Equivalent metric for Euclidean space in 3dimension other than usual Euclidean metric

Question

I am from a physics background. I really want to know whether there is any equivalent metric for the Euclidean metric in $\mathbb{R^{3}}$. If there are two points in $\mathbb{R^{3}}$,$(x_{1},y_{1},z_{1})$ and$(x_{2},y_{2},z_{2}) $, The distance between the two points is,

$$s=\sqrt{(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}+(z_{1}-z_{2})^{2}}$$

Is there any other metric, which is equivalent to this euclidean metric (one to one and on-to),looks simpler than this as this involved square root? The reason why I am asking is that if I want to define distances for vertices of polygonal systems, this metric makes it much messier.So please suggest some other metric which looks simpler and yet equivalent.

Answer 1

You can take, for instance,$$d\bigl((x_1,x_2,x_3),(x_1,y_2,y_3)\bigr)=\lvert x_1-y_1\rvert+\lvert x_2-y_2\rvert+\lvert x_3-y_3\rvert$$or$$d\bigl((x_1,x_2,x_3),(x_1,y_2,y_3)\bigr)=\max\bigl\{\lvert x_1-y_1\rvert,\lvert x_2-y_2\rvert,\lvert x_3-y_3\rvert\bigr\}.$$