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Turns are better than radians

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Re: Turns are better than radians

#101

Post by ncmncm »

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:/…

Wait, like instead of 1/6283.185307 of a turn? That's wild!

(It is off by 1.86%. That much error matters, nowadays, though it wouldn't have, back when.)

Re: Turns are better than radians

#102

Post by quickthrower2 »

tromp wrote: post #18

> Some time ago, much effort was expended to convince people to replace approximations of “pi” (3.14159…) with approximations of “tau” (6. 28318…). IMO the effort was simply to replace the use of pi with the use of tau. What does approximation have to do with it?

The point is PI and it’s brother TAU, being transcendental, is always approximated. Which is important to think about if you care about accuracy.

Re: Turns are better than radians

#105

Post by red_trumpet »

mjw_byrne wrote: post #93
Double_a_92 wrote: post #34

Earlier quoted context omitted.

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 int…

> or the imaginary and real parts of e^ix

For that particular application, x is exactly the argument[1] of your complex number, though!

[1] https://en.wikipedia.org/wiki/Argument_(complex_analysis)

Re: Turns are better than radians

#106

Post by tromp »

version_five wrote: post #73
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…

Right, radians are the "natural" units of angle, others generally just make a circle into some integral number of units for convenience, but you always have to go back to radians to actually do calculation. In the next installment, maybe he'll propose that turns can be limiting because diving up a circle requires the use of fractions, and suggest instead of 1 turn per circle, we make a number that's easily divisible…

> you always have to go back to radians to actually do calculation.

The article actually argues the opposite: that the common implementations of sine and cosine start by converting their radian based arguments to turns or halfturns by dividing by pi.

Re: Turns are better than radians

#107

Post by meheleventyone »

WastingMyTime89 wrote: post #14

I’m surprised game engines do conversion from degrees to radian to call trigonometric functions. I would have called that bad code in an industrial context. We did everything in radians and converted to degrees only for displaying and input. It was a lot more natural from us because we are a lot more familiar with radians anyway. I don’t think I have used degrees often since starting high school twenty years ago.

Most game engines work in radians under the hood and expose degrees in the UI but still need to glue the UI values to the API which is the code seen in Casey’s Godot example the full context of which is a color picker and h has already been converted from degrees. Godot itself has a bunch of the API in degrees which is pretty bleugh.

Re: Turns are better than radians

#108

Post by nkurz »

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).

It apparently was chosen because it's the starting sound of "turn": Hartl chose tau to represent 2pi because it nicely ties in with the Greek word “tornos,” meaning “turn,” and “looks like a pi with one leg instead of two.”

https://blogs.scientificamerican.com/observations/the-tao-of...

There was an earlier effort that used a new "two pi" symbol consisting of a "π" with an extra leg in the middle: https://www.math.utah.edu/~palais/pi.pdf.

Re: Turns are better than radians

#109

Post by 11235813213455 »

WastingMyTime89 wrote: post #67
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 writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1. Reading the submission and the comments here, I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explai…

any angle unit is dimensionless, radian is no exception

Re: Turns are better than radians

#110

Post by genewitch »

WastingMyTime89 wrote: post #67
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 writer don’t seem to realise that radian is not an arbitrary unit but a dimensionless one which is defined so that 1rad is actually just 1. Reading the submission and the comments here, I’m under the impression that trigonometry is not extensively taught in middle schools and high schools in the USA. While I’m slightly envious you might not have to suffer developing powers of cosine and sine but that would explai…

I was a bad student through 8th grade, but managed to get selected for a STEM magnet school. I was supposed to enter 9th grade with Geometry, then algebra II, trig, Calc for the 4 years. But they discovered i'd never passed algebra prior, they put me in algebra, which means i would have finished in trig. Due to a crazy 3.5 years, i never got a high school math education. Calculus makes my eyes glaze over, and all i know about triangles is sohcahtoa.

Every couple of years i try to get some higher math education, but nothing makes sense. It's one of the reasons i [think] i suck at programming - i should note that another reason is i first learned BASIC, then qbasic, then fortran, and then C never made sense to me. At least i can putter around with python and R.

however i can do "basic" math things that generally everyone else has to dig out a calculator app for in my head, percentages, fractions, moving decimals, "making change". Since i suck at higher math, i'm only able to help my kids with basic math, and i try to ensure that they know it fairly well.

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