Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Turns are better than radians

computerenhance.com

Post reply on HN
Print view View original

91–100 of 494 posts

Re: Turns are better than radians

#91

Post by throwaway9870 »

Aardwolf wrote: post #80
doliveira wrote: post #52

Earlier quoted context omitted.

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

Those perfect radians use 2*pi, aka tau, though, a different math notation issue, where mathematicians have chosen the wrong option (imho) and a case for disrupting that part of math notation, to make radians easier to teach: 1/4th of a circle could be tau/4 radians, 1/8th could be tau/8, etc..., instead of confusing halved factors with radians expressed as amount of pi. Regarding long variable names: I'd rather have…

Just out of curiosity, where did tau come from? I never heard of it used for 2pi, and frankly, it seems like a poor choice because in engineering it is one of the most common symbols used (time constant tau).

Re: Turns are better than radians

#92

Post by ithkuil »

adrian_b wrote: post #63
fbanon wrote: post #32

>But math never decreed that sine and cosine have to take radian arguments! Ummm, actually it did. The Taylor-series of sine and cosine is the simplest when they work with radians. Euler's formula (e^ix = cosx + isinx) is the simplest when working with radians. Of course you can work in other units, but you'll need to insert the appropriate scaling factors all over the place. "Turns" don't generalize to higher dimens…

The simplicity of the Taylor series of sine and cosine is irrelevant, there are no important applications for those series. There is only one consequence of those series that matters in practice, which is that when the angles are expressed in radians, for very small angles the angle, its sinus and its tangent are approximately equal. While this relationship between small angles, sinuses and tangents looks like an arg…

> So the use of the radian can simplify a lot many pen and paper symbolic transformations, but it is rarely, if ever, beneficial in numeric algorithms.

If only computers could do a bit of symbolic algebraic manipulations before issuing the machine code.

Wait, isn't that what optimizing compilers can do? That requires an optimization across library calls and thus a form of inlining, which doesn't see far fetched for a math library call. Or some optimizations can't be done due to floating point error propagation (which could be relaxed)?

Re: Turns are better than radians

#93

Post by mjw_byrne »

Double_a_92 wrote: post #34
mjw_byrne wrote: post #25

"math never decreed that sine and cosine have to take radian arguments!" This is at best questionable and at worst false. If you only want to use sin and cos as functions for doing trigonometry, it is true that you can choose whatever angle unit you like and stick with it and it will be fine. For most other stuff, e.g. differential equations, complex analysis, signal processing and mechanics, it's pretty much inescap…

Why do they need to be at multiple integers of pi, instead of at half turns?

The main thing to realise is that sin and cos are not fundamentally tools for doing geometry. The fact that you can use them for working out side lengths of triangles or converting polar to cartesian coordinates is somewhat incidental.

It doesn't help that at school our first look at sin and cos is all about adjacent sides and opposite sides in right-angled triangles. It's understandable, because jumping straight into the deep end would be too hard, but it's a bit misleading.

In most mathematical applications, the x in "sin(x)" doesn't even represent an angle, so it doesn't make sense to talk about whether sin and cos are "in degrees or in radians". They're simply functions that crop up as solutions to the differential equation that describes harmonic oscillation; or the imaginary and real parts of e^ix; or exponentiation of certain matrices; or a whole load of other stuff I haven't thought of.

In all those settings, it turns out that sin has zeroes at integer multiples of pi, which forces the convention that a half-turn is an angle of pi, and the definition of radians follows from there. But as I said, for the specific case of basic trig, carrying around a scaling factor and doing everything in degrees is easy enough. Carrying that same scaling factor around in pretty much any other application of sin, cos and related functions would be hell.

Re: Turns are better than radians

#94

Post by ncmncm »

layer8 wrote: post #41
adrian_b wrote: post #26

While I agree that "turns" are much more convenient in some applications than radians, there is no need to invent a new terminology. For a long time, including the 19th century, the plane angle measurement unit corresponding with 4 right angles, i.e. a complete rotation around a point, has been named "cycle". That is why in many old physics or engineering books one will find wave numbers measured in "cycles per meter…

> there is no need to invent a new terminology. It's not a new terminology: https://en.wikipedia.org/wiki/Turn_(angle)

I am partial to the term "windings", defined as not modulo 1 turn.

I e., we are talking here about two very different quantities. Angle is always within a turn, but rotation, using the same units, is not.

Re: Turns are better than radians

#95

Post by coldacid »

hutzlibu wrote: post #76
doliveira wrote: post #52

Earlier quoted context omitted.

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

Whats wrong with long variable names?

Nothing, but their use in mathematical equations will certainly conflict with the implicit multiplication in equations (i.e. `abc` in a formula means `a * b * c`, not a variable abc).

Re: Turns are better than radians

#96

Post by Dan42 »

jjgreen wrote: post #3
ClassyJacket wrote: post #2

I'm so glad someone else finally said this. This article takes the approach of simplicity of code, which I do agree with, but additionally I've been thinking turns would have to be more intuitive and easier to teach (particularly to disinterested teenagers) for YEARS, and I finally feel vindicated at least one person agrees with me. Turns are really the most neutral way to count an angle. We don't use them for everyd…

... and the derivative of sin(x) is ... ?

> and the derivative of sin(x) is ... ?

I believe it's cos(x). And I fail to see how that would change regardless of what unit x is expressed in. Sorry, my trig is veeery rusty.

Re: Turns are better than radians

#97

Post by doliveira »

Aardwolf wrote: post #80
doliveira wrote: post #52

Earlier quoted context omitted.

sin(x) ~~ x only in radians, so honestly that's reason enough. Once in a while we get programmers wanting to disrupt mathematical notation for whatever reason... Worst I've seen so far was one arguing that equations should be written with long variable names (like in programming) instead of single letters and Greek letters. Using turns because it's a little easier in specific programming cases is just as short-sighte…

Those perfect radians use 2*pi, aka tau, though, a different math notation issue, where mathematicians have chosen the wrong option (imho) and a case for disrupting that part of math notation, to make radians easier to teach: 1/4th of a circle could be tau/4 radians, 1/8th could be tau/8, etc..., instead of confusing halved factors with radians expressed as amount of pi. Regarding long variable names: I'd rather have…

As far as I know, the whole tau disruption wasn't proposed by programmers, so I think we're safe on that.

And proposing to write equations in books and articles with long variable names... Well, Algebra was invented for a reason.

Re: Turns are better than radians

#98

Post by quickthrower2 »

fmajid wrote: post #29
lordnacho wrote: post #8

That was quite convincing actually. I guess we all have this realization at some point in early math education. Why is it 360 degrees? Mainly because that's a nicely divisible number, no other good reason. Sometimes you find a 400 degree system on calculators but it doesn't seem to be taught anywhere (is it a French thing?) Then at some point you get shown radians, which relates the arc length to the radius. That som…

360 comes from the Babylonians, who used base-60 for numbers much for the reasons you describe (and who gave us the 24-hour day, 60 minute hour and 60-minute second, not to mention the 7-day week). NATO forces have compasses labelled in mils or milliradians, which are not actually 1/1000 of a radian but as an approximation 1/6400 of a full turn. I still have my Silva military compass from 1989 graduated thus. https:/…

I heard that the reason is 1km away it makes a difference of 1m, so easy to figure out how far enemy troops are from each other.

Re: Turns are better than radians

#99

Post by airblade »

throwaway9870 wrote: post #91
Aardwolf wrote: post #80

Earlier quoted context omitted.

Those perfect radians use 2*pi, aka tau, though, a different math notation issue, where mathematicians have chosen the wrong option (imho) and a case for disrupting that part of math notation, to make radians easier to teach: 1/4th of a circle could be tau/4 radians, 1/8th could be tau/8, etc..., instead of confusing halved factors with radians expressed as amount of pi. Regarding long variable names: I'd rather have…

Just out of curiosity, where did tau come from? I never heard of it used for 2pi, and frankly, it seems like a poor choice because in engineering it is one of the most common symbols used (time constant tau).

Look up the Tau Manifesto: it’s all explained there.

Re: Turns are better than radians

#100

Post by grey_earthling »

throwaway9870 wrote: post #91
Aardwolf wrote: post #80

Earlier quoted context omitted.

Those perfect radians use 2*pi, aka tau, though, a different math notation issue, where mathematicians have chosen the wrong option (imho) and a case for disrupting that part of math notation, to make radians easier to teach: 1/4th of a circle could be tau/4 radians, 1/8th could be tau/8, etc..., instead of confusing halved factors with radians expressed as amount of pi. Regarding long variable names: I'd rather have…

Just out of curiosity, where did tau come from? I never heard of it used for 2pi, and frankly, it seems like a poor choice because in engineering it is one of the most common symbols used (time constant tau).

https://tauday.com/ is a good entrance to this particular rabbit-hole.
Post reply on HN