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// SPDX-License-Identifier: CC-BY-SA-4.0
// Copyright © Paweł Redman
#import "/common.typ": post, static, img, anchor
#show: post.with(
title: "atan2",
date: datetime(year: 2018, month: 5, day: 13),
)
$
op("atan2"): RR^2 without (0, 0) -> (-pi, pi]
$
$
op("atan2")(y, x) = cases(
+pi / 2, (x = 0) and (y > 0),
-pi / 2, (x = 0) and (y < 0),
tan^(-1)(y / x), x > 0,
tan^(-1)(y / x) + pi, (x < 0) and (y >= 0),
tan^(-1)(y / x) - pi, (x < 0) and (y < 0),
)
$
where $tan^(-1) x$ is the principal value of the arctangent, ie. the one in the interval $(-pi, pi]$.
= Properties
#figure(
img(static("math/atan2-plot.png")),
caption: [A 3D plot of atan2(y, x).],
)
- $op("atan2")(-y, x) = -op("atan2")(y, x)$
- $op("atan2")(c y, c x) = op("atan2")(y, x)$ for any $c > 0$.
- Discontinuous along $(x < 0) and (y = 0)$: For any $x < 0$:
- $lim_(y -> 0^-) op("atan2")(y, x) = -pi$
- $lim_(y -> 0^+) op("atan2")(y, x) = pi$
== Partial derivatives
The partial derivatives, just like the function itself, are defined everywhere except at the origin. Furthermore, unlike the function, they're not discontinuous along $y = 0$ (except at the origin).
$
& (partial)/(partial y) = x / (x^2 + y^2) \
& (partial)/(partial x) = -y / (x^2 + y^2)
$
Interestingly, the function $tan^(-1)(y / x)$ has the same partial derivatives.
#anchor("systems")
= Systems of equations involving the cosine and the sine of an unknown <systems>
#figure(
img(static("math/polar-coords.png")),
caption: [A geometric interpretation: θ = atan2(y,x).],
)
Let $theta, x, y in RR$. Consider the following system of equations:
$
cases(
sin theta = y,
cos theta = x,
)
$
If $x^2 + y^2 = 1$, this system has a unique (up to a multiple of $2 pi$) solution:
$
theta = op("atan2")(y, x) + 2 pi k
$
for any $k in ZZ$. For simplicity k is often set to zero.
Because $op("atan2")(sin theta, cos theta) = theta$ (up to a multiple of $2 pi$), atan2 can be thought of as the "inverse" of the cosine and sine of the same argument.
= C's atan2
The definition at the beginning of this article is suitable for mathematical analysis. However, a more intricate definition is needed for floating-point computation, one that takes signed zeros and infinities into account. The following appears in the C standard:
$
op("atan2")(y, x) = cases(
op("NaN") & "if" space (y = op("NaN")) or (x = op("NaN")),
plus.minus pi & "if" space (x < +0) and (y = plus.minus 0),
plus.minus 0 & "if" space (x > +0) and (y = plus.minus 0),
-pi / 2 & "if" space (x = plus.minus 0) and (y < -0),
pi / 2 & "if" space (x = plus.minus 0) and (y > +0),
plus.minus pi & "if" space (x = -0) and (y = plus.minus 0),
plus.minus 0 & "if" space (x = +0) and (y = plus.minus 0),
pi & "if" space (x = -infinity) and (y > +0) and (y < +infinity),
-pi & "if" space (x = -infinity) and (y < -0) and (y > -infinity),
+0 & "if" space (x = +infinity) and (y > +0) and (y < +infinity),
-0 & "if" space (x = +infinity) and (y < -0) and (y > -infinity),
plus.minus pi / 2 & "if" space (x > -infinity) and (x < +infinity) and (y = plus.minus infinity),
plus.minus (3 pi) / 4 & "if" space (x = -infinity) and (y = plus.minus infinity),
plus.minus pi / 4 & "if" space (x = +infinity) and (y = plus.minus infinity),
tan^(-1)(y / x) & "if" space x > +0,
tan^(-1)(y / x) + pi & "if" space (x < -0) and (y >= 0),
tan^(-1)(y / x) - pi & "if" space (x < -0) and (y < -0),
)
$
In this definition $op("atan2")(0, 0)$ is *not* undefined.
= See also
- #link("https://linux.die.net/man/3/atan2")[atan2(3)]
- #link("https://en.wikipedia.org/wiki/Atan2")[Atan2]
- #link("https://en.wikipedia.org/wiki/Argument_(complex_analysis)")[Argument (complex analysis)]
- #link("https://en.wikipedia.org/wiki/Inverse_trigonometric_functions")[Inverse trigonometric functions]
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