Turns are Better than Radians (2022)
computerenhance.com
computerenhance.com
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.
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.
The derivative of log_b x is 1/(x ln b), where ln b is 1 if b is e.
The computational aspect of it is totally compelling. The library routines are already using turns internally, so it is wasteful to go from/to radians when the caller doesn't require it, and many callers can be rewritten not to. Plus the part argument about common angles like multiples of the right angle having to be irrational numbers under radians is also compelling.
Assume people have read the article and understood it.
If your code doesn't look like the math it's "ported" from, the odds of it being bad code go up like 100x
It's important to note that such cases are not always clearly signaled as being humor or untrue. It is a part of the joke's effect that the reader or listener will not at first know it is a joke, but will realize it after noticing an absurdity.
A related concept is "dry humor".
And for all the people who are concerned about how sin' 2πx = 2π cos 2πx, in actual code, it doesn’t matter. Let’s say that I’m writing a basic graphing function and I want to be able to display the slope of the sin curve at any point.
I am not going to expose the turn-based units to the user. Caring about slopes implies that I’m doing calculus and thus assuming radians. So even though my internal values are [0,1], I will label them as [0, 2π] (and the actual numeric values on the display may actually be something like [50,450] which is yet another numeric value we don’t display). So to get the slope at π/4, I’ll calculate cos_t 0.125 and display that value.
We do all kinds of unit translations in computing without worrying about it. This is just another case of that which observes that numerically speaking, using turns is better aligned with the underlying numerical algorithm for calculating trig values.
Haha !
I have been coding for much shorter time but having done some ML on orientations and on spheres in my time, I have had to take their derivatives all the time.
It will be interesting to consider folks who do machine learning on robot trajectories or analysing dynamics of robotic arms.
You could be using the results of calculus, which became frazzled with gratuitous constants because of poor angle units before anyone wrote any code.
You want to keep all the math in radians until you code the calculations; then figure out how to optimize it with turns where possible.
It sounds like discussions about "porting from math" do not pertain to you then?
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.
There are alternative trigonometric functions with the argument measured in cycles that can be used instead of those with argument measured in radians. All formulae written with sin, cos and atan can be rewritten with sin2pi, cos2pi and atan2pi, or whatever names you want for those functions.
Moreover, before the electronic computers, nobody in the entire history of mankind had used trigonometric functions with arguments measured in radians for any practical problem, like astronomical or geodesy problems.
The radians became popular in the 19th century, only for symbolic computations, because they simplify the formulae for derivatives and primitives.
For all numeric computations, even in the 19th century, nobody used trigonometric functions with arguments measured in radians, but only trigonometric functions with arguments measured in sexagesimal degrees, which behave much more similarly to the functions with arguments measured in cycles, than to the functions with arguments measured in radians, which cannot be computed exactly for any rational argument, and where the argument can never be reduced exactly to a value in the first quadrant.
While the trigonometric functions with arguments in radians are harmful, the constant 2Pi remains an essential constant, which must be used to convert between the length and the central angle that correspond to a circular arc, and also to compute the derivatives and primitives of the exponential 1^x and of the trigonometric functions.
The sin function has Indian and Greek roots. Part of that history is in the name 'sin' itself. I will let you all look that up. It's quite a funny story especially the jya to jayb bit.
This comment is about computation of sin in Indian mathematics and its definition.
The analogue of the sin function in Indian mathematics was a function from length to length. It was defined as the length of the half chord that corresponds to the angle, not of a unit radius circle but of a circle of radius 3438 units.
But why 3438 ?
That requires understanding the Indian system of measuring angles by length.
They wanted to have a numeric precision of 1 arc minute but weren't very fond of manipulating fractions. So they incorporated their requirement in how they standardized their circle -- Not by radius or diameter but by arc length. They chose that circle to be the standard for which the arc length of one arc minute is unit length.
If one does the math, it is 360 * 60 / 2π. Plugin the accuracy of π that was known to Indian mathematicians of that time you get a radius of 3438 units.
If you are curious about their numeric calculation of sin look up Aryabhatta and Bhaskara. Wikipedia is quite informative.
https://en.wikipedia.org/wiki/%C4%80ryabha%E1%B9%ADa's_sine_...
https://en.wikipedia.org/wiki/Bh%C4%81skara_I%27s_sine_appro...
Something akin to radians was widely used in ancient India. That is, sine tables constructed for a circle with circumference 360·60 (angular minutes) and radius 3438 ≈ 360·60/2π.
In Europe, radians per se developed in the early 18th century. What is true is that the name radian came about in the late 19th century.
> trigonometric functions with arguments in radians are harmful
Trigonometric functions are not inherently "harmful", but they are sometimes cumbersome. Changing the units doesn't make all that much practical difference.
Avoiding transcendental functions in favor of vector methods is often a good strategy though, especially in code.
No discussion of why you'd use this over for example the Haversine function, of course, just a straight out copypasta and a demonstration of how clever someone is at the mathematical notation markup.
Incidentally you'd use this instead of Haversine if you wanted to be really really accurate down to millimetres across a long baseline, as opposed to lots and lots of "it's ten miles that way in a straight line" very quickly and simply.
I already edit quite a bit on Wikipedia. I stay away from the maths bits, because it's a little outside my area of expertise - and because it's just things like "it's a function to calculate distance, and here's ten pages of algebra with no explanation" I can't really improve that on Wikipedia.
Haversine is strongly deprecated in even short distance applications like planning a 35 km point to point licensed band radio microwave link, because you risk having both slightly the wrong distance and the azimuth figure being off by a tenth of a degree.
Vincenty in python form is quite compact. Karney by comparison is something like 1000 lines of code. And the difference is going between like a few mm precision to nanometers.
I should try it, right?
Edit: I also wonder what's to be gained by being accurate to nanometres over the surface of a fairly inaccurate planet.
My application was to locate movable things to identify which was nearest to a moorland wildfire, where you need to be accurate to within about 1km at best - you don't need to drop the pin right on it, you'll know where you're going when you see it ;-)
Huh?? "Euler angles" are one of the most popular representations in computer graphics! The only other good alternative is quaternions, where as you say this also matters!
I think what they're getting at is "why are we scaling everything so one full turn is sin(6.28ish) when we can just say sin(1)?"
You could easily try it out by defining a function that replaces sin(), cos(), and tan() with their "turn" equivalents, and seeing how you get on.
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.
If we want to get real interesting with it, this could also motivate an explicitly geometric "unit" aware exp operation, and depending on the defaults we use for the angular scale and the linear scale, 2 pi could be the conversion factor, making exp(2 pi i x) = cos x + i sin x actually interesting and useful and clarifying.
[0] pedantically, they're not units, or at least not dimensional units, so whatever the word for dimensionless units are, as in degrees vs radians vs turns.
When you multiply two complex numbers z1 and z2, their angles add: Arg(z1 z2) = Arg(z1) + Arg(z2).
That carries into exponentiation: Arg(z^2) = 2 Arg(z). The exponent 2 has an interpretation as doubling the angle.
In other words, e^2πix has an interpretation as working with angles.
When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos.
Your left side shows that you are sticking with Arg(z) being in radians!
But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number <cos x, sin x>.
But your left side essentially says that the Arg of this point: Arg(cos x + i sin x) is not x, but 2πx!
When we have a point on a unit circle whose Arg is x, then if we raise e to the power of ix, we get that point. That's what the original left hand says, without the pi.
You have a "trigonometric angle" and "Arg" that are separate, right in a fundamental equation.
> But on your right side, you have turns: the expression cos x + i sin x is literally saying that the point whose angle is x on the unit circle in the complex plane is the complex number <cos x, sin x>.
I totally agree here, and that's purely geometric, regardless of what we express x in. We can talk in terms of abstract points without specific coordinates/embeddings in R.
> When you have that 2π in there, but not in the sin and cos expressions, you're using different angles for multiplication/exponentiation and for sin/cos.
This part I'm not following. When we talk about a scale, any purely universal identity like Arg(z^2) = 2 Arg(z) is going to hold regardless of the scale. I agree that the Arg stuff nicely motivates interpreting it as an angle, but don't see how it says anything at all about the scale in question. Like, we get an interpretation of e^2πix as working in angles from the Arg reasoning, but we don't get a scale for those angles from it, do we? We'd only get Arg if we impose a scale on Arg itself, right?
So if we take e^2πix at x=1/2 turn=1/2, we get e^πi=-1, which gets us Arg(-1)=1/2 turn=pi rads=180 degrees, and we can work from there, but I still don't see how it imposes a unique scale that we can say is still radians and thus incompatible with the RHS's scale of turns.
What you really mean is that there are certain mathematical problems where the complex exponential function is useful and for the complex exponential it is more convenient to measure the real part in nepers and the imaginary part in radians (in order to have a simple formula for computing its derivative and its primitive).
However, the problems where the complex exponential is truly useful are at least an order of magnitude less numerous than it appears from the manner in which mathematics is taught in schools, following a tradition from the 19th century, when symbolic computations done with pen and paper were more important than numeric computations.
For the vast majority of practical problems, the complex exponential is not useful at all (i.e. using it introduces unnecessary complications, without providing any advantage), but a pair of other exponential functions is much more useful, because they ensure computations that are both faster and more accurate: the binary exponential 2^x, with real argument and value, and the exponential 1^x, with argument measured in cycles and a value that is a complex number of unit norm (unit modulus).
Using the pair of exponentials from above, removes computations that are slow and inaccurate (for the reduction of the arguments) from each function evaluation, by moving them to the computation of derivatives or primitives, which are operations that happen much more seldom and which also can frequently be done at compile time, instead of at run time.
Any algorithm that is described by using complex exponentials can be rewritten to use only 2^x and 1^x, and this normally allows various simplifications in the numeric computations. Even the formulae for Fourier transforms are simpler.
The author of the TFA is perfectly right and the use of nepers and radians is a very bad habit, which is a handicap with which most people remain after being taught mathematics in schools, with antiquated methods.
While in TFA the author uses the term "turn" for the unit of plane angle, the traditional name, which was used until some time after WWII, was "cycle", from which various other unit names where derived, e.g. "cycles per meter" or "cycles per second".
1^x is multi-valued, and like with all other multi-valued complex functions it is possible to select a branch of the function that is a proper function.
Defined correctly, the value of 1^x for any rational x is the corresponding smallest root of unity, and for irrational arguments it is defined by continuity.
Therefore 1^x rotates on the unit circle for increasing x.
The standard complex exponential and complex logarithm functions are defined exactly in the same way, because they are also multi-valued, so the same kind of equalities like yours would also be true for them.
With the correct definition, 1^x = 1 only for integer x, not for any x. As I have written above, the useful 1^x is defined only for real arguments, not for complex arguments. Only its values are complex numbers of unit modulus.
Example values of e^z and 1^z:
e^2 ~= 7.389
1^2 = 1
e^(i\pi) =-1
1^(i\pi) = 1
e^-7 ~= 0.000912
1^(-7) = 1
e^(1+i) ~= 1.46869394 + 2.28735529 i
1^(1+i) = 1
Unfortunately, this requires the use of a custom library of transcedental functions, because the standard libraries of most programming languages provide only the functions with arguments in nepers and radians.
The standard C library has attempted to add the missing functions, but for a completely impossible to understand reason the standard has defined wrongly the trigonometric functions, requiring arguments measured in half cycles, instead of cycles.
This is an extraordinarily stupid mistake, but at least if a standard C library implementation has all the functions, it is easy to modify the sources of the trigonometric functions with arguments in half cycles, to take arguments in cycles instead.
Do you have a reference?
First sentence of the Wikipedia link:
> The number τ (spelled as tau)
> 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?
You were asking why they used the symbol τ rather than the words "rev" or "turn". You didn't give a clue that you knew any of the rest, including what the symbol meant.
the derivative of ln(x) is 1/x
> 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.
the derivative (with respect to x) of sin(x) is cos(x), regardless of units for x. Otherwise the chain rule wouldn't work
Not if you use "turn"-trigonometric functions, as the author suggests.
turn-sin `t sin(x) = sin(2pi * x)` has `d/dx tsin(x) = 2pi * tcos(x)`.
but basically one would be differentiating to 2pi x rather than x. and things will work out
However,
Proportion of code where radians are used to represent rotation in 2D (i.e. turns): 99.9999%
Proportion of code where radians are used to perform higher degree derivatives, symbolic computation, etc: 00.0001%
The use of the trigonometric functions with arguments in radians, which are unfortunately standard in most programming languages, wastes time and introduces unnecessary rounding errors at each function evaluation.
The functions with arguments in cycles perform exact argument reduction instead of approximate argument reduction, and which is also faster (omitting a multiplication with an approximate value).
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 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 + ...
cos(2 pi t) = 1 - (2 pi t)^2/2 + ...
sin(2 pi t) = (2 pi t) - (2 pi t)^3/6 + ...
Closely related to this, if you ever need the derivative of a function that does trig (e.g., in numerical optimization), you may as well use radians because if you don't, any extra factors you apply will appear in the expressions for the derivatives and you'll have to deal with them there anyway.
Basically for that reason, it's pretty clear that trigonometry in terms of radians is the "correct" convention mathematically speaking (away from computers), since derivatives of the radian-based trig functions are so easy to express. Given that, if we have to pick one convention...isn't it less confusing to use the same thing everywhere? That said, there are interfaces that provide both versions, and since as the article points out there are cases where the turn-based versions can be more efficient, that's probably the right way to go.
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.
So mathematical libraries in WASM reimplement the trigonometric functions using series.
Example: https://github.com/WebAssembly/wasi-libc/blob/2e6fb9d8ee0cdf...
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.
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.
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.
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.
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
> 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
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.
I remember things the best (only) when I discover them for myself. Slow progress but high retention.
But that's more for analysis of your code / formulas than when you actually go and compute things.
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.
Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?
Uh, that is not what the article you linked is saying. Natural units don’t make speed dimensionless, nor allow you to use the same unit for length and time. Natural units remove the conversion constants, not the units or dimensions.
‘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.
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.
If instead we start defining ‘amounts of distance’ as some set D and ‘amounts of time’ as some set T, I have all sorts of extra work to do to make it so that products of amounts of distance are ‘amounts of area’ and amounts of distance over amounts of time are ‘amounts of speed’.
‘Dimension’ is the mathematical tool that lets us bundle all that up into something that we can deal with separately, alongside a real number. And of course you can totally make functions that are dimensional - but it affects what you can do with your functions, like composition and differentiation.
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.
https://www.nist.gov/publications/dimensionless-units-si
A key takeaway:
> one conclusion that is not optional is that the unit hertz cannot be regarded as a coherent unit of the SI, in contrast to its designation in the current form of the SI, where cycles are ignored and Hz may be replaced by s^−1
One thing I found especially interesting: They argue that you can express the (complex) exponential function exp(x) as a power series with powers x^k. They do not say it explicitly, but if we assume the power series comes from a Taylor series, then the k-th factor 1/(k!) is the derivative evaluated at x=0. And the k-th derivative has exactly the unit needed to cancel the unit of x^k. So, all summands of the series are unitless and hence the exponential function's argument is unitless.
This argument would hold for any function which can be written as a series like this. I am wondering whether this is actually a "problem" of the derivative operator.
Case in point:
Edit: Apparently "same dimension" doesn't imply "same unit".
Perhaps not in mathematics, but in programming that's clearly possible. I guess programming is more general than mathematics.
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.
The answer is not, and the result of 3 multiplied by m is 3m. Just like 3 multiplied by pi is 3pi; or, perhaps more accurately, you can view m as a kind of vector unit, and 3m as the scalar product. Of course, none of this is exactly matching - dimensions are different from irrationals, vectors, complex numbers, etc, they are mostly a thing of their own.
> What this gets you in the end is a type algebra, but that is also not exactly a new concept.
Sure, that's why I said specifically programming language types. I am aware that type theory has way more complex operations on types. I think some of these may even be expressible in Idris or Haskell + some appropriate extension. But in almost all programming languages, even ones like OCaml, SML, plain Haskell, Rust, C++ with template magic, Scala, F# and what have you, there is no way to specify that the result of multiplying two values of type A is of type "A * A", especially not in a way that then allows you specify that the division of a value of type "A * A" by A has type A. So types as exposed in any of the common programming languages are horrible for modelling dimensions as used in even high school physics.
I harbor a terrible internal mental model of dimensions which I have never really validated or explored fully, where I like to think they might be vector exponents, or something vaguely similar. If we assign each dimension to be a dimension of a vector - (length, mass, time, etc…) then a ‘distance’ might be e^((1,0,0,…)); a ‘duration’ e^((0,0,1,…)).
These have the requisite properties that when we multiply and divide them, we end up adding and subtracting these vectors.
So a distance times a distance is e^((2,0,0,…)) and a distance over a duration (a speed) is e^((1,0,-1,…))
They have the right basic algebraic behavior but who knows what terrible consequences they would have.
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)
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.
That is one fantastic page. It's a pity if it's sitting there as a draft. At the least have it on your blog.
Half angles substitution is of course a common exercise trying out symbolic integration of many trigonometric integrands. That will s where I encountered it most frequently.
The similarity between hyperbolic and trigonometric angle is also one of my favorite things.
What I found interesting is that we think in terms of slightly different but equivalent pictures. Mine is
https://upload.wikimedia.org/wikipedia/commons/1/18/Versin.p...
I think in terms of angle CAD and the angle subtended on the circumference on the opposite side of the center by the arc AB.
Rotations tickle me just right. One thing I had thought of is turtle graphics motion on the sphere over great circles. Turns out it is not an original thought at all.
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.
My friend,
it's called a complex number. A unit complex number, to be specific.
>Recall rotation is a linear transformation with a matrix whose entries are in terms of sin and cosine.
That's what multiplication of complex numbers is.
Can't wait till people here rediscover that a 3D rotation matrix has an axis-angle representation whose composition is given by quaternion multiplication.
At that, children, is why learning mathematics is important.
You have to wait and inordinate amount of negative time then.
Or maybe it's us who has to do the waiting for you to catch up on what HN folks in general are knowledgeable about.
I have a suggestion. Scroll to the bottom of this page. You will find a search box. Plug in the names of concepts you were thinking of. Enjoy.
To be honest, I too have come close to sounding a smug clueless smartass in my youth - lectured a guy on longest common subsequence not knowing that he was the original author of the file compression utility on AT&T Unix that used the same several decades ago.
If you also incorporate scale then it is, otherwise they need to be unit normalised.
The unnormalized case is essentially a complex number.
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.
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.
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.
Great idea, I will definitely do this!
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.
With C++ template libraries one can do that but compile times can become a hell.
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.
- 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.
This is very interesting to me, but it would help your argument if you provided how the original Gogot example would be rewritten.
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?
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?
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 assume you're saying something other than this though?
For more, there's a good post on this kind of flag in Rust: https://pythonspeed.com/articles/faster-float-math-rust/
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.
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.
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.
This is not a naivety problem, it's as I said, that floating point numbers cannot represent some numbers. In practical use, we see this as a precision problem, thus requiring tolerances.
Yes, that discussion is precisely what I'm talking about. That tolerance is required because the bits below the tolerance are not stable. They will practically be, after some practical operations, random which is why you need to ignore them. As I said, yes you will know what those meaningless lower bits will be, at any point in time, if you know the algs, which is why I said "considered to be" and "practical".
And yes, old code will produce the same numbers as new code, if the same hardware/libraries/implementations are used. But, since even the order of operation will change the lower bits, due to precision limits. This will usually NOT be true across libraries. If you're using hardware sin/cos, then it will not be true even if the same library is used, since different platforms can use different implementations/approximations of hardware sin/cos. This is all fine, because if you're actually doing this type of work, the fact that floating point numbers have limited precision, therefore requiring a tolerance for comparison, is well understood (see your discussion link).
Cheers!
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.
"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)."
Although new to me this seems to be an old trick. The unit has a name -- BRAD.
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.
[†] 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.
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.
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.
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...
Thanks, I was waiting for this pun the moment turns were introduced in the article.
This means 100 gradians to a right angle, so arbitrary small angles feel more like percentages of a right angle.
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.
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.
Also, sin is probably used so much that inlinung would probably inflate binary size significantly.
Hamilton's theory of turns revisited
sin(1*pi) = 1.2246e-16
I’ve had a brief moment of hope, forgetting the point was about mathematics.
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.
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.
e^(i*x) = cos(x) + i*sin(x)
-1^(2x) = cost(x) + i*sint(x)
(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)
e^(i*pi)+1 = 0
e^(i*x) = cos(x) + i*sin(x)
-1^(2x) = cost(x) + i*sint(x)
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 )
Yeah I get it now, a "turn" acts like a dimensionless unit to the circle/cycle.
Please stick to radians.
For example when you do a Taylor series expansion the cos/sin are well approximated by x.
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.
the article acts like radians are arbitrary without discussing this key property.
Gradians exist because "let's change everything, even things that aren't broken".
a^b = exp (b ln a)
One can change based of the exponentiation operation. Exp happens to be a convenient base.
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.
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.
Imagine it like someone suggesting (understandably) that you express memory sizes in hex: no push to make everybody stop using decimal numbers!
s/trigonometry/calculus
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.
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).
Same advantages as here but multiplied times 400...