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Turns are Better than Radians (2022)

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Re: Turns are Better than Radians (2022)

#61

Post by djmips »

aldonius wrote: post #42
em3rgent0rdr wrote: post #29

And could use fixed-point decimal for more efficiency since can store as integers and use integer hardware for them. So for instance with 32-bits, the 16 most-sig bits store the number of turns and the 16 least-significant bits store the fraction of a turn. Then if you want to wrap angles that exceed 360 degrees back around the circle, you can simply Logical_AND with 0x0000FFFF. And while you are at it, you could jus…

Binary fractions of a turn are also a nice intuition pump for two's complement in general. Let's keep it simple and use just 8 bits. 0° is 0x00, 180° is 0x80, and 255/256ths of 360° is 0xFF. And if we wanted to use signed integers, then 0x80 through 0xFF - the high-bit half of the range - now represent the negative quadrants just as they represent negative integers.

and that's exactly what we did in the old days of 8 bit games. We called them BRADs but others had their own names.

Re: Turns are Better than Radians (2022)

#62

Post by ant6n »

thyristan wrote: post #57
dahart wrote: post #52

Earlier quoted context omitted.

> all angles are without a unit. Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?

In a very awkward way: rad is m/m, which is 1...

Perhaps in the physics sense, but in computer science we do have the notion of types which does allow us to model the difference between an angle and other numerics.

Re: Turns are Better than Radians (2022)

#63

Post by setopt »

jameshart wrote: post #26
mayoff wrote: post #2

I like to store angles as turns in my own code, because (as noted) it makes quarter-turns computable without rounding. OTOH if you need, say, twelfths of a turn, you might want to just store angles as degrees since that’s already common. Michael Spivak, in Calculus (3rd ed p. 301) considers the unit choice to be a property of the function and initially defines sin° and sinʳ (before settling on sin meaning sinʳ) and c…

You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars. ‘But wait!’ You may cry: ‘the formula for a tra…

> You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars.

I mostly agree with your explanation, but would like to emphasize that this is just a convention from mathematics which mostly carries over into physics and engineering. We like to define functions that are R -> R and similar, instead of defining special sets like R° = { r * 360° | r \in R }, corresponding to "real numbers with unit degrees", and then defining functions like sin: R° -> R. It’s just simpler to define and analyze most functions from R -> R and so we mostly do that.

But if you look up physics papers, it’s not uncommon to define functions that require unitful inputs as well. For example, the wave function in the Schrödinger equation maps a position r (3D vector with unit meter) and time t (scalar with unit seconds), to a probability amplitude (complex number with unit m^-3/2), so that \int |ψ(r,t)|^2 d3r becomes a scalar (a probability). Up wave function is still considered a function by all physicists.

Re: Turns are Better than Radians (2022)

#64

Post by thaumasiotes »

jameshart wrote: post #26
mayoff wrote: post #2

I like to store angles as turns in my own code, because (as noted) it makes quarter-turns computable without rounding. OTOH if you need, say, twelfths of a turn, you might want to just store angles as degrees since that’s already common. Michael Spivak, in Calculus (3rd ed p. 301) considers the unit choice to be a property of the function and initially defines sin° and sinʳ (before settling on sin meaning sinʳ) and c…

You generally can’t apply functions to dimensional units. The only thing units can do is be multiplied or divided together. So I can multiply a mass by a distance or divide a distance by a speed, and I can multiply the result by a scalar; but I can’t take the sine of a distance or the logarithm of a time or exponentiate a mass. Those are things I can only do to scalars. ‘But wait!’ You may cry: ‘the formula for a tra…

You can apply functions to anything. That's the only thing "function" means. They transform values into other values, and there is no limit on what kind of values you might want to talk about.

Re: Turns are Better than Radians (2022)

#65

Post by srean »

math-man wrote: post #6
mayoff wrote: post #2

I like to store angles as turns in my own code, because (as noted) it makes quarter-turns computable without rounding. OTOH if you need, say, twelfths of a turn, you might want to just store angles as degrees since that’s already common. Michael Spivak, in Calculus (3rd ed p. 301) considers the unit choice to be a property of the function and initially defines sin° and sinʳ (before settling on sin meaning sinʳ) and c…

It's because both radians and degrees are are a ratio of a length to another length and are thus dimensionless. No matter how you measure it, all angles are without a unit. It's most obvious with radians but it's also the case with degrees. Using radians, you are guaranteed to not introduce unusual extra terms to rescale angles, if you use any other scale of angle you will have to keep track of extra terms. That may…

It is dimensionless by fiat and convention. It clearly has units such as degrees, grad and radians. Just like other quantities that have units, a specific measurement is expressed as a pure numerical multiple of an unit which may be radians, degrees etc.

This is a known wrinkle in dimensional analysis and people have considered making angles a fundamental quantity such as mass, length and time but have not done so because of the disruption it would cause.

More details here

https://en.wikipedia.org/wiki/Radian#Dimensional_analysis

https://en.wikipedia.org/wiki/Angle#Dimensional_analysis

Re: Turns are Better than Radians (2022)

#66

Post by lefra »

thrtythreeforty wrote: post #45

Here's another good reason to think in turns: it turns Euler's formula from this Eldritch Terror: e^(i*x) = cos(x) + i*sin(x) into something you can kinda understand by staring at the complex plane: -1^(2x) = cost(x) + i*sint(x) Credit to justinpombrio for this: https://news.ycombinator.com/item?id=32986869

Now define exponentiation by a non-integer.

Re: Turns are Better than Radians (2022)

#69

Post by srean »

ttoinou wrote: post #67

Even better : did you know (-1)^x draws the unit circle in the complex plane ? No need for complex exp and i*pi

That's because

   a^b = exp (b ln  a)
That's equivalent to saying, no need for -1 because we have exp.

One can change based of the exponentiation operation. Exp happens to be a convenient base.

Re: Turns are Better than Radians (2022)

#70

Post by kazinator »

The 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 formula, shows that radians are special: like what binary is to computers.

The natural logarithm being its own derivative is in fact directly linked to the derivative a radians-based sin(x) being cos(x) and so on. Make it any other unit, and you have a mess of conversion factors worse than 2pi.

Imagine complex chained derivatives, double and triple derivative, chain and product rules, all stuffed with trig functions and generating gratuitous piles of cascaded conversion constrants because radians were not used.

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