51–60 of 225 posts
The floating point expressions needed to represent the math library functions with decent precision and performance becomes significantly more weird and complex with turns. Please stick to radians.
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…
Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?
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…
On the other hand I am still unhappy with calling the ratio of two quantities, that happen to have the same units, "dimensionless". This way, you could create any two "dimensionless" quantities and try to compare them or use one in place of the other which might be meaningless.
Rather than trying to agree on the best meaning the various integers or floats that we're passing around, maybe we should instead build a more complex angle type that doesn't force callers to conform. Like, I can pass minutes or seconds to functions that accept a time type and it just works because they're not being collapsed to numbers. Is there any reason we couldn't do that with angles too?
I’ve had a brief moment of hope, forgetting the point was about mathematics.
Earlier quoted context omitted.
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…
> all angles are without a unit. Dimensionless, sure, but what do you mean here? Radians and degrees are units, are they not?
The problem is that trigonometric functions are used in many more fields beyond geometry. The input is not always an angle around a point in euclidean space, it could be phase angle of a periodic signal. You can make an alternative set of trig functions that take turns, but you will anger a lot of people if you mess with the vanilla trig functions.
When dealing with waves you often are dealing with turns - or, as they’re called in that world, cycles. A cycle is a turn is tau is 2pi. The SI unit for frequency after all is Hertz - cycles per second - which should really be considered equal to 2pi s^-1, but for complicated reasons, often isn’t, and most formulae that involve frequency ignore the ‘cycle’ - or it’s also hiding inside the definition of something like…
Great idea, I will definitely do this!
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