# $\mathrm{arctan}$ with two arguments

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

One usually sees expressions like $\mathrm{arctan}(y/x)$,
which is equal to $\mathrm{arctan}(x,y)$ when $(x,y)$ is in the first quadrant^{}.
However, $\mathrm{arctan}(y/x)$ does not give the correct angle when $(x,y)$
is in the third quadrant (since $y/x=(-y)/(-x)$).
Also, the quotient $y/x$ involves a division by zero when $x=0$,
which is damaging both numerically and mathematically.

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

In Python language^{} the functions atan(x) and atan2(x,y) are the respective one and two argument versions of $\mathrm{arctan}$. The point of having the two argument version is to determine the correct quadrant of the point. For instance, $1/1=1=-1/-1$, so atan(x) cannot distinguish between $(1,1)$ and $(-1,-1)$, but atan2(x,y) can, as the following Python code illustrates:

\PMlinkescapetext{ >>> from math import * >>> print atan(1) 0.785398163397 >>> print atan2(1,1) 0.785398163397 >>> print atan2(-1,-1) -2.3619449019 }

because $(1,1)$ has argument $\pi /4=0.7853\mathrm{\dots}$ but $(-1,-1)$ has argument $-3\pi /4=-2.3619\mathrm{\dots}$.

## Analytic properties

In mathematical works, $\mathrm{arctan}(x,y)$ is simply denoted by $\theta (x,y)$. The symbol $\theta $ obviously refers to the angle, but it is really the function ${h}_{2}$, where

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

The function $g:{\mathbb{R}}^{2}\to {\mathbb{R}}^{2}$ is the polar-to-Cartesian coordinate transformation^{}.
By the inverse function theorem^{}, the function $h$ (the Cartesian-to-polar coordinate transformation) exists and is smooth wherever it is defined.
Note that $h$ 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 $g$.
(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 $h$ (and $\theta $) can only be taken to be some simply connected open subset of ${\mathbb{R}}^{2}\setminus \{0\}$. (Note: ${\mathbb{R}}^{2}\setminus \{0\}$ itself is not simply connected.) For example, such a domain might be ${\mathbb{R}}^{2}\setminus \{(x,y):x\le 0\}$, i.e. delete the negative real axis from ${\mathbb{R}}^{2}$.

The exterior derivative of $\theta $ is

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

(found by implicit differentiation^{}),
and hence

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

(which can also be found by differentiating $\mathrm{arctan}(y/x)$ 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 $\gamma :[a,b]\to {\mathbb{R}}^{2}\setminus \{0\}$:

$${\int}_{\gamma}\mathit{d}\theta ={\int}_{a}^{b}{\gamma}^{*}\mathit{d}\theta ={\int}_{a}^{b}\left(\frac{-y\dot{x}}{{x}^{2}+{y}^{2}}+\frac{x\dot{y}}{{x}^{2}+{y}^{2}}\right)\mathit{d}t.$$ |

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

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

Title | $\mathrm{arctan}$ with two arguments |
---|---|

Canonical name | operatornamearcTanWithTwoArguments |

Date of creation | 2013-03-22 15:18:19 |

Last modified on | 2013-03-22 15:18:19 |

Owner | matte (1858) |

Last modified by | matte (1858) |

Numerical id | 11 |

Author | matte (1858) |

Entry type | Definition |

Classification | msc 51M04 |

Classification | msc 51-01 |

Synonym | angle function |