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.
The author's not talking about doing math, but about porting math into code. Counting turns is the same as counting cycles. People do that all the time. It works fine.
And this math is kind of a mess. exp(x) is its own derivative but the log is not. (d/dx)log(x) = 1/x
But, agreed, if you're going to do calculus, use radians.
Also with radians the differential equation x''''(t)=x(t) has {exp(t), exp(-t), sin(t), cos(t)} as the (real) canonical base for its solution space. And x''(t)=-x(t) gets {sin(t), cos(t)} where they even result from the simplest possible (non-trivial) initial conditions (x(0)=0,x'(0)=1 and x(0)=1,x'(0)=0).
If you look at all the simplest differential equations you can think of, the sin(t)/cos(t) functions in radians are almost inevitable independent from their geometric usage.
For graphics rendering Euler equation doesnt matter. Colours are 0.0-1.0 and have no relation to reality, but it works. Same with rotations (if we’re not using Quaternikns)
I can only imagine what a ridiculous problem it would be to try to re-do, for example, the Vincenty formula for distance between two latitude/longitude points on an oblate spheroid (the earth) if it couldn't use radians.
Further, inverse vincenty is pretty much an essential in anything that needs to find the azimuth between two points on a map. Such as for microwave radio link planning purposes.
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?
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.
It doesn't really change Euler's formula. It just adds a factor for x to it (or an additional term to hide inside sin and cos as functions). That's less convenient, not less true.
However, I don't really see a benefit gained out of doing this to keep your theta between -1 and 1 instead of between -2pi and 2pi. Like it's not a difficult thing to estimate or convert in your head to get a close enough estimation.
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.
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.
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).
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.
I have used the Taylor series approximations to produce the LUT over a defined interval. This may be generated pre-complication or at startup with a defined precision depending on the destination signed type.
Tend to use radians because we're moving from written proofs or simulations into embedded code in such systems. The code needs to read and work the same as those.
I've used Taylor series in numerical optimization. A function we were implementing needed to be differentiable (for automatic differentiation), but its definition had a special case, so we used a couple terms of the Taylor series in the special case.
edit: Sorry, to clarify, this was a function involving trigonometry but not simply vanilla sine or cosine. However, angular values being represented in radians did help in the same way I described in the parent post.
Our favorite WebAssembly is an example! It specifically excludes trigonometry from the spec, because real hardware doesn't produce exactly the same results.
So mathematical libraries in WASM reimplement the trigonometric functions using series.
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.
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.
> 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.
I mostly agree with your explanation, but would like to emphasize that this is just a convention from mathematics which mostly carries over into physics and engineering. We like to define functions that are R -> R and similar, instead of defining special sets like R° = { r * 360° | r \in R }, corresponding to "real numbers with unit degrees", and then defining functions like sin: R° -> R. It’s just simpler to define and analyze most functions from R -> R and so we mostly do that.
But if you look up physics papers, it’s not uncommon to define functions that require unitful inputs as well. For example, the wave function in the Schrödinger equation maps a position r (3D vector with unit meter) and time t (scalar with unit seconds), to a probability amplitude (complex number with unit m^-3/2), so that \int |ψ(r,t)|^2 d3r becomes a scalar (a probability). Up wave function is still considered a function by all physicists.
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.
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".
You can apply functions to anything. That's the only thing "function" means. They transform values into other values, and there is no limit on what kind of values you might want to talk about.
This is precisely why (programming language) types are poor model of physics units, despite often being touted for this exact use case. 3m is not the same thing as "the value 3 of type meter". It is the multiplication of the dimensionless scalar 3 with the special "m" constant for meters.
That's why pow(3m, 2) = 9 m^2, and not `the value 9 of type meter`. Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`. However this quickly becomes overwhelming once you start doing more complex expressions with multiple types. What is the type of `pow (3kg^2 * m/s, 3/2)`?
Edit to add: also, there is a simple fact that "sin(pi/2 kg)" is just not defined, in programming or math or physics or any other useful system. It's definitely not 1kg, just like sin ( (pi/2) * 2) is not sin (pi/2) * sin (2).
Isn't the "special constant" exactly "value 1, type meters", defined as equal to "value <...very large number...> type atoms" etc?
If not, then what would be the result of the multiplication of 3 with "m"?
> Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`
As long as your power is an integer, you can reduce it to multiplication. So what you'd really want to define is the result of "<value1 of type meter> * <value2 of type meter>", "(<value1 of type meter> * <value2 of type meter>) * <value3 of type meter>" etc.
What this gets you in the end is a type algebra, but that is also not exactly a new concept.
No, it's definitely possible in mathematics, they've left out some details as to what the units are doing that makes them unable to be assigned to functions. I mean a regular ODE that you get from newtons laws is a set of functions that take position and time as inputs, which all have units. What they mean should be "dimensionless functions cannot be applied to dimensional variables". These are commonly functions like sin cos exp log and so on.
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.
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.
> 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 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.
> 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).
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.
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"..
Ensuring units agree is indeed a form of type checking. A more thorough procedure for the former is dimensional analysis.
I think the fact that action and angular momentum have the same units played an illuminating role in making sense in Bohr's model regarding why only certain orbits are allowed.
Not sure how that would play out once angle is considered a fundamental entity.
In another comment I made in this thread https://news.ycombinator.com/item?id=49373317 I think it came to the understanding that a "turn" is similar to these dimensionless physical constants, as it takes the full circle/cycle as a fundamental constant. Apparently, using the turn as a dimensionless unit allows one to get rid of pi and e in Euler's formula, in favor of 1 and -1.
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.
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).
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.
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.
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.
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.
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.
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.
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.
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.
> 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.
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.
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?
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?)
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.
This statement is a little too strong. It's a mistake to care about the equality of floating point numbers after subjecting them to irrational operations. On the other hand, the entire internet runs on the fact that doubles exactly represent the integers up to 2^53.
Huh, what? Floating point numbers have a standard, you know. They aren't non-deterministic YOLO numbers.
By default, the compiler has to stick to what the standard requires, and can't just say add arbitrary imprecision.
Your epsilon is what you get when you try to analyse floating point numbers as approximations of real numbers. But they also have an independent life as bit patterns, and the compiler can't just willy-nilly muck around with these bit patterns.
Please look at the link where context is clear here. Floating point has limited precision and the complete inability to exactly represent some numbers.
Example, this equality check is false:
0.1 + 0.2 == 0.3
Because the last bits of a floating point number, after any practical chain of operations, is practically random, do to errors from limited precision. Yes you can always know what the result will exactly be for any operation, if you know the exact number and operations used. Good luck with something like sin/cos though, where implementations can vary wildly depending on the platform/library.
Your problem only occurs when you try to naively transport equality of real numbers into equality of floating point numbers.
https://www.netlib.org/fp/dtoa.c is how eg CPython parses literals like 0.3 into floating point numbers and how it converts floating point numbers back to strings. Lo and behold: these algorithm compare floating point numbers for equality, and would break catastrophically, if the compiler were allowed to willy-nilly fiddle with the bit patterns.
The authors of these algorithm did care about floating point equality, and that is not a mistake. (However it would be a mistake to assume that equality of mathematical real numbers translates to equality of floating point numbers.)
> Yes you can always know what the result will exactly be for any operation, if you know the exact number and operations used.
Yes, and for some algorithms like illustrated above this is exactly what people do, and have to do.
And the lower bits of 0.1 + 0.2 ain't random: they are the same on your computer as on mine, whether we run the code in 1999 or in 2029.
The short answer is you need fast-math flags to allow optimizations that may change floating-point results, and you also need to guarantee an implementation of sinpi/cospi (these were added in C23, so they're not all that common in host library implementations yet).
It's possible if you had the implementation of the math library visible to the compiler that it could do inlining and then simplify expressions, but honestly most math library function implementations are going to be the kind of function that doesn't get picked up by inline heuristics, as there's a pile of if statements (handling special cases and range reduction) that the compiler can't eliminate due to there not really existing a sufficiently powerful FP range analysis.
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?
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.
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.
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/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.
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.
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.
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)."
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.
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.
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?
The floating point expressions needed to represent the math library functions with decent precision and performance becomes significantly more weird and complex with turns.
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.)
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.
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
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.
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.
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.
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.
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.
Funny how the so-called eldritch terror is another face of what is widely considered one of the most beautiful equations in mathematics, Euler's identity that unites five fundamental constants.
e^(i*pi)+1 = 0
..which is a result of the more general formula.
e^(i*x) = cos(x) + i*sin(x)
Pi is hiding there in the sin and cos functions implicitly, because the unit radian is defined by 2*pi. In comparison, the version you mentioned that takes x in "turns".
-1^(2x) = cost(x) + i*sint(x)
It got rid of pi and e, which already is a win for simplicity. i is still there for the imaginary component, or y in the complex plane. So the need for pi was removed thanks to the "turn", defined by 1 as the whole circle or cycle.
Multiplying -1 to itself every half turn makes it an alternating series of 1 and -1.. Weird, but it is visually clear to understand, without involving e. Though I still don't see where e went. Oh, this comment explains:
> If we rearrange the products in the exponent we get
2πix πi2x ( πi ) 2x
e -> e -> (e )
> Where e^(πi) is -1. That shows there is something to the turns units; we can express the analog of the Euler identity using exponentiation using a base and factor which are integers.
Yeah I get it now, a "turn" acts like a dimensionless unit to the circle/cycle.
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.
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.
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.
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.
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.
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.
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.
The author's not talking about doing math, but about porting math into code. Counting turns is the same as counting cycles. People do that all the time. It works fine.
And this math is kind of a mess. exp(x) is its own derivative but the log is not. (d/dx)log(x) = 1/x
But, agreed, if you're going to do calculus, use radians.
Also with radians the differential equation x''''(t)=x(t) has {exp(t), exp(-t), sin(t), cos(t)} as the (real) canonical base for its solution space. And x''(t)=-x(t) gets {sin(t), cos(t)} where they even result from the simplest possible (non-trivial) initial conditions (x(0)=0,x'(0)=1 and x(0)=1,x'(0)=0).
If you look at all the simplest differential equations you can think of, the sin(t)/cos(t) functions in radians are almost inevitable independent from their geometric usage.
For graphics rendering Euler equation doesnt matter. Colours are 0.0-1.0 and have no relation to reality, but it works. Same with rotations (if we’re not using Quaternikns)
I can only imagine what a ridiculous problem it would be to try to re-do, for example, the Vincenty formula for distance between two latitude/longitude points on an oblate spheroid (the earth) if it couldn't use radians.
https://en.wikipedia.org/wiki/Vincenty%27s_formulae
https://www.johndcook.com/blog/2018/11/24/spheroid-distance/
Further, inverse vincenty is pretty much an essential in anything that needs to find the azimuth between two points on a map. Such as for microwave radio link planning purposes.
Karney (2013) is also radian dependent.
https://github.com/pbrod/karney
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?
I think it's Tau[0].
[0] https://en.wikipedia.org/wiki/Tau_(mathematics)
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.
It doesn't really change Euler's formula. It just adds a factor for x to it (or an additional term to hide inside sin and cos as functions). That's less convenient, not less true.
However, I don't really see a benefit gained out of doing this to keep your theta between -1 and 1 instead of between -2pi and 2pi. Like it's not a difficult thing to estimate or convert in your head to get a close enough estimation.
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: 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.
The time where "turns" are really great is when a whole lot of what you're doing is a phase accumulator.
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.
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... for the implementation in musl, or 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.
I have used the Taylor series approximations to produce the LUT over a defined interval. This may be generated pre-complication or at startup with a defined precision depending on the destination signed type.
Tend to use radians because we're moving from written proofs or simulations into embedded code in such systems. The code needs to read and work the same as those.
I've used Taylor series in numerical optimization. A function we were implementing needed to be differentiable (for automatic differentiation), but its definition had a special case, so we used a couple terms of the Taylor series in the special case.
edit: Sorry, to clarify, this was a function involving trigonometry but not simply vanilla sine or cosine. However, angular values being represented in radians did help in the same way I described in the parent post.
Our favorite WebAssembly is an example! It specifically excludes trigonometry from the spec, because real hardware doesn't produce exactly the same results.
So mathematical libraries in WASM reimplement the trigonometric functions using series.
Example: https://github.com/WebAssembly/wasi-libc/blob/2e6fb9d8ee0cdf...
“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.
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.
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.
> 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.
I mostly agree with your explanation, but would like to emphasize that this is just a convention from mathematics which mostly carries over into physics and engineering. We like to define functions that are R -> R and similar, instead of defining special sets like R° = { r * 360° | r \in R }, corresponding to "real numbers with unit degrees", and then defining functions like sin: R° -> R. It’s just simpler to define and analyze most functions from R -> R and so we mostly do that.
But if you look up physics papers, it’s not uncommon to define functions that require unitful inputs as well. For example, the wave function in the Schrödinger equation maps a position r (3D vector with unit meter) and time t (scalar with unit seconds), to a probability amplitude (complex number with unit m^-3/2), so that \int |ψ(r,t)|^2 d3r becomes a scalar (a probability). Up wave function is still considered a function by all physicists.
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.
> 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.
Case in point:
https://trac.ffmpeg.org/ticket/11279
https://trac.ffmpeg.org/ticket/11284
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".
You can apply functions to anything. That's the only thing "function" means. They transform values into other values, and there is no limit on what kind of values you might want to talk about.
Well you can also square root etc.
> You generally can’t apply functions to dimensional units.
Perhaps not in mathematics, but in programming that's clearly possible. I guess programming is more general than mathematics.
This is precisely why (programming language) types are poor model of physics units, despite often being touted for this exact use case. 3m is not the same thing as "the value 3 of type meter". It is the multiplication of the dimensionless scalar 3 with the special "m" constant for meters.
That's why pow(3m, 2) = 9 m^2, and not `the value 9 of type meter`. Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`. However this quickly becomes overwhelming once you start doing more complex expressions with multiple types. What is the type of `pow (3kg^2 * m/s, 3/2)`?
Edit to add: also, there is a simple fact that "sin(pi/2 kg)" is just not defined, in programming or math or physics or any other useful system. It's definitely not 1kg, just like sin ( (pi/2) * 2) is not sin (pi/2) * sin (2).
> with the special "m" constant for meters.
Isn't the "special constant" exactly "value 1, type meters", defined as equal to "value <...very large number...> type atoms" etc?
If not, then what would be the result of the multiplication of 3 with "m"?
> Of course, you can define the type `square meter` as well, and define `pow -> meters -> positive integer -> square meters`
As long as your power is an integer, you can reduce it to multiplication. So what you'd really want to define is the result of "<value1 of type meter> * <value2 of type meter>", "(<value1 of type meter> * <value2 of type meter>) * <value3 of type meter>" etc.
What this gets you in the end is a type algebra, but that is also not exactly a new concept.
No, it's definitely possible in mathematics, they've left out some details as to what the units are doing that makes them unable to be assigned to functions. I mean a regular ODE that you get from newtons laws is a set of functions that take position and time as inputs, which all have units. What they mean should be "dimensionless functions cannot be applied to dimensional variables". These are commonly functions like sin cos exp log and so on.
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.
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/Angle#Dimensional_analysis
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 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
> 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
---
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_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
Ensuring units agree is indeed a form of type checking. A more thorough procedure for the former is dimensional analysis.
I think the fact that action and angular momentum have the same units played an illuminating role in making sense in Bohr's model regarding why only certain orbits are allowed.
Not sure how that would play out once angle is considered a fundamental entity.
This sure is a rabbit hole.
Thanks for your submission
https://news.ycombinator.com/item?id=49372847
hope it gets picked up.
In another comment I made in this thread https://news.ycombinator.com/item?id=49373317 I think it came to the understanding that a "turn" is similar to these dimensionless physical constants, as it takes the full circle/cycle as a fundamental constant. Apparently, using the turn as a dimensionless unit allows one to get rid of pi and e in Euler's formula, in favor of 1 and -1.
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.
> all angles are without a unit.
Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?
In a very awkward way: rad is m/m, which is 1...
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.
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
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.
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.
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.
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.
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.
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.
> 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!
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.
Functions are free. Create new ones. Sin1 instead of Sin, Cos1 instead of Cos.
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.
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.
> 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.
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.
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)
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?)
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
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/
This statement is a little too strong. It's a mistake to care about the equality of floating point numbers after subjecting them to irrational operations. On the other hand, the entire internet runs on the fact that doubles exactly represent the integers up to 2^53.
Huh, what? Floating point numbers have a standard, you know. They aren't non-deterministic YOLO numbers.
By default, the compiler has to stick to what the standard requires, and can't just say add arbitrary imprecision.
Your epsilon is what you get when you try to analyse floating point numbers as approximations of real numbers. But they also have an independent life as bit patterns, and the compiler can't just willy-nilly muck around with these bit patterns.
Please look at the link where context is clear here. Floating point has limited precision and the complete inability to exactly represent some numbers.
Example, this equality check is false:
0.1 + 0.2 == 0.3
Because the last bits of a floating point number, after any practical chain of operations, is practically random, do to errors from limited precision. Yes you can always know what the result will exactly be for any operation, if you know the exact number and operations used. Good luck with something like sin/cos though, where implementations can vary wildly depending on the platform/library.
Your problem only occurs when you try to naively transport equality of real numbers into equality of floating point numbers.
https://www.netlib.org/fp/dtoa.c is how eg CPython parses literals like 0.3 into floating point numbers and how it converts floating point numbers back to strings. Lo and behold: these algorithm compare floating point numbers for equality, and would break catastrophically, if the compiler were allowed to willy-nilly fiddle with the bit patterns.
The authors of these algorithm did care about floating point equality, and that is not a mistake. (However it would be a mistake to assume that equality of mathematical real numbers translates to equality of floating point numbers.)
> Yes you can always know what the result will exactly be for any operation, if you know the exact number and operations used.
Yes, and for some algorithms like illustrated above this is exactly what people do, and have to do.
And the lower bits of 0.1 + 0.2 ain't random: they are the same on your computer as on mine, whether we run the code in 1999 or in 2029.
https://news.ycombinator.com/item?id=47767398 has a discussion.
The short answer is you need fast-math flags to allow optimizations that may change floating-point results, and you also need to guarantee an implementation of sinpi/cospi (these were added in C23, so they're not all that common in host library implementations yet).
It's possible if you had the implementation of the math library visible to the compiler that it could do inlining and then simplify expressions, but honestly most math library function implementations are going to be the kind of function that doesn't get picked up by inline heuristics, as there's a pile of if statements (handling special cases and range reduction) that the compiler can't eliminate due to there not really existing a sufficiently powerful FP range analysis.
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?
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.
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.
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.
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/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.
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.
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.
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)."
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.
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.
> It turns out (pun intended!)
Thanks, I was waiting for this pun the moment turns were introduced in the article.
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
Or you could use 1/360 of a turn.
degrees were primarily chosen due to many integer divisors - likely for applications of time and seasons.
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?
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.
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
Even better : did you know (-1)^x draws the unit circle in the complex plane ? No need for complex exp and i*pi
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.)
I agree. I just meant the 2D function cos(pix),sin(pix) is quite natural to work with and it can be reflected easily in the formulation of (-1)^x
That's because
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.
Yeah bad formulation on my part
Not bad at all, just equivalent.
If you plot it over which domain?
Complex domain
In the old days of making 8 bit video games we used BRADs of 0-255 - worked well and the wrap was easy.
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.
Are there any c/c++ libs / headers that use this (without converting to radians in the background). I like this idea.
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
Fails to mention that radians relates angle to arc length.
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.
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.
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.
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.
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.
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!
> Radians really are the natural unit for trigonometry.
s/trigonometry/calculus
I was hoping for some code examples but got none. Can anyone help?
> 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.
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
Here's another good reason to think in turns: it turns Euler's formula from this Eldritch Terror:
into something you can kinda understand by staring at the complex plane: Credit to justinpombrio for this: https://news.ycombinator.com/item?id=32986869Funny how the so-called eldritch terror is another face of what is widely considered one of the most beautiful equations in mathematics, Euler's identity that unites five fundamental constants.
..which is a result of the more general formula. Pi is hiding there in the sin and cos functions implicitly, because the unit radian is defined by 2*pi. In comparison, the version you mentioned that takes x in "turns". It got rid of pi and e, which already is a win for simplicity. i is still there for the imaginary component, or y in the complex plane. So the need for pi was removed thanks to the "turn", defined by 1 as the whole circle or cycle.Multiplying -1 to itself every half turn makes it an alternating series of 1 and -1.. Weird, but it is visually clear to understand, without involving e. Though I still don't see where e went. Oh, this comment explains:
> If we rearrange the products in the exponent we get
> Where e^(πi) is -1. That shows there is something to the turns units; we can express the analog of the Euler identity using exponentiation using a base and factor which are integers.Yeah I get it now, a "turn" acts like a dimensionless unit to the circle/cycle.
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.
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.
Now define exponentiation by a non-integer.
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.
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.
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...
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!
Maybe related
Hamilton's theory of turns revisited
https://arxiv.org/abs/0904.4787
The title should say (2022)
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.
At least we got metric units out of the French Revolution.
Gradians exist because "let's change everything, even things that aren't broken".
Norman Wildberger has an alternative system for trigonomtry:
Understanding uniform motion: are radians really necessary? | WildTrig
https://youtu.be/CnQXRdgN_7I?si=EiYY99i6mBOIyczI
Wild Trig: An introduction to Rational Trigonometry
https://youtube.com/playlist?list=PLIljB45xT85CyF_7bKd6y36VA...
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.