8 ms·

Turns are Better than Radians (2022)

• mayoff 1mo ago
  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.
  • math-man 1mo ago
    It's because both radians and degrees are are a ratio of a length to another length and are thus dimensionless. No matter how you measure it, all angles are without a unit. It's most obvious with radians but it's also the case with degrees. Using radians, you are guaranteed to not introduce unusual extra terms to rescale angles, if you use any other scale of angle you will have to keep track of extra terms. That may be useful in whatever you're doing. I work in degrees quite often and I'm careful to keep track of the 2pi/360 terms that crop up all over the place. With grade measure you have to keep track of 2pi/400 terms and with turns you have to keep track of 2pi/1 terms that will repeatedly show up. Again, depending on what you're doing, this may or may not make sense to do. In general, mathematics works out easier when the scaling term is 2pi/2pi because then you have a lovely 1 scale factor you don't have to keep track of.
    • eru 1mo ago
      Agreed. Though sometimes it's useful to keep track of 'fake' units like for angles, to make something like dimensional analysis work for you. But that's more for analysis of your code / formulas than when you actually go and compute things.
    • dahart 1mo ago
      > all angles are without a unit. Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?
      • thyristan 1mo ago
        In a very awkward way: rad is m/m, which is 1...
        • ant6n 1mo ago
          Perhaps in the physics sense, but in computer science we do have the notion of types which does allow us to model the difference between an angle and other numerics.
        • aidenn0 1mo ago
          It is only equal to 1 by convention. If we instead considered the ratio of the diameter to the arc-length then rad would be 1/2.
        • simiones 1mo ago
          Dimensions and units are separate things, though. For example, 1 minute and 1 second are different units of the same time dimension. Similarly, 1 rad and 1 degree are both dimensionless, but they are both different units.
          • thyristan 1mo ago
            Theoretically, you can define systems of measurement where a lot of seemingly separate things fall on the same units and dimensions. There are "natural units" in physics where you take the fundamental nature of particles and relativity into account and make some convenient choices for some physical constants such as the speed of light c := 1. Then the speed becomes dimensionless, length and time have the same unit and dimension (a length of 1eV is a time period of 1eV), and almost all units are simply derived from a measurement of energy (electron volt, not as basic as people would like, but useful enough). https://en.wikipedia.org/wiki/Natural_units https://en.wikipedia.org/wiki/Natural_units
    • srean 1mo ago
      It is dimensionless by fiat and convention. It clearly has units such as degrees, grad and radians. Just like other quantities that have units, a specific measurement is expressed as a pure numerical multiple of an unit which may be radians, degrees etc. This is a known wrinkle in dimensional analysis and people have considered making angles a fundamental quantity such as mass, length and time but have not done so because of the disruption it would cause. More details here https://en.wikipedia.org/wiki/Radian#Dimensional_analysis https://en.wikipedia.org/wiki/Radian#Dimensional_analysis https://en.wikipedia.org/wiki/Angle#Dimensional_analysis https://en.wikipedia.org/wiki/Angle#Dimensional_analysis
      • lioeters 1mo ago
        That's my rabbit hole of the week. > The current state of affairs leads inevitably to ghostly appearances and disappearances of the radian in the dimensional analysis of physical equations. In "A spectral unit", Nature Physics (2020) - https://www.nature.com/articles/s41567-020-0997-3.pdf https://www.nature.com/articles/s41567-020-0997-3.pdf Giacomo Prando summarizes the troubled history of the radian, a unit with the odd property of appearing and disappearing seemingly at will in dimensional formulas It led me to reading about "dimensionless quantity". > There have been periodic proposals to "patch" the SI system to reduce confusion regarding physical dimensions. For example, a 2017 op-ed in Nature argued for formalizing the radian as a physical unit. SI units need reform to avoid confusion (2017) - https://doi.org/10.1038%2F548135b https://doi.org/10.1038%2F548135b > The idea was rebutted on the grounds that such a change would raise inconsistencies for both established dimensionless groups, like the Strouhal number, and for mathematically distinct entities that happen to have the same units, like torque (a vector product) versus energy (a scalar product). Don't tamper with SI-unit consistency (2017) - https://doi.org/10.1038%2F549160d https://doi.org/10.1038%2F549160d --- What's surprising is that these discussions are so recent, considering the wide implications on so much practical work in mechanics and engineering. Maybe people got so used to working around the question, that any proposed solution would be too disruptive - so it's better to keep things backward compatible rather than conceptually simple/clear and explicit. In another comment someone said, "types" (in programming and type theory, I'd guess) are a poor model of physics units. But I wonder about that, it seems "units" are something like types with associated quantities, like degrees with 360, or meters with the speed of light. This line of inquiry also led me to "dimensionless physical constants". https://en.wikipedia.org/wiki/Dimensionless_physical_constant https://en.wikipedia.org/wiki/Dimensionless_physical_constan... Certainly pi (and e) must be one of those fundamental constants regardless of any unit of measurement. A "turn", on the other hand - well, if we consider 1 as a dimensionless constant that means a "whole".. How Many Fundamental Constants Are There? (2011) John Baez https://math.ucr.edu/home/baez/constants.html https://math.ucr.edu/home/baez/constants.html
    • srean 1mo ago
      You might find the following interesting. It is about trigonometry as practiced by early Indian mathematicians. Rather than using an unit circle they used a circle of 3438 units. https://news.ycombinator.com/item?id=45129081 https://news.ycombinator.com/item?id=45129081 Now it is customary to standardized on the radius. Early Indian astronomers and mathematicians standardize on the arc length of a minute.
  • jameshart 1mo ago
    You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars. ‘But wait!’ You may cry: ‘the formula for a transverse wave varies with the sine of a distance!’ To which I would say no: it varies with the sine of a distance (the horizontal displacement), divided by another distance (the wavelength), divided by 2pi. The distances cancel out and leave a scalar. The sine is taken of that pure scalar; it results in a pure scalar; and then it’s multiplied by another distance (the amplitude) to give you a vertical displacement. Sine is a pure function. Something else to consider is that the way we combine units with scalars to create dimensional quantities is through multiplication - and it’s not like there’s a simple formula for what a sine of a product is - I can’t determine sin(ab) in terms of sines or other functions of a and b. So if, say, a ‘degree’ were some dimensional unit, sin(90°) would not be something I could calculate - despite knowing sin(90) I don’t know sin(°) - whatever that would mean - and even if I did it gets me no closer to figuring out sin(90°) Realizing that ° is just a mathematical constant equal to pi/180 solves a lot here.
    • cozzyd 1mo ago
      Well you can also square root etc.
    • hasley 1mo ago
      I basically agree, at least for standard functions like sin, cos, tan, exp etc. It is even possible to see mistakes in equations just by checking that all the units to standard functions cancel out making the arguments dimensionless. On the other hand I am still unhappy with calling the ratio of two quantities, that happen to have the same units, "dimensionless". This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.
      • cubefox 1mo ago
        Yeah, "dimensionless" would mean they have equal dimension, which would mean they are comparable, which isn't necessarily the case. E.g. both radians and degrees are called "dimensionless". Edit: Apparently "same dimension" doesn't imply "same unit".
  • Timwi 1mo ago
    I don't think of it as units (as the sibling comment pointed out, angles are dimensionless); I think of ° as a postfix unary operator that does the conversion. In other words, I read sin(x°) as a shorthand for sin(x*Pi/180).
• jp57 1mo ago
  Or you could use 1/360 of a turn.
  • math-man 1mo ago
    [dead]
  • groundzeros2015 1mo ago
    degrees were primarily chosen due to many integer divisors - likely for applications of time and seasons.
  • randusername 1mo ago
    I always liked gradians [0]. 400 gradians to a turn. This means 100 gradians to a right angle, so arbitrary small angles feel more like percentages of a right angle. [0]: https://en.wikipedia.org/wiki/Gradian https://en.wikipedia.org/wiki/Gradian
• chabska 1mo ago
  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.
  • sriku 1mo ago
    You'll have to bring in the 2π factor somewhere. Cant escape it. If sint is the sin function but with angle give in turns, then d/dx sint(x) = 2π cost(x). sin(x) ~ x for small x but sint(x) ~ 2πx for small x.
  • jameshart 1mo ago
    When dealing with waves you often are dealing with turns - or, as they’re called in that world, cycles. A cycle is a turn is tau is 2pi. The SI unit for frequency after all is Hertz - cycles per second - which should really be considered equal to 2pi s^-1, but for complicated reasons, often isn’t, and most formulae that involve frequency ignore the ‘cycle’ - or it’s also hiding inside the definition of something like the wavelength or the Planck constant where it cancels out. Meanwhile the SI unit for angular velocity is radians per second which is dimensionally equivalent to s^-1. That said a becquerel, which measures rate of discrete events, is also dimensionally s^-1. (Next time you are measuring traffic to your website consider using the appropriate SI unit for measuring requests per second: the Becquerel.) - so dimensional equivalence isn’t the same as equivalence. You wouldn’t add a rate to a frequency, same as you probably shouldn’t add a torque to an amount of energy.
    • thyristan 1mo ago
      > Next time you are measuring traffic to your website consider using the appropriate SI unit for measuring requests per second: the Becquerel. Great idea, I will definitely do this!
  • otikik 1mo ago
    Functions are free. Create new ones. Sin1 instead of Sin, Cos1 instead of Cos.
• slwvx 1mo ago
  Yes, the idea of a turn [1] is interesting. And maybe useful. I have a different question: What would it take for a compiler to remove (elide) the multiply by pi + divide by pi that the author uses as an example? I guess one would not have to go as far as a Lean proof that two bits of code produce the same result? [1] https://en.wikipedia.org/wiki/Turn_(angle) https://en.wikipedia.org/wiki/Turn_(angle)
  • eru 1mo ago
    Well, they don't produce the same result in floating point math, I'm afraid. So you'd need to teach your compiler about what your formulas mean and what context you are using them in. (Ie are you actually doing geometry, or is your AI coding agent just trying arbitrary activation functions for your neuronal net experiments and some of them happen to look like geometry?)
    • nomel 1mo ago
      It's a mistake to care about equality of floating point numbers [1]. You must usually consider the lower bits of the number as random. I assume you're saying something other than this though? [1] https://en.wikipedia.org/wiki/Machine_epsilon https://en.wikipedia.org/wiki/Machine_epsilon
      • ainch 1mo ago
        I think the point is that, from a compiler's perspective, it's not obvious how much you should be allowed to optimise code at the cost of changing the outcomes of floating points maths - do you allow 1e-10, or 1e-6, or 1e-4 level changes? Does your compiler have to run some test calcs to bound the scale of the change introduced by rewriting fp maths? Some compilers will let you opt in to rewriting floating point maths, but that's opt in so users understand that their numeric outputs might change between optimisation levels. For more, there's a good post on this kind of flag in Rust: https://pythonspeed.com/articles/faster-float-math-rust/ https://pythonspeed.com/articles/faster-float-math-rust/
      • eru 1mo ago
• zahrevsky 1mo ago
  > It turns out (pun intended!) Thanks, I was waiting for this pun the moment turns were introduced in the article.
• traes 1mo ago
  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.
  • deleted 1mo ago
    [deleted]
  • HWR_14 1mo ago
    I feel like that approximates how I learned math. In geometry or trig you can use degrees or turns or any other unit, but almost never radians because that's harder write. As soon as you learn calculus, you switch to radians and never go back.
  • dhosek 1mo ago
    It depends on your context, and is mentioned in the article. The advantage of turns comes from the fact that the implementation of sinᵣ etc. is internally doing a conversion to turns, so by using sin_turns directly, you avoid calculating π/π with every call. It’s not a call for someone doing calculus or solving differential equations to abandon radians, just for the particular case of getting a numerical value for sin, cos, etc. from a library, having direct access to a turns-based function would produce faster code and also avoid some of the rounding errors that come from that π/π not to mention the imprecision of any angle that isn’t 0.
  • itemize123 1mo ago
    but surely u differentiate by turns?
• traes 1mo ago
  The title should say (2022)
• groundzeros2015 1mo ago
  Fails to mention that radians relates angle to arc length.
  • HWR_14 1mo ago
    There are valid reasons to prefer radians, especially in calculus. The fact that it's related to arc length is something that never (directly) comes up.
    • groundzeros2015 1mo ago
      Every part of calculus with trig functions relies on this fact! The rate of motion along a circle is approximately linear at the same speed when described in radians. For example when you do a Taylor series expansion the cos/sin are well approximated by x.
      • HWR_14 1mo ago
        That's why I put "directly" in my original post. All the nice functions in calculus rely on that fact, but that fact itself is almost never used or useful by itself . If I were writing the article I would focus on the benefits for derivatives and integration and other things that are slipping my mind at the moment. I wouldn't waste time going down the rabbit hole of why. At least not for an article aimed at this type of audience.
        • groundzeros2015 1mo ago
          Your awareness of a key relationship does not make it irrelevant. It happens all the time in math. the article acts like radians are arbitrary without discussing this key property.
• WCSTombs 1mo ago
  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.: cos(x) = 1 - x^2/2 + ... sin(x) = x - x^3/6 + ... If you've committed to representing all trigonometry in "turn" units, then you instead need to use: cos(2 pi t) = 1 - (2 pi t)^2/2 + ... sin(2 pi t) = (2 pi t) - (2 pi t)^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. 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.
  • Analemma_ 1mo ago
    I don't have a super-wide gamut of experience here and numerical analysis isn't my specialty, but nearly all trig implementations I've looked into (in both software and hardware) make heavy use of lookup tables and other shortcuts. I've never seen a Taylor series used in a general implementation - not saying it doesn't exist anywhere, but in most cases that I'm familiar with you could support turns just as easily with a different lookup table.
    • jcranmer 1mo ago
      If you're being technical, it's usually not a Taylor series, it's a minimax series. (The difference is that Taylor series minimize error at a given value, whereas minimax is trying to minimize maximum error in a range). In most general math library implementations (e.g., the library in glibc, musl, etc.), the implementation of sin, as with most functions, is going to be a polynomial evaluation. See, e.g., https://github.com/kraj/musl/blob/kraj/master/src/math/__cos.c https://github.com/kraj/musl/blob/kraj/master/src/math/__cos... for the implementation in musl, or https://github.com/bminor/glibc/blob/master/sysdeps/ieee754/dbl-64/s_sin.c https://github.com/bminor/glibc/blob/master/sysdeps/ieee754/... for glibc's implementation. Of course, if you're not using a standard math library implementation, you're probably preferring speed over accuracy, and so you might use a lookup table and linear interpolation to get a very coarse approximation instead.
• ethanlipson 1mo ago
  I think the author is either being disingenuous or doesn’t understand the subject if they don’t honestly address the reason radians are used in the first place. I’m leaning towards the latter, because I can’t imagine someone having an ulterior motive for pushing for trig reform like this, lol. Radians really are the natural unit for trigonometry. With that said, I certainly agree that a lot of code would be simplified by using turns over radians, especially outside the context of numerical methods. I could see myself supporting the addition of sint(x) and cost(x) functions to the math standard library, where sint = “sine turns”. While not a strict rule, Chesterton’s fence is a good heuristic: before we change something, we should first attempt to understand why it is the way it is.
  • teo_zero 1mo ago
    You might have misunderstood TFA. No push for trig reform, just a consideration on what internal representation is optimal in code. Imagine it like someone suggesting (understandably) that you express memory sizes in hex: no push to make everybody stop using decimal numbers!
    • ethanlipson 1mo ago
      TFA states “the less tau and pi, the better” and calls the radian-oriented functions “legacy”. I understand that a title like “Turns are Better than Radians” is intentionally inflammatory to get clicks, but I’d expect a more calibrated take in the article body. Phrasing like the above indicates the author doesn’t know what they’re talking about, even though I do agree turn-oriented functions would be useful.
  • srean 1mo ago
    > Radians really are the natural unit for trigonometry. s/trigonometry/calculus
• oliculipolicula 1mo ago
  Maybe related Hamilton's theory of turns revisited https://arxiv.org/abs/0904.4787 https://arxiv.org/abs/0904.4787
• zarzavat 1mo ago
  > But math never decreed that sine and cosine have to take radian arguments! If you don't use radians you have to add to add conversion factors everywhere to do calculus. Radians are the natural unit for sin/cos just as E is the natural base of the logarithm and exponential functions.
• em3rgent0rdr 1mo ago
  And could use fixed-point decimal for more efficiency since can store as integers and use integer hardware for them. So for instance with 32-bits, the 16 most-sig bits store the number of turns and the 16 least-significant bits store the fraction of a turn. Then if you want to wrap angles that exceed 360 degrees back around the circle, you can simply Logical_AND with 0x0000FFFF. And while you are at it, you could just use fixed-point decimal for sine and cos, whereby the maximum of +1 or -1 map to the most positive and most negative integer value. These type of optimizations were common before FPUs were cheap and fast.
  • aldonius 1mo ago
    Binary fractions of a turn are also a nice intuition pump for two's complement in general. Let's keep it simple and use just 8 bits. 0° is 0x00, 180° is 0x80, and 255/256ths of 360° is 0xFF. And if we wanted to use signed integers, then 0x80 through 0xFF - the high-bit half of the range - now represent the negative quadrants just as they represent negative integers.
    • djmips 1mo ago
      and that's exactly what we did in the old days of 8 bit games. We called them BRADs but others had their own names.
• stephenlf 1mo ago
  I was hoping for some code examples but got none. Can anyone help?
• kens 1mo ago
  One weird unit for angles is the mil, defined as 6400 mils in a circle. This unit is very useful for artillery, since 1 meter displacement at a distance of 1 km is 1 mil [†]. Thus, you can see how much you missed by, divide by the distance, and easily determine how much you need to adjust your aim in mils. Another interesting thing about artillery is they traditionally do a binary search to get the distance correct, which they call "bracketing". Link: https://unitedtaskforce.net/training/sop/communication/artillery-control-bracketing https://unitedtaskforce.net/training/sop/communication/artil... [†] Note that this isn't exactly correct since it corresponds to pi = 3.2. A mil is almost the same as a milliradian, but 6400 mils in a circle is much more convenient than 6283.18... milliradians in a circle.
  • kqr 1mo ago
    It's also useful for sighting distances when the width or height of something is known. A knuckle on your outstretched arm is roughly 30 mils, so you cover the thing with your hand, count knuckles, multiply by 30, then divide the size by that number to get the distance. You can calibrate your knuckles by doing this is reverse. Put up a target 1 cm wide and back up until it's just covered by a knuckle. Measure how far you got and divide. It was when I thought about why this works I started really understanding radians.
    • kqr 1mo ago
      Oh, and I forgot and now it's too late to edit my comment. 6400 has a bunch of nice divisors too. A half-turn is 3200 mils, a quarter is 1600, a quarter of a quarter is 400, etc. A sixth of a turn is nearly 1000 mils. A tenth is obviously 640 mils.
• smallstepforman 1mo ago
  Are there any c/c++ libs / headers that use this (without converting to radians in the background). I like this idea.
  • teo_zero 1mo ago
    The C standard defines the functions sinpi(), cospi(), etc. that act on half-turns. If you have a modern compiler, all you have to do is to include math.h
• thrtythreeforty 1mo ago
  Here's another good reason to think in turns: it turns Euler's formula from this Eldritch Terror: e^(i*x) = cos(x) + i*sin(x) into something you can kinda understand by staring at the complex plane: -1^(2x) = cost(x) + i*sint(x) Credit to justinpombrio for this: https://news.ycombinator.com/item?id=32986869 https://news.ycombinator.com/item?id=32986869
  • judofyr 1mo ago
    I'm confused. How is this simpler? Is there something in (-1)^(2x) that can easily understood by staring at the complex plane? It seems mostly that you've gotten rid of "e", but one of the goals of Euler's formula IMO is to explain what "e^(i …)" means so I'm not sure how this variant is useful.
    • voidmain 1mo ago
      I'll defend i^4x since I like it better. (cost x, sint x) is a point on the unit circle x turns counterclockwise from (1,0). cost x + i sint x is a point in the complex plane x turns counterclockwise from 1. Now look at integer powers of i, a point in the complex plane 1/4 turn from 1: i^0 = 1 (0 turns from 1) i^1 = i (1/4 turn from 1) i^2 = -1 (2/4 turn from 1) i^3 = -i (3/4 turn from 1) and we define complex exponentiation such that, for all real x, i^x = cost (x/4) + i sint (x/4) (x/4 turn from 1)
  • lefra 1mo ago
    Now define exponentiation by a non-integer.
  • WCSTombs 1mo ago
    Sorry but this is pretty bogus. (-1)^x is only well defined when x is an integer. This is generally the case for r^x whenever r isn't a positive real number. For example, when x = 0.5, r has two distinct square roots. Sure, you can choose one of them arbitrarily and declare it to be the value of r^0.5 (and math libraries typically do this), but there's unfortunately no good way to make this arbitrary choice consistently for all values of r simultaneously.
  • 1mo ago
• mattmcal 1mo ago
  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.
• rajnathani 1mo ago
  Dumb question: For multiplying for smaller turns such as 1 arc-second (1,296,000 in 1 turn), that would floating point precision issues be a tiny slight issue (22619.4671 arc-seconds in 2pi radians), or is it just a coding convention change?
• nyc111 1mo ago
  Norman Wildberger has an alternative system for trigonomtry: Understanding uniform motion: are radians really necessary? | WildTrig https://youtu.be/CnQXRdgN_7I?si=EiYY99i6mBOIyczI https://youtu.be/CnQXRdgN_7I?si=EiYY99i6mBOIyczI Wild Trig: An introduction to Rational Trigonometry https://youtube.com/playlist?list=PLIljB45xT85CyF_7bKd6y36VArOy3p2oh&si=8-oPc61g_senEQD8 https://youtube.com/playlist?list=PLIljB45xT85CyF_7bKd6y36VA...
• fooker 1mo ago
  The floating point expressions needed to represent the math library functions with decent precision and performance becomes significantly more weird and complex with turns. Please stick to radians.
  • andrepd 1mo ago
    Posit arithmetic requires not only sin(x), but also sin(2πx), correctly rounded that is. I wish IEEE floats had that as well. https://posithub.org/docs/posit_standard-2.pdf https://posithub.org/docs/posit_standard-2.pdf
• __MatrixMan__ 1mo ago
  This seems to be mostly from the perspective of what makes the most sense to use at an API boundary. Rather than trying to agree on the best meaning the various integers or floats that we're passing around, maybe we should instead build a more complex angle type that doesn't force callers to conform. Like, I can pass minutes or seconds to functions that accept a time type and it just works because they're not being collapsed to numbers. Is there any reason we couldn't do that with angles too?
• Juliate 1mo ago
  > There are many implementations of sin, but no matter which one you look at… I’ve had a brief moment of hope, forgetting the point was about mathematics.
• zkmon 1mo ago
  I think it misses the whole point of Pi. Turns are for angles. Pi is not a measure of angle. It is a number that can be used to find the length of an arc. For example, it gives half-length of an arc, given an angle in Turns. So it deals with lengths, not strictly angles. Turns deal with angles only.
• djmips 1mo ago
  In the old days of making 8 bit video games we used BRADs of 0-255 - worked well and the wrap was easy.
  • burnt-resistor 1mo ago
    Oh yeah, in the era of ¼ circle trig tables (cos and maybe tan; inverse (arc) versions as needed) in ROM or Taylor/Maclaurin approximation (with fast integer division) when FPUs were rare. Such tables and tricks mostly fell by the wayside when the 80486DX, 68040, and N64 (VR4300) arrived and SIMD/MIMD systems followed. I miss strict, deterministic unsigned addition overflow. In many modern languages, all kinds of verbose hoops are required to get this behavior and there's a chance it will generate terrible machine code.
    • kbolino 1mo ago
      > I miss strict, deterministic unsigned addition overflow. In many modern languages, all kinds of verbose hoops are required to get this behavior and there's a chance it will generate terrible machine code. Most modern (post-2000) languages actually handle this perfectly fine. They have both signed and unsigned types, with exact bit widths and fully specified semantics, which generally match what the hardware is natively capable of, at least on modern processors. It's languages from the 1990s that make this complicated. There must have been something in the water that motivated language designers in that decade to "simplify" the number system. Maybe this was an overcorrection from even older languages, which generally weren't trying to be clever but were trying to be portable, at a time when a lot of the basics hadn't been nailed down yet, like 8-bit bytes and two's complement.
      • burnt-resistor 1mo ago
        Go read K&R 1ed, squirt.
        • deleted 1mo ago
          [deleted]
• ttoinou 1mo ago
  Even better : did you know (-1)^x draws the unit circle in the complex plane ? No need for complex exp and i*pi
  • srean 1mo ago
    That's because a^b = exp (b ln a) That's equivalent to saying, no need for -1 because we have exp. One can change based of the exponentiation operation. Exp happens to be a convenient base.
    • ttoinou 1mo ago
      Yeah bad formulation on my part
      • srean 1mo ago
        Not bad at all, just equivalent.
        • ttoinou 1mo ago
          We do need complex exp to define my formula, you’re right
  • fph 1mo ago
    If you plot it over which domain?
    • ttoinou 1mo ago
      Complex domain
  • WCSTombs 1mo ago
    You do in fact need the complex exponential to define this correctly because the function a^x for nonintegers x is only unambiguously defined when a is a positive real number. For example, your function could be either e^(pi i x) or e^(-pi i x), which trace the circle in opposite directions as x varies over the reals. (They happen to agree when x is an integer.)
• burnt-resistor 1mo ago
  At least we got metric units out of the French Revolution. Gradians exist because "let's change everything, even things that aren't broken".
• kazinator 1mo ago
  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.
  • ogogmad 1mo ago
    In another comment, I asked why people chose to use the symbol τ over just writing turn or "rev(olution)" (defined to be the constant ≈ 6.28318530718) given how unambiguous the latter is as a name for 2π. And why not just write sinrev() or sinturn(), and leave the symbols sin() and rev (defined to be ≈ 6.28318530718) alone?
    • simiones 1mo ago
      The naming is irrelevant here. The point is that sin(x) ~ x for small x, whereas sinrev(x) ~ rev * x for small x, which is much uglier. And similar things happen to the derivative of sinrev() vs regular sin() and so on. So switching to preferring to express angles in revs instead actually complicates most formulas, at least in some domans.
    • robertlagrant 1mo ago
      I think it's Tau[0]. [0] https://en.wikipedia.org/wiki/Tau_(mathematics) https://en.wikipedia.org/wiki/Tau_(mathematics)
      • ogogmad 1mo ago
• fragmede 1mo ago
  It's similar to why taxicab distance is better for distance measurement on limited hardware where sqrt() costs precious cycles. The reason to use sin() though is because it's a lookup table (where it counts) and not a bit of math, so moving to turns isn't necessarily a win.
• moffkalast 1mo ago
  I'm not super versed on the subject, but I think there's a case where using radians allows you to do direct multiplication without any conversion when trig isn't even involved, for rotation or transformation matrices? In which case this would fall apart rather completely if that doesn't work anymore and wouldn't be any different than switching to degrees, a convenience fix that requires conversion anyway.
• trklausss 1mo ago
  Wait until you discover gradians: centesimal system applied to angles. A turn is 400 gradians, right angles are 100 gradians. Same advantages as here but multiplied times 400...
  • womble2 1mo ago
    You could even pick a highly compound number like 2^3*3^2*5 then you could divide it neatly into whole numbers for lots of divisors!
• fph 1mo ago
  This alone should be a reason to drop the pi factor: it's basically impossible to get an exact zero for the sine of a half-turn: sin(1*pi) = 1.2246e-16
• ogogmad 1mo ago
  Turn is a measurement unit, and measurement units are just numbers. So turn ≈ 6.28318530718. You're welcome. That should put to bed that whole τ crap. "But the symbol τ is used for other things!" Yeah, yeah, yeah, just write turn. Even better, because it's more international and has more precedent, write rev for revolution.
• otikik 1mo ago
  Indeed, this is what Pico-8 uses for its trigonometric functions[1] (angles go from 0 to 1, instead of from 0 to 2*Pi). I was surprised by this at first, but then I found it is very convenient and simplifies a bunch of stuff. http://pico8wiki.com/index.php?title=Sin http://pico8wiki.com/index.php?title=Sin
• boomlinde 1mo ago
  This can be useful for some geometry, but pi isn't a completely arbitrary choice and some useful relationships are lost when not using radians. I use different angle units depending on the application. On a platform with 8-bit index registers, 1/256 of a turn can be useful. IIRC Pico-8 uses turns.
• srean 1mo ago
  Rather than sin(), cos() and motion on a circle it is fun to consider uniform speed motion along the perimeter of a regular polygon and its projection hor() and ver() along horizontal and vertical directions. You can parameterized the motion in terms of the time T to complete one period and consider it's horizontal (or vertical) shadow at any t mod T. This is related to DFT. As one increases the number of vertices of the regular polygon we will recover sin and cos in the limit. 2 \pi will show up in the ratio of the distance covered in one period of the uniform speed motion and the extents of the projected motion. Another interesting (and fundamental) construction is to forget about circles and polygons entirely. Simply consider a periodic function over a bounded length L. Consider first the discrete case where the domain is divided into k parts. We want to find an orthonormal basis for all nicely behaved (smooth) periodic functions on this domain. But there are infinitely many orthonormal basis sets for periodic functions on this domain. We are free to choose any. One choice is that adjacent values do not have large adjacent differences. This can be measured by squared adjacent differences. We choose that basis set that minimizes this quantity. For the discrete case we recover DFT basis and taking limits carefully we end up with sinusoids. \Pi will show up because of the requirement of orthonormality.
• theodorethomas 1mo ago
  The Fortran 2023 Standard introduces new intrinsics: "The intrinsic functions ACOSPI, ASINPI, ATANPI, ATAN2PI, COSPI, SINPI, and TANPI are trigonometric functions in which angles are specified in halfrevolutions (that is, as multiples of π)."
• amelius 1mo ago
  The problem with this is that when I see pi I know we're talking about an angle; when you use turns it's just some number. Maybe in typed languages it would work better.
• beeforpork 1mo ago
  Well, \tau vs \pi is a question of taste, but 1 vs. \tau (or \pi) is not. Because you don't get rid of these weird constants, because \pi (or \tau) is, as a fact, in the circumference and area of circles and in surface and volume of spheres, and in other places. There jus is a weird constant. And for APIs, you could reasonably well have turns or radians or degrees or even percentage of turns, whatever -- it depend on the context what is 'better'. What's really missing, I think, is the support of units in programming languages (in the type system) so that you cannot mess up when invoking sin()/cos(), because you would be forced to provide a unit.
• binarymax 1mo ago
  “The math is easy!” …proceeds to provide no examples of implementation. This is very interesting to me, but it would help your argument if you provided how the original Gogot example would be rewritten.
• srean 1mo ago
  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)
  • kps 1mo ago
    HN has clumsy filtering, but you're allowed to write π and τ.
    • srean 1mo ago
      Thanks for the push. I was being lazy on my mobile phone. Note to readers who maybe confused by the parent comment. I was using \pi and \tau. I just noticed now that my default mobile keyboard has π.
  • jacobolus 1mo ago
    If you want a single number (e.g. you are trying to serialize a lot of data), in many contexts you can replace the coordinates (cos θ, sin θ) with the stereographic projection h = tan ½θ = sin θ / (1 + cos θ). Converting back and forth between these representations is cheap and easy: cos θ = (1 − h²) / (1 + h²) and sin θ = 2h / (1 + h²).
    • srean 1mo ago
      Damn :) why did I not think of that. This would be useful for serialization deserialization. I have used half angles ½θ because there is no ambiguity about the full angle θ if I know it's sin value (and of course this holds for tan ½θ). Tan works better because one does not have to remember to take the correct branch of sqrt(1 - sin^2 θ). I learned two clever tricks in this discussion: (i) your tan ½θ and (ii) free modular arithmetic by embedding turns in signed integers.
• jerf 1mo ago
  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.
  • srean 1mo ago
    One would reap the most benefit if libraries were compiled late for the compiler to optimize away adjacent conversions and back. With C++ template libraries one can do that but compile times can become a hell.
• _verandaguy 1mo ago
  I mean, it's just shifting the value conversion in the other direction, isn't it? Pi is a naturally-emerging concept: it's the ratio of a circle's circumference to its diameter. It just so happens that a lot of useful stuff we do in math operate on that ratio, not on either on the individual values (at least, not those alone).
• jpopesculian 1mo ago
  The nice thing about turns is that they can be represented by an unsigned integer. For example with an 8 bit unsigned integer: - 0000_0000 = 0 or tau - 1000_0000 = pi - 1100_0000 = 3*pi/4 And the addition and multiplication by integer scalars all apply and the overflows work naturally. I wrote a little library in rust [0] to help with this as you can define the operations */+-% etc. [0] https://docs.rs/turns/latest/turns/ https://docs.rs/turns/latest/turns/
  • srean 1mo ago
    Ooh! that overflow is so clever. I would have been proud to have thought of that. Perfect fit for modular arithmetic.
• cbondurant 1mo ago
  My gut instinct was that operations like sin, cos, tan, etc, were the kind that would get aggressively inlined by any modern compiler, and then after inlining the redundant conversions between radian/turns would be canceled out by optimization passes. Of course, gut instinct can be wrong, so I checked in godbolt and was surprised to find out that even if I try to force lto, it still doesn't get inlined and remains a call.
  • aerzen 1mo ago
    I don't think compilers are smart enough to realize that multiplication with tau, followed by multiplication of 2/pi should be simplified away. Also, sin is probably used so much that inlinung would probably inflate binary size significantly.
• VonTum 1mo ago
  If you're going for "turns", then I would go even further and argue that it shouldn't be represented as a float, but rather a fixed point (signed?) integer. With 0x00000000 being 0°, 0x40000000 90°, etc. You get modular overflow for free, and the precision is consistent all around the circle.
  • srean 1mo ago
    There are a couple of comments buried in this discussion about this. It is really cute and clever. Although new to me this seems to be an old trick. The unit has a name -- BRAD.
• gweinberg 1mo ago
  Here's an even better idea: instead of turns, use degrees!
• qubex 1mo ago
  Them there is fightin’ words. Also, as pointed out by an illustrious colleague: Euler’s formula doesn’t work, and pretty much all of complex analysis breaks along with it.
• ocodo 1mo ago
  multiples, fractions of Pi, Tau are better ... mathematically and in code.
• jffhn 1mo ago
  >Want to store 90 degrees in radians? No matter how many bits you use, it will never be exact. That can be seen as a feature: cos(angRad) will never return zero, avoiding edge cases, and same with sin(angRad) if argument is not zero.