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Turns are better than radians

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Re: Turns are better than radians

#51

Post by anonymoushn »

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.

The post says that programmers convert to radians to call APIs that take radians and then immediately divide by pi before computing the sine. So doing everything in radians would still involve an extra floating point divide compared to the alternative.

Re: Turns are better than radians

#52

Post by doliveira »

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…

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-sighted, I'd say, it doesn't "scale out" to the myriad of other applications of angles.

Re: Turns are better than radians

#53

Post by ncmncm »

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?

Right, it is a funny way to say it.

Computationally, we all only ever work with approximations, but when doing mathematics, pi is exact all the way out to the infinity-th digit. To multiply by pi (or any irrational, but particularly transcendental) in a pure mathematical context is to audaciously specify an infinitely long computational process. It is dizzying to contemplate, almost mystical. Sort of like modular arithmetic with an infinitely-precise irrational modulus.

Re: Turns are better than radians

#54

Post by SuchAnonMuchWow »

eesmith wrote: post #23
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?

It's a correct description. (Perhaps further refined as "technically correct".) The following are not exact values of π or τ, only approximate ones: >>> import math >>> math.pi 3.141592653589793 >>> math.tau 6.283185307179586

The effort wanted to replace all uses of Pi (including the ones in math formulas) and not just their use in (floating-point) computer programs where we implement approximations of those formulas.

Re: Turns are better than radians

#55

Post by api »

ncmncm wrote: post #53
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?

Right, it is a funny way to say it. Computationally, we all only ever work with approximations, but when doing mathematics, pi is exact all the way out to the infinity-th digit. To multiply by pi (or any irrational, but particularly transcendental) in a pure mathematical context is to audaciously specify an infinitely long computational process. It is dizzying to contemplate, almost mystical. Sort of like modular ari…

Is any number mod infinity equal to itself?

Re: Turns are better than radians

#56

Post by bmacho »

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 sine and cosine that are defined with Taylor series are not the same sine and cosine that are defined for right triangles.

The former are R->R functions, while the latter are defined on Angles (Angle is unfortunately not an SI physical dimension yet, but I expect it soon to change), and they don't care about the measurement unit.

I have no idea what you mean by radians generalizing for higher dimensions, but not turns.

Re: Turns are better than radians

#57

Post by vimsee »

severak_cz wrote: post #20
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…

It's 360 degrees because of stars. One degree is how much stars move by one day.

> One degree is how much stars move by one day.

I was going to look that up to confirm it, but then I realized I could prove that statement true using some simple logic I already know. Earth does one cycle around the sun in 365 days. So at midnight looking straight up on a specific star (that is angled perpendicular of the rotating poles of earth) in the sky, the star you would have spotted on that day would appear slightly off the next day at midnight. It would only end up on the same spot on midnight after 365 days. So we are 5 days off, but I am going to believe it is true until someone is correcting me.

Re: Turns are better than radians

#58

Post by WastingMyTime89 »

anonymoushn wrote: post #51
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.

The post says that programmers convert to radians to call APIs that take radians and then immediately divide by pi before computing the sine. So doing everything in radians would still involve an extra floating point divide compared to the alternative.

No, you are confused. That’s a conversion back to degree after computing the sine. You don’t randomly divide by pi for no reason.

Re: Turns are better than radians

#59

Post by jayjader »

the_other wrote: post #48

I'm interested in this from the perspective of learning maths, rather than writing code (for now). I've wondered for a few years now whether teaching angles and trig using turns, rather than degrees or radians, would be better from the very beginning. Degrees are arbitrary and based on the numeric preferences of a dead culture, rather than on what's happening on the page or in 3d space. Radians seem better because th…

> but they're hard to visualise and reason about because they don't fit a circle in whole numbers.

I think, unfortunately, that you can't avoid encountering irrational numbers in trig. You would need to constrain yourself to working only with right angles, but in those cases sin and cos are trivial[0] so there would be no need to use trig in the first place.

[0]: 1 or 0, you're either on the right axis or you're not. No circles involved.

Re: Turns are better than radians

#60

Post by ncmncm »

pestatije wrote: post #5
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…

So now the diameter is 1/pi turns?

No, 2/tau turns, philistine. ;-)
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