that it defines a dot product. We will now refer to $\mathcal{M}$ as a metric tensor or metric.

Canonical Euclidean metric space $({\mathbb{R}}^3,{\mathcal{I}}_3)$

The most familiar example of Euclidean space is the one defined by an identity matrix. The 2D canonical one is defined by the 2x2 identity matrix ${\mathcal{I}}_2$ and the 3D by the 3x3 identity matrix ${\mathcal{I}}_3$.

Geometric definitions in Euclidean spaces

The dot product defined by $\mathcal{M}$ allows to define the distance notion in ${\mathbb{R}}^d$. In the following $d$ denotes the dimension and is equal either to 2 or 3. We can then define norm and distance definitions:

• $\forall \mathbf{u}\in {\mathbb{R}}^d,||\mathbf{u}|{|}_{\mathcal{M}}=\sqrt{⟨\mathbf{u},\mathcal{M}\mathbf{u}⟩}$
• $\forall (\mathbf{a},\mathbf{b})\in {\mathbb{R}}^d\times {\mathbb{R}}^d,d_{\mathcal{M}}(\mathbf{a},\mathbf{b})=||\mathbf{b}-\mathbf{a}|{|}_{\mathcal{M}}=||\mathbf{ab}|{|}_{\mathcal{M}}$

The length $l_{\mathcal{M}}$ of a segment $\mathbf{ab}=[\mathbf{a},\mathbf{b}]$ is then given by the distance between its extremities:

$$l_{\mathcal{M}}(\mathbf{ab})=d_{\mathcal{M}}(\mathbf{a},\mathbf{b})=||\mathbf{ab}|{|}_{\mathcal{M}}$$

Moreover, given a bounded subset $K$ of ${\mathbb{R}}^d$, the volume $|K{|}_{\mathcal{M}}$ is given by:

$$|K{|}_{\mathcal{M}}={\int }_K\sqrt{\det \mathcal{M}}d\mathbf{x}=\sqrt{\det \mathcal{M}}|K{|}_{{\mathcal{I}}_d}$$

where $|K{|}_{{\mathcal{I}}_d}$ is the Euclidean volume of $K$.

Finally, as metric tensor $\mathcal{M}$ is symmetric, it is diagonalizable. It thus admits the following spectral decomposition:

$$\mathcal{M}=\mathcal{R}\Lambda {\mathcal{R}}^{\top }$$

where $\mathcal{R}$ is an orthonormal matrix composed of the eigenvectors $({\mathbf{v}}_{\mathbf{i}}{)}_{i=1,d}$ of $\mathcal{M}$:

$$\mathcal{R}=({\mathbf{v}}_1\dots {\mathbf{v}}_{\mathbf{d}})$$

verifying $\mathcal{R}{\mathcal{R}}^{\top }={\mathcal{R}}^{\top }\mathcal{R}={\mathcal{I}}_d$. $\Lambda =\text{diag}({\lambda }_i)$ is a diagonal matrix composed of the eigenvalues of $\mathcal{M}$, denoted $({\lambda }_i{)}_{i=1,d}$, which are strictly positive.

Natural metric mapping

The metric $\mathcal{M}$ can be seen as a mapping from the unit ball ${\mathcal{B}}_{{\mathcal{I}}_d}$ of identity matrix ${\mathcal{I}}_d$ onto the unit ball ${\mathcal{B}}_{\mathcal{M}}$ of $\mathcal{M}$, where these unit balls are defined as:

$$\begin{array}{rl}{\mathcal{B}}_{{\mathcal{I}}_d}& =\left\{\mathbf{x}\in \mathcal{V}(\mathbf{a})\mid (\mathbf{x}-\mathbf{a}{)}^{\top }{\mathcal{I}}_d(\mathbf{x}-\mathbf{a})=1\right\}\\ {\mathcal{B}}_{\mathcal{M}}& =\{\mathbf{x}\in \mathcal{V}(\mathbf{a})\mid (\mathbf{x}-\mathbf{a}{)}^{\top }\mathcal{M}(\mathbf{x}-\mathbf{a})=1\}\end{array}$$

Where $\mathcal{V}(\mathbf{a})$ denotes the vicinity of point $\mathbf{a}$. A visual representation of these unit balls is given in Figure 1. Notice that ${\mathcal{B}}_{{\mathcal{I}}_d}$ defines the unit circle in 2D and the unit sphere in 3D.

Figure 1: Geometric interpretation of the unit ball ${\mathcal{B}}_{\mathcal{M}}$ of $\mathcal{M}$. ${\mathbf{v}}_i$ are the eigenvectors of $\mathcal{M}$ and ${\lambda }_i=h_i^{-2}$ are the eigenvalues of $\mathcal{M}$. (Loseille 2011)

${\mathcal{M}}^{-\frac{1}{2}}$ is defined by the spectral decomposition:

$${\mathcal{M}}^{-\frac{1}{2}}=\mathcal{R}{\Lambda }^{-\frac{1}{2}}{\mathcal{R}}^{\top }\text{where}{\Lambda }^{-\frac{1}{2}}=\text{diag}({\lambda }_i^{-\frac{1}{2}})$$

Where $\text{diag}$ denotes a 2D or 3D diagonal matrix. ${\Lambda }^{-\frac{1}{2}}$ defines the diagonal matrix of the sizes $h_i$ along directions as $h_i={\lambda }_i^{-\frac{1}{2}}$. This gives the relation between the metric $\mathcal{M}$ and the size it enforces.

The mapping defined above is written in the form of the following application:

$$\begin{array}{rl}{\mathcal{M}}^{-\frac{1}{2}}:{\mathbb{R}}^d& ⟼{\mathbb{R}}^d\\ \mathbf{x}& ⟶{\mathcal{M}}^{-\frac{1}{2}}\mathbf{x}\end{array}$$

Figure 2: Natural mapping associated with metric $\mathcal{M}$. (Loseille 2008)

for more details, see A. Loseille and F. Alauzet, Continuous mesh framework. Part I: well-posed continuous interpolation error, SIAM J. Numer. Anal., Vol. 49, Issue 1, pp. 38-60, 2011. PDF