Join the discussion
Write your take first — we'll ask for email only when you're ready to publish.
- Hacker News
- Having read over this entire conversation I feel that people are almost uniformly missing the practical impact of this, which is simply that you can write your own math in terms of "turns" as much as you like. Nothing stops you right now. Defining a sinT function that takes turns is trivial. And so on for all the functions.
I haven't done much graphics programming, but what I did I did with tau rather than pi. You all seem to be arguing about whether or not you need to enter some parallel universe where all the math is completely rewritten or something, but you don't. It was easy. It didn't clash with the universe at all. I just used "tau" instead of "2 * pi". That's, like, it. That's all there is to it. "const TAU = 2 * PI;" and I was done with the conversion work. Similarly, all that is being suggested is that instead of "sin(2 * pi * (1/4))" you define something like sinT and write "sinT(1/4)".
Seriously. That's it. You don't need to rewrite every math library in the world. You don't even need your math library to support it at all, these are not complicated wrappers. You don't need to redo the whole of calculus to worry about taking derivatives of it or whatever. Besides, degrees already has all the same problems and we use them a lot too anyhow. Having trig function variants that take degrees isn't that uncommon, this is just another variant. You don't need to go in to your math library and remove everything that isn't based on turns. You don't need to force it in the face of the user of your code.
It's true that the benefits of this approach are modest but the costs are way, way less than a lot of the posts seem to be arguing about. The costs are a few local function definitions and a new possible unit for the programmer to have to think about. How expensive that is depends on the local programming language and whether or not you can press the type system into service to represent units in a sane way, and even that is not a problem created by this proposal because I'd want radians and degrees isolated in the same way already.
If you are programming in a language or environment that has no (practical) way to encode the units into the type system, and you had a program that settled on "every angle is radians" I don't think I'd introduce this into my code base. But if you have something where you can very easily integrate it with the existing code and get very solid guarantees that my new "turns" unit is compile-time guaranteed to never mix with "radians" or "degrees" I'd definitely consider it.
by jerf - The problem is that trigonometric functions are used in many more fields beyond geometry. The input is not always an angle around a point in euclidean space, it could be phase angle of a periodic signal. You can make an alternative set of trig functions that take turns, but you will anger a lot of people if you mess with the vanilla trig functions.by chabska
- I argued this idea to a couple of my classmates when I was a physics undergrad, and they agreed. However, I later changed opinions because of what this does to the derivatives/integrals of your trig functions.
For general periodic functions, [0, 1) is a good domain. But circles and spheres are geometric objects, and radians/steradians are geometrically significant units that are well suited for general purposes.
I do remember that Doom uses an interesting alternative representation where an angle is a u16 multiple of `(2 * pi) / 65536`. Fixed point is sometimes a good choice in games and simulations due to having uniform precision.
by mattmcal - Very bold title! Turns are very convenient until you need to calculate a rate of change, as of course d/dx sin(2pi x) = 2pi cos(2pi x). Unfortunately this is a common enough problem that I will be sticking with the radian.by traes
- Let's do a full circle.
It all began with replacing frequent occurrence of 2π in calls of sin and cos functions with τ.
This post suggests an optimisation by getting rid of τ by getting rid of radians. That way one can get rid of frequent and adjacent radians to degrees conversions and back.
I say, let's get rid of sin and cos itself !
Of course I am being over the top here. However, if you represent angle not as a scalar in degrees, radians or turns but as a tuple (sin, cos), one can usually dramatically reduce the number of calls to trigonometric functions. Rational polynomials and square root suffices. Recall rotation is a linear transformation with a matrix whose entries are in terms of sin and cosine.
As an API it might not be convenient but consider converting angles internally into a tuple of sin and cosine and keep it that way if your code frequently calls trigonometric functions.
(Aside: Sometimes I prefer keeping the tuple in terms of half angles. Tan half theta is nice to have. And I am embarrassed by the number of comments I have made on this post)
by srean - I like to store angles as turns in my own code, because (as noted) it makes quarter-turns computable without rounding. OTOH if you need, say, twelfths of a turn, you might want to just store angles as degrees since that’s already common.
Michael Spivak, in Calculus (3rd ed p. 301) considers the unit choice to be a property of the function and initially defines sin° and sinʳ (before settling on sin meaning sinʳ) and considers “sin x°” and “sin x radians” to be misleading, saying that ‘a number x is simply a number—it does not carry a banner indicating that it is “in degrees” or “in radians”’. I don’t really understand this argument, since in science and engineering we constantly carry units around with our quantities.
by mayoff - I think I cautiously agree with this notion to some extent, but IMHO the real answer is that it's application-dependent, and if you're writing a low-level trig library and you have to pick one or the other, it really isn't clear to me that turns should win over radians.
I expect many systems that use trigonometry would sometimes use small-angle approximations either for efficiency or to bootstrap to the general case. It'd be natural to use Taylor series here, i.e.:
If you've committed to representing all trigonometry in "turn" units, then you instead need to use:cos(x) = 1 - x^2/2 + ... sin(x) = x - x^3/6 + ...
In this case it would be less accurate and efficient to force everything into turns if you ever need to work with radians.cos(2 pi t) = 1 - (2 pi t)^2/2 + ... sin(2 pi t) = (2 pi t) - (2 pi t)^3/6 + ...Closely related to this, if you ever need the derivative of a function that does trig (e.g., in numerical optimization), you may as well use radians because if you don't, any extra factors you apply will appear in the expressions for the derivatives and you'll have to deal with them there anyway.
Basically for that reason, it's pretty clear that trigonometry in terms of radians is the "correct" convention mathematically speaking (away from computers), since derivatives of the radian-based trig functions are so easy to express. Given that, if we have to pick one convention...isn't it less confusing to use the same thing everywhere? That said, there are interfaces that provide both versions, and since as the article points out there are cases where the turn-based versions can be more efficient, that's probably the right way to go.
by WCSTombs - The math is definitely not fine with turns, because your Euler formula e^ix = cos x + i sin x no longer holds. We can use a base other than e, namely B = e^2pi which around 535.4916. This doesn't have the nice e properties like d/dx e^x = e^x.
The elegant fact that the base of the natural logarithm, which produces an exponential function that is its own derivative, also shows up as the basis for the above Euler's formula, shows that radians are special: like what binary is to computers.
The natural logarithm being its own derivative is in fact directly linked to the derivative a radians-based sin(x) being cos(x) and so on. Make it any other unit, and you have a mess of conversion factors worse than 2pi.
Imagine complex chained derivatives, double and triple derivative, chain and product rules, all stuffed with trig functions and generating gratuitous piles of cascaded conversion constrants because radians were not used.
by kazinator