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$\operatorname{arc tan}$ with two arguments

(Definition)

When inverting the polar coordinates, one needs the arc tan function $\arctan$ with two arguments. If $(x,y)\in \mathbbmss{R}^2\setminus\{0\}$ , then $\arctan(x,y)$ is defined as the angle makes with the positive -axis.

One usually sees expressions like $\arctan(y/x)$ , which is equal to $\arctan(x,y)$ when is in the first quadrant. However, $\arctan(y/x)$ does not give the correct angle when is in the third quadrant (since ). Also, the quotient involves a division by zero when , which is damaging both numerically and mathematically.

In most mathematical software and programming languages the two-argument $\arctan$ is directly implemented.

In Python language the functions atan(x) and atan2(x,y) are the respective one and two argument versions of $\arctan$ . The point of having the two argument version is to determine the correct quadrant of the point. For instance, , so atan(x) cannot distinguish between and , but atan2(x,y) can, as the following Python code illustrates:

>>> from math import *
>>> print atan(1)
0.785398163397
>>> print atan2(1,1)
0.785398163397
>>> print atan2(-1,-1)
-2.3619449019

because

has argument $\pi/4=0.7853\ldots$ but

has argument $-3\pi/4=-2.3619\ldots$ .

Analytic properties

In mathematical works, $\arctan(x,y)$ is simply denoted by $\theta(x, y)$ . The symbol $\theta$ obviously refers to the angle, but it is really the function

, where

$\displaystyle g(r, \theta) = (r \cos \theta, r \sin \theta)\,, \quad h(x,y) = g^{-1}(x, y) = (r, \theta)\,.$

The function $g \colon \mathbbmss{R}^2 \to \mathbbmss{R}^2$ is the polar-to-Cartesian coordinate transformation. By the inverse function theorem, the function

(the Cartesian-to-polar coordinate transformation) exists and is smooth wherever it is defined. Note that

cannot be defined continuously everywhere, because of the multi-valued nature of $\theta$ -- $(r, \theta)$ and $(r, \theta + 2\pi n)$ always map to the same point under

. (Similarly, $\theta$ cannot defined when $r = \sqrt{x^2 + y^2} = 0$ .) This means, if one chases a loop (say a circle) around the origin, $\theta$ would move from 0 to $2\pi$ , even though the image point $g(r, \theta)$ winds back to the starting point.

Technically, a “largest” possible domain of (and $\theta$ ) can only be taken to be some simply connected open subset of $\mathbbmss{R}^2 \setminus \{0\}$ . (Note: $\mathbbmss{R}^2 \setminus \{0\}$ itself is not simply connected.) For example, such a domain might be $\mathbbmss{R}^2 \setminus \{ (x, y) : x \leq 0 \}$ , i.e. delete the negative real axis from $\mathbbmss{R}^2$ .

The exterior derivative of $\theta$ is

$\displaystyle d\theta = \frac{-y}{x^2 + y^2} \, dx + \frac{x}{x^2 + y^2} \, dy \,,$

(found by implicit differentiation), and hence

$\displaystyle \frac{\partial \theta}{\partial x} = \frac{-y}{x^2 + y^2}\,, \quad \frac{\partial \theta}{\partial y} = \frac{x}{x^2 + y^2}$

(which can also be found by differentiating $\arctan(y/x)$ directly and piecing the results for each quadrant).

Of course, the formulas above are only valid wherever $\theta$ is defined, but the analytical expressions do not change no matter which domain of definition is taken for $\theta$ . This allows for the following neat formula to find the total variation of angle of a smooth curve $\gamma \colon [a, b] \to \mathbbmss{R}^2 \setminus \{0 \}$ :

$\displaystyle \int_{\gamma} d\theta = \int_a^b \gamma^* d\theta = \int_a^b \left( \frac{-y \dot{x} }{x^2 + y^2} + \frac{x \dot{y}}{x^2 + y^2} \right) \, dt\,.$

(This is related to the formula for the winding number and the argument principle in complex analysis.)

For example, if $\gamma(t) = (r \cos t, r \sin t)$ , for $t \in [0, 2\pi n]$ , is the circle that winds around the origin times, then $\int_{\gamma} d\theta = 2\pi n$ .