Polar coordinates in taxicab geometry

Question

We know that in euclidean $\mathbb{R}^2$ space polar coordinates are defined by $$r = \sqrt{x^2 + y^2}$$ $$\theta = \arctan\frac{y}{x}\text{.}$$ Now, geometrically we can think of it as of point, which is located somewhere in a distance $r$ from origin under angle $\theta$. Applying this logic to different metrics (taxicab for example) will definition of polar coordinates look like this $$r = |x| + |y|$$ $$\theta = \arctan\frac{y}{x}\text{?}$$

Answer 1

As far as I know, there is no commonplace generalization of polar coordinates to non-Euclidean metrics.

If I was to define one, I might use the fact that the angle is proportional to the curvilinear abscissa along the iso-$r$ curves (circles in the corresponding geometry).

In the case of the taxicab distance, these curves are tilted squares (diamonds) of diagonal $r$ (and side $r/\sqrt2$), and in the first quadrant the curvilinear abscissa from the $x$ axis is just $\sqrt2y$, ranging in $[0,\sqrt2r]$.

Hence, per quadrants, $\theta=$

$$\sqrt2\dfrac yr,\\ \sqrt2(1-\frac xr),\\ \sqrt2(2-\frac yr),\\ \sqrt2(3+\frac xr).$$

The "sine" and "cosine" functions would be triangular waves, with period $4\sqrt2$.

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These formulas were obtained by using the Euclidean metric to measure the curvilinear abscissa. It seems more coherent to use the taxicab metric, resulting in the factor $\sqrt 2$ being replaced by $2$.