Even better : did you know (-1)^x draws the unit circle in the complex plane ? No need for complex exp and i*pi
If you plot it over which domain?
Turns are Better than Radians (2022)
81–90 of 225 posts
Re: Turns are Better than Radians (2022)
#82The 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…
In another comment, I asked why people chose to use the symbol τ over just writing turn or "rev(olution)" (defined to be the constant ≈ 6.28318530718) given how unambiguous the latter is as a name for 2π. And why not just write sinrev() or sinturn(), and leave the symbols sin() and rev (defined to be ≈ 6.28318530718) alone?
Re: Turns are Better than Radians (2022)
#83Are there any c/c++ libs / headers that use this (without converting to radians in the background). I like this idea.
The C standard defines the functions sinpi(), cospi(), etc. that act on half-turns. If you have a modern compiler, all you have to do is to include math.h
Re: Turns are Better than Radians (2022)
#84I think I cautiously agree with this notion to some extent, but IMHO the real answer is that it's application-dependent, and if you're writing a low-level trig library and you have to pick one or the other, it really isn't clear to me that turns should win over radians. I expect many systems that use trigonometry would sometimes use small-angle approximations either for efficiency or to bootstrap to the general case.…
I don't have a super-wide gamut of experience here and numerical analysis isn't my specialty, but nearly all trig implementations I've looked into (in both software and hardware) make heavy use of lookup tables and other shortcuts. I've never seen a Taylor series used in a general implementation - not saying it doesn't exist anywhere, but in most cases that I'm familiar with you could support turns just as easily wit…
Our favorite WebAssembly is an example! It specifically excludes trigonometry from the spec, because real hardware doesn't produce exactly the same results.
So mathematical libraries in WASM reimplement the trigonometric functions using series.
Example: https://github.com/WebAssembly/wasi-libc/blob/2e6fb9d8ee0cdf...
Re: Turns are Better than Radians (2022)
#85One weird unit for angles is the mil, defined as 6400 mils in a circle. This unit is very useful for artillery, since 1 meter displacement at a distance of 1 km is 1 mil [†]. Thus, you can see how much you missed by, divide by the distance, and easily determine how much you need to adjust your aim in mils. Another interesting thing about artillery is they traditionally do a binary search to get the distance correct,…
It's also useful for sighting distances when the width or height of something is known. A knuckle on your outstretched arm is roughly 30 mils, so you cover the thing with your hand, count knuckles, multiply by 30, then divide the size by that number to get the distance. You can calibrate your knuckles by doing this is reverse. Put up a target 1 cm wide and back up until it's just covered by a knuckle. Measure how far…
Oh, and I forgot and now it's too late to edit my comment. 6400 has a bunch of nice divisors too. A half-turn is 3200 mils, a quarter is 1600, a quarter of a quarter is 400, etc. A sixth of a turn is nearly 1000 mils. A tenth is obviously 640 mils.
Re: Turns are Better than Radians (2022)
#86I think the author is either being disingenuous or doesn’t understand the subject if they don’t honestly address the reason radians are used in the first place. I’m leaning towards the latter, because I can’t imagine someone having an ulterior motive for pushing for trig reform like this, lol. Radians really are the natural unit for trigonometry. With that said, I certainly agree that a lot of code would be simplifie…
You might have misunderstood TFA. No push for trig reform, just a consideration on what internal representation is optimal in code.
Imagine it like someone suggesting (understandably) that you express memory sizes in hex: no push to make everybody stop using decimal numbers!
Re: Turns are Better than Radians (2022)
#87Indeed, this is what Pico-8 uses for its trigonometric functions[1] (angles go from 0 to 1, instead of from 0 to 2*Pi). I was surprised by this at first, but then I found it is very convenient and simplifies a bunch of stuff.
Re: Turns are Better than Radians (2022)
#88This can be useful for some geometry, but pi isn't a completely arbitrary choice and some useful relationships are lost when not using radians.
I use different angle units depending on the application. On a platform with 8-bit index registers, 1/256 of a turn can be useful. IIRC Pico-8 uses turns.
Re: Turns are Better than Radians (2022)
#89I think the author is either being disingenuous or doesn’t understand the subject if they don’t honestly address the reason radians are used in the first place. I’m leaning towards the latter, because I can’t imagine someone having an ulterior motive for pushing for trig reform like this, lol. Radians really are the natural unit for trigonometry. With that said, I certainly agree that a lot of code would be simplifie…
> Radians really are the natural unit for trigonometry.
s/trigonometry/calculus
Re: Turns are Better than Radians (2022)
#90The 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…
For graphics rendering Euler equation doesnt matter. Colours are 0.0-1.0 and have no relation to reality, but it works. Same with rotations (if we’re not using Quaternikns)