The Universe of Discourse

Sun, 07 Jul 2019

Calculating π with atan2()

[ I wrote this in 2007 and forgot to publish it. Or maybe I was planning to finish it first. But if so I have no idea what I was originally planning to say, so here we are. ]

In computer programs, it's quite common to need a numerical value for π. Often you see something like:

        #define PI 3.141592654

This has the drawback of not representing π as exactly as possible. But to do that in C probably requires putting in 16 digits after the decimal point, and most people don't have so much memorized. And anyway, you don't really know at compile time what the floating-point precision will be; some platforms support quad-width floats. So you can do better, maybe, by using the math library to calculate π. And people do:

        static double pi = 4*atan2(1,1);

The atan2(yx) function produces the (almost-)unique value θ from the range !![-\pi, \pi]!! such that a ray from the origin, passing through point (x, y), makes angle θ with the x-axis.

Diagram showing
the atan2() function.  A ray from the origin through the point (1,1)
is labeled “atan2(1,1) = π/4”.  A ray through an arbitrary point (x,y)
has the angle θ with the positive x-axis.  And a ray pointing along
the negative x-axis is labeled “atan2(0,-1) = π”.

Note that the arguments have y first and x second. For example, atan2(17, 0) returns !!\frac\pi 2!!, because a line at angle !!\frac\pi 2!! passes through the point (0, 17). Similarly, atan2(-17, 0) returns -!!\frac\pi 2!!.

You can use atan2 to calculate π, by using !!4·{\operatorname{atan}}2(1,1)!!, as I mentioned above. Many people do; Google searching finds hundreds of examples. The manual for the standard Perl module mentions this example.

But this is a bit strange. Why is this so well-known? Why calculate 4*atan2(1,1) when $$\pi = {\operatorname{atan}}2(0,-1)$$ produces the same result and is simpler?

(Obligatory IEEE 754 complaining: atan2 should return an always-unique value from !!(-\pi, \pi]!!, but I have to say “almost-unique” because as usual IEEE 754 fucks everything up, this time with its stupid distinction between 0 and -0.)

[ Addendum: Leah Neukirchen suggests that the atan2(1,1) is a translation from earlier systems that provide a single-argument atan function but no atan2. In those systems, there is no workable analogue of atan2(0, -1) because the transformation !!{\operatorname{atan}}2(y, x)\Rightarrow {\operatorname{atan}}\left(\frac yx\right)!! gives !!{\operatorname{atan}}(0)!!, which doesn't work for this application as it yields !!0!! instead of the desired !!\pi!!. And similarly in languages with atan but not atan2 there is no analogue of !!\pi = 2·{\operatorname{atan}}2(1, 0)!!. So the simplest thing you can do is pi = 4 * atan(1), and after the transformation above one gets !!\pi = 4·{\operatorname{atan}}2(1,1)!!. ]

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