Why is the Euclidean metric the natural choice?

I'm trying to come up with a rationalisation for using the Euclidean distance in an application of mine. Any thoughts on why it is the fundamental choice?

Thanks

Solution 1:

Beginning with the one-dimensional case: On ${\mathbb R}$ we have translations $t\mapsto t+a$, $\, a$ fixed, and reflexions $t\mapsto -t$ as natural "geometric" isomorphisms. A translation and reflexion invariant metric then necessarily is of the form $d(x,y)=\phi\bigl(|x-y|\bigr)$ where $\phi$ should satisfy some "technical" conditions to make $d$ a metric which in addition is compatible with the inborn topological structure of ${\mathbb R}$. There are many such $\phi$; e.g. the definition $d(x,y):=\tanh\bigl(|x-y|\bigr)$ would turn ${\mathbb R}$ into a bona-fide metric space where all distances are $<1$.

But in ${\mathbb R}$ we have an additional set of "geometric" isomorphisms, namely scalings. If we want that our metric behaves in a reasonable way under scalings $T_\lambda: \ x\mapsto \lambda x$, $\,\lambda>0$ fixed, then the only $\phi$s left are the functions $\phi(u)= cu$, $\, c>0$ fixed, and we may as well choose $c=1$, so that we arrive at $d(x,y)=|x-y|$.

Now the two-dimensional case: Symmetry considerations like the above imply that we should choose a direction dependent $\phi:\ S^1\to {\mathbb R}_{>0}$ which is even and satisfies a certain convexity condition; then we should put $$d(x,y):=\phi\left({x-y\over |x-y|}\right)\ |x-y|\ .$$ This metric is translation invariant and behaves correctly under scalings.

But again, in ${\mathbb R}^2$ new sets of "geometric" isomorphisms are available, namely compact one-parameter groups of "rotations". If we want our $d$ to be invariant under such a group the only candidates left are of the form $$\|x\|^2:=\bigl(d(x,0)\bigr)^2= x'Qx\ ,$$ where $Q$ is a positive definite quadratic form of the coordinate variables $x_1$, $x_2$. Introducing a coordinate system adapted to the $Q$ at hand we then arrive at $\|x\|^2=x_1^2+x_2^2$, i.e., the euclidean distance function.

Solution 2:

The Euclidean Norm, $\|\cdot\|:\mathbb{R}^{n}\to\mathbb{R}$ is the most intuitive of the norms, it gives us the straight line distance (as we are used to thinking about it), from the origin to the position defined by the vector in question. The Euclidean norm is defined as:

$$\|\vec{v}\|=\sqrt{\sum_{i=1}^{n}{|v_{i}|^{2}}}$$

There are of course other norms, such as the Taxicab norm, $\|\cdot\|_{1}:\mathbb{R}^{n}\to\mathbb{R}$ defined as:

$$\|\vec{v}\|_{1}=\sum_{i=1}^{n}|v_{i}|,$$

Which as stated by wikipedia:

The name relates to the distance a taxi has to drive in a rectangular street grid to get from the origin to point $\vec{v}$.

Indeed we can define any norm $\|\cdot\|_{x}:\mathbb{R}^{n}\to\mathbb{R}$:

$$\|\vec{v}\|_{x}=\sqrt[x]{\sum_{i=1}^{n}{|v_{i}|^{x}}}$$

Which of these norms is best for your application depends on what exactly you are trying to measure. We need more information to give you a better idea of which of these to use for your application. For most applications (such as games, CAD packages etc. we are interested in the real-world distance, so we will use the Euclidean norm).