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Turns are Better than Radians (2022)

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Re: Turns are Better than Radians (2022)

#171

Post by lowbloodsugar »

walrus01 wrote: post #94
kazinator wrote: post #70

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 fo…

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 t…

Latitude and Longitude? By definition it’s not starting in radians. After that, pi is involved but that doesn’t mean you have to use radians, unless you define “anything that uses pi is radians”. The discussion TFA started was about programming computers optimally. If you want the code to be readable by humans with a math background, or Wikipedia background, sure, use the algorithms as written down. TFA was not targeting that audience. It was targeting folks who care about two pointless multiplies in their (checks notes) game engine.

Re: Turns are Better than Radians (2022)

#172

Post by voidmain »

thrtythreeforty wrote: post #45

Here's another good reason to think in turns: it turns Euler's formula from this Eldritch Terror: e^(i*x) = cos(x) + i*sin(x) into something you can kinda understand by staring at the complex plane: -1^(2x) = cost(x) + i*sint(x) Credit to justinpombrio for this: https://news.ycombinator.com/item?id=32986869

How about i^4x? It makes it clearer which direction we are rotating (vs -1 which is a 0.5 turn rotation in either direction) and avoids the garden path confusion that could arise from it not being obvious from the left side of the equation that we are working in the complex plane at all.

Re: Turns are Better than Radians (2022)

#173

Post by jacobolus »

srean wrote: post #167
jacobolus wrote: post #165

Earlier quoted context omitted.

If you want a single number (e.g. you are trying to serialize a lot of data), in many contexts you can replace the coordinates (cos θ , sin θ ) with the stereographic projection h = tan ½ θ = sin θ / (1 + cos θ ). Converting back and forth between these representations is cheap and easy: cos θ = (1 − h ²) / (1 + h ²) and sin θ = 2 h / (1 + h ²).

Damn :) why did I not think of that. This would be useful for serialization deserialization. I have used half angles ½θ because there is no ambiguity about the full angle θ if I know it's sin value (and of course this holds for tan ½θ). Tan works better because one does not have to remember to take the correct branch of sqrt(1 - sin^2 θ). I learned two clever tricks in this discussion: (i) your tan ½θ and (ii) free m…

For more: https://en.wikipedia.org/wiki/User:Jacobolus/HalfTan

Re: Turns are Better than Radians (2022)

#174

Post by voidmain »

judofyr wrote: post #59
thrtythreeforty wrote: post #45

Here's another good reason to think in turns: it turns Euler's formula from this Eldritch Terror: e^(i*x) = cos(x) + i*sin(x) into something you can kinda understand by staring at the complex plane: -1^(2x) = cost(x) + i*sint(x) Credit to justinpombrio for this: https://news.ycombinator.com/item?id=32986869

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.

I'll defend i^4x since I like it better.

(cost x, sint x) is a point on the unit circle x turns counterclockwise from (1,0).

cost x + i sint x is a point in the complex plane x turns counterclockwise from 1.

Now look at integer powers of i, a point in the complex plane 1/4 turn from 1:

i^0 = 1 (0 turns from 1)

i^1 = i (1/4 turn from 1)

i^2 = -1 (2/4 turn from 1)

i^3 = -i (3/4 turn from 1)

and we define complex exponentiation such that, for all real x,

i^x = cost (x/4) + i sint (x/4) (x/4 turn from 1)

Re: Turns are Better than Radians (2022)

#175

Post by tigen »

bee_rider wrote: post #147
titzer wrote: post #136

Earlier quoted context omitted.

If you look at the implementation of sqrt for a computer, it's usually implemented with Newton's algorithm, which is an iterative numerical method with high speed convergence. It is computationally efficient and looks approximately zero how √ looks.

Right. I was hoping to highlight that difference with a joke.

Sometimes a person will tell a story which is untrue, with the purpose of bringing levity to a conversation. This may be termed a joke. These stories may also be used to illustrate a point.

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".

Re: Turns are Better than Radians (2022)

#176

Post by NyxWulf »

kazinator wrote: post #70

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 fo…

The author didn't argue that you should never use radians, or tau. Just that for some use cases there is an alternative that is simpler using turns. Your argument can be true and his proposal can also be true, it just depends what your use case looks like.

Re: Turns are Better than Radians (2022)

#177

Post by zardo »

srean wrote: post #145

Let's do a full circle. It all began with replacing frequent occurrence of 2π in calls of sin and cos functions with τ. This post suggests an optimisation by getting rid of τ by getting rid of radians. That way one can get rid of frequent and adjacent radians to degrees conversions and back. I say, let's get rid of sin and cos itself ! Of course I am being over the top here. However, if you represent angle not as a s…

That will introduce an interesting new bit-bending avenue since almost all tuples of floats are not valid angle representations in that form.

Re: Turns are Better than Radians (2022)

#178

Post by adrian_b »

kazinator wrote: post #70

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 fo…

That formula as such has no importance at all, it is just a correspondence between different notations.

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".

Re: Turns are Better than Radians (2022)

#179

Post by srean »

jacobolus wrote: post #173
srean wrote: post #167

Earlier quoted context omitted.

Damn :) why did I not think of that. This would be useful for serialization deserialization. I have used half angles ½θ because there is no ambiguity about the full angle θ if I know it's sin value (and of course this holds for tan ½θ). Tan works better because one does not have to remember to take the correct branch of sqrt(1 - sin^2 θ). I learned two clever tricks in this discussion: (i) your tan ½θ and (ii) free m…

For more: https://en.wikipedia.org/wiki/User:Jacobolus/HalfTan

Oh! I did not know you were a contributing author to wikipedia.

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.

https://news.ycombinator.com/item?id=48732233

Re: Turns are Better than Radians (2022)

#180

Post by 6gvONxR4sf7o »

kazinator wrote: post #70

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 fo…

Why not e^{2 pi i x} = cos x + i sin x then? We already handle e^{2 pi/360 i x} = cos x + i sin x for x in degrees just fine. It's not that euler no longer holds, it's that you just have to be clear about what units[0] you use when comparing the explicitly angular/geometric cos and sin with the numeric exponential, and then deciding on a default numeric cos and sin/a default "unit" for angles.

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.

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