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Turns are better than radians

computerenhance.com

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Re: Turns are better than radians

#151

Post by mihaaly »

This constant need to redefine the known world around a favourite detail amazes me. Are people that bored?!

(also I itch hearing the idea of redefining interface - and the world - to fit the implementation detail. how about reimplementing using the [0...0.7854] domain instead of the [0...1] if this is such a huge worry after decades of computing - on slower machines - with the natural radian (arc_length/radius) values? I feel Godot engine should fit the world and not the other way around.)

> Math doesn’t require radians. ....What?!?! Circumference, radius and volume, just to name some, try calculate those easily on turns only (without a new constant introduced!).

Re: Turns are better than radians

#152

Post by smcameron »

otikik wrote: post #117

Fun fact: PICO-8 only provides turn-based trigonometric functions https://pico-8.fandom.com/wiki/Sin > PICO-8 uses an input range of 0.0 to 1.0 to represent the angle, a percentage of the unit circle. Some refer to these units as "turns". For instance, 180° or π (3.14159) radians corresponds to 0.5 turns in PICO-8's representation of angles. In fact, for fans of τ (tau), it's just a matter of dropping τ from your exp…

Back in the early 80's a common thing to do in games on 8 bit computers was to implement sin and cos as lookup tables with the angles being 0-255 or 0-128 or something like that and the result also an integer that was some fixed point representation, so you'd do something like:

  vx = (cos[angle] * speed) >> 8;
  vy = (sin[angle] * speed) >> 8;

Re: Turns are better than radians

#153

Post by happimess »

bluGill wrote: post #144
WastingMyTime89 wrote: post #67

Earlier quoted context omitted.

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…

Trig is generally called per-calculus in US high schools. It is not a required course, but it is one of the courses everyone on the college track is expected to take. Though most people haven't used any of that since college and so don't know it very well anymore. I smelled BS when I read the blog, but couldn't put my finger on why - the comment you replied to explained what I knew was the case but couldn't remember.

    s/per-calculus/pre-calculus/g

Re: Turns are better than radians

#154

Post by insightcheck »

genewitch wrote: post #110
WastingMyTime89 wrote: post #67

Earlier quoted context omitted.

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 k…

It's time-consuming, but there are great resources to learn high school math to a very high level (likely much more effectively in many cases, than actually taking a high school math course, due to thoughtful exercises and more control over the pace of learning).

I learned a lot from the Art of Problem Solving book series because they're highly focused on the reader solving problems to learn, versus giving explanations. Even if you don't finish all of it, you can strengthen any problem areas.

For a less-comprehensive but still great introduction to precalculus (with a great section on trigonometry in particular from memory), Simmons' Precalculus in a Nutshell has a great introduction to this. Then you can read a book like Thomas Calculus, which has a great introduction to trigonometry in the first review chapter.

I would even say that you would be better off working through the books above than if you had the high school classes; the best math students probably took the same approach too (working through books instead of focusing just on the class material). The main obstacle is time, because it's hard to find time when you have work and children to take care of.

Re: Turns are better than radians

#157

Post by insightcheck »

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’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 explain the lack of familiarity with radian I see here. Am I wrong?"

It varies by school, but overall I think this prediction is incorrect. Trigonometry was an important subject in high school — for all of the math, physics, and possibly chemistry courses — and then if you take calculus in university, it's very, very important to learn trigonometry well (or you'll really struggle as a student).

So, even on the off-chance that trigonometry is not taught in high school (which I predict is rare), a first-year student taking calculus in university must learn it on their own time. Good calculus textbooks (e.g. Thomas Calculus) even account for this, having fairly comprehensive textbook sections on what you need to know about trigonometry to succeed in the calculus course.

Most students who therefore took math to pre-calculus or calculus (or physics and possibly chemistry), should therefore have a good exposure to the definition of the radian.

Re: Turns are better than radians

#159

Post by dosshell »

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…

I was taught that Eulers formula defined complex exponents?

If we used turns for cos and sin we could redefine what e^ix means so it works without radians. From the other answer I guess this is completely wrong...

(I do understand it is nuts to redefine, i'm just interested as a theoretical thought)

Now, how is Eulers formula is deduced? How did we figure out what e^ix means?

Re: Turns are better than radians

#160

Post by weinzierl »

ascar wrote: post #64
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…

I already learnt in school to calculate trigonometry using radians or turns depending on the situation. It was part of the general math curriculum in Bavaria. As far as I am aware both are mathematically sound and there is no reason to religiously use one of them over the other. Let your use-case or input parameters decide. The examples given in the article definitely make no sense in radians.

"I already learnt in school to calculate trigonometry using radians or turns depending on the situation. It was part of the general math curriculum in Bavaria. "

Out of interest, when did you go to school in Bavaria and in which grade did you learn about turns? I was in school in Bavaria a long time ago and I don't remember learning about turns there. Could very well be that I forgot or our teacher forgot to teach it.

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